### The Mizar article:

### Ring Ideals

**by****Jonathan Backer,****Piotr Rudnicki, and****Christoph Schwarzweller**

- Received November 20, 2000
Copyright (c) 2000 Association of Mizar Users

- MML identifier: IDEAL_1
- [ MML identifier index ]

environ vocabulary RLVECT_1, ALGSTR_1, VECTSP_1, VECTSP_2, BINOM, BINOP_1, LATTICES, REALSET1, FINSEQ_1, FILTER_2, ALGSTR_2, COLLSP, BOOLE, ARYTM_1, GROUP_1, RELAT_1, FUNCT_1, FUNCT_7, ARYTM_3, FINSEQ_4, PRELAMB, MCART_1, SETFAM_1, RLVECT_3, SUBSET_1, GCD_1, LATTICE3, SQUARE_1, NEWTON, FINSET_1, INT_3, WAYBEL_0, TARSKI, NORMSP_1, GR_CY_1, IDEAL_1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, CQC_SIM1, FINSET_1, FUNCT_1, FINSEQ_1, VECTSP_1, INT_3, NORMSP_1, NUMBERS, XCMPLX_0, XREAL_0, NAT_1, RELSET_1, RLVECT_1, PARTFUN1, FUNCT_2, ALGSTR_1, PRE_TOPC, GR_CY_1, PRE_CIRC, MCART_1, DOMAIN_1, FINSEQ_4, POLYNOM1, SETFAM_1, STRUCT_0, GROUP_1, BINOP_1, VECTSP_2, FUNCT_7, REALSET1, GCD_1, BINOM; constructors INT_3, CQC_SIM1, PRE_CIRC, GROUP_2, ALGSEQ_1, BINOM, GCD_1, ALGSTR_2, DOMAIN_1, POLYNOM1, MONOID_0, BINOP_1, PRE_TOPC; clusters INT_3, RELSET_1, FINSET_1, SUBSET_1, VECTSP_2, STRUCT_0, PRE_TOPC, FINSEQ_1, FINSEQ_5, XBOOLE_0, POLYNOM1, INT_1, GCD_1, BINOM, REALSET1, VECTSP_1, NAT_1, XREAL_0, MEMBERED, RELAT_1, FUNCT_2, PRE_CIRC, NUMBERS, ORDINAL2; requirements NUMERALS, REAL, SUBSET, BOOLE, ARITHM; definitions TARSKI, XBOOLE_0; theorems INT_3, CQC_SIM1, TARSKI, ZFMISC_1, RLVECT_1, REALSET1, VECTSP_1, GCD_1, FUNCT_1, FUNCT_2, POLYNOM1, VECTSP_2, PRE_TOPC, RELAT_1, FINSET_1, PRE_CIRC, NAT_1, GR_CY_1, ALGSTR_1, NORMSP_1, REAL_1, AXIOMS, MCART_1, FINSEQ_1, FINSEQ_2, FINSEQ_3, JORDAN3, SETFAM_1, BINOP_1, INT_1, FUNCT_7, FINSEQ_4, FINSEQ_5, POLYNOM2, BINOM, XBOOLE_0, XBOOLE_1, XCMPLX_0, XCMPLX_1, ORDINAL2; schemes NAT_1, RECDEF_1, FUNCT_2, POLYNOM2, BINOM, FINSEQ_2; begin :: Preliminaries definition cluster add-associative left_zeroed right_zeroed (non empty LoopStr); existence proof consider R being non degenerated comRing; take R; thus thesis; end; end; definition cluster Abelian left_zeroed right_zeroed add-cancelable well-unital add-associative associative commutative distributive non trivial (non empty doubleLoopStr); existence proof consider R being non degenerated comRing; take R; thus thesis; end; end; theorem Th1: for V being add-associative left_zeroed right_zeroed (non empty LoopStr), v,u being Element of V holds Sum <* v,u *> = v + u proof let V be add-associative left_zeroed right_zeroed (non empty LoopStr), v,u be Element of V; <* v,u *> = <* v *> ^ <* u *> by FINSEQ_1:def 9; then Sum<* v,u *> = Sum<* v *> + Sum<* u *> by RLVECT_1:58 .= v + Sum <* u *> by BINOM:3 .= v + u by BINOM:3; hence thesis; end; begin :: Ideals definition let L be non empty LoopStr, F being Subset of L; attr F is add-closed means :Def1: for x, y being Element of L st x in F & y in F holds x+y in F; end; definition let L be non empty HGrStr, F be Subset of L; attr F is left-ideal means :Def2: for p, x being Element of L st x in F holds p*x in F; attr F is right-ideal means :Def3: for p, x being Element of L st x in F holds x*p in F; end; definition let L be non empty LoopStr; cluster add-closed (non empty Subset of L); existence proof set M = the carrier of L; for u being set holds u in M implies u in the carrier of L; then reconsider M as Subset of L by TARSKI:def 3; reconsider M as non empty Subset of L; take M; for x, y being Element of L st x in M & y in M holds x+y in M ; hence thesis by Def1; end; end; definition let L be non empty HGrStr; cluster left-ideal (non empty Subset of L); existence proof set M = the carrier of L; for u being set holds u in M implies u in the carrier of L; then reconsider M as Subset of L by TARSKI:def 3; reconsider M as non empty Subset of L; take M; for p, x being Element of L st x in M holds p*x in M; hence thesis by Def2; end; cluster right-ideal (non empty Subset of L); existence proof set M = the carrier of L; for u being set holds u in M implies u in the carrier of L; then reconsider M as Subset of L by TARSKI:def 3; reconsider M as non empty Subset of L; take M; for p, x being Element of L st x in M holds x*p in M; hence thesis by Def3; end; end; definition let L be non empty doubleLoopStr; cluster add-closed left-ideal right-ideal (non empty Subset of L); existence proof set M = the carrier of L; for u being set holds u in M implies u in the carrier of L; then reconsider M as Subset of L by TARSKI:def 3; reconsider M as non empty Subset of L; take M; A1: for x,y being Element of L st x in M & y in M holds x+y in M; A2: for p,x being Element of L st x in M holds p*x in M; for p, x being Element of L st x in M holds x*p in M; hence thesis by A1,A2,Def1,Def2,Def3; end; cluster add-closed right-ideal (non empty Subset of L); existence proof set M = the carrier of L; for u being set holds u in M implies u in the carrier of L; then reconsider M as Subset of L by TARSKI:def 3; reconsider M as non empty Subset of L; take M; A3: for x,y being Element of L st x in M & y in M holds x+y in M; for p, x being Element of L st x in M holds x*p in M; hence thesis by A3,Def1,Def3; end; cluster add-closed left-ideal (non empty Subset of L); existence proof set M = the carrier of L; for u being set holds u in M implies u in the carrier of L; then reconsider M as Subset of L by TARSKI:def 3; reconsider M as non empty Subset of L; take M; A4: for x,y being Element of L st x in M & y in M holds x+y in M; for p, x being Element of L st x in M holds p*x in M; hence thesis by A4,Def1,Def2; end; end; definition let R be commutative (non empty HGrStr); cluster left-ideal -> right-ideal (non empty Subset of R); coherence proof let I be non empty Subset of R; assume I is left-ideal; then for p,x being Element of R st x in I holds x*p in I by Def2; hence thesis by Def3; end; cluster right-ideal -> left-ideal (non empty Subset of R); coherence proof let I be non empty Subset of R; assume I is right-ideal; then for p,x being Element of R st x in I holds p*x in I by Def3; hence thesis by Def2; end; end; definition let L be non empty doubleLoopStr; mode Ideal of L is add-closed left-ideal right-ideal (non empty Subset of L); end; definition let L be non empty doubleLoopStr; mode RightIdeal of L is add-closed right-ideal (non empty Subset of L); end; definition let L be non empty doubleLoopStr; mode LeftIdeal of L is add-closed left-ideal (non empty Subset of L); end; theorem Th2: for R being right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), I being left-ideal (non empty Subset of R) holds 0.R in I proof let R be right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr); let I be left-ideal (non empty Subset of R); consider a being Element of I; 0.R*a in I by Def2; hence thesis by BINOM:1; end; theorem Th3: for R being left_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr), I being right-ideal (non empty Subset of R) holds 0.R in I proof let R be left_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr); let I be right-ideal (non empty Subset of R); consider a being Element of I; a*0.R in I by Def3; hence thesis by BINOM:2; end; theorem Th4: for L being right_zeroed (non empty doubleLoopStr) holds {0.L} is add-closed proof let L be right_zeroed (non empty doubleLoopStr); let x,y be Element of L; assume x in {0.L} & y in {0.L}; then x = 0.L & y= 0.L by TARSKI:def 1; then x + y = 0.L by RLVECT_1:def 7; hence x + y in {0.L} by TARSKI:def 1; end; theorem Th5: for L being left_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr) holds {0.L} is left-ideal proof let L be left_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr); let p,x be Element of L; assume x in {0.L}; then A1: x = 0.L by TARSKI:def 1; reconsider p' = p as Element of L; p'*x = 0.L by A1,BINOM:2; hence p*x in {0.L} by TARSKI:def 1; end; theorem Th6: for L being right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr) holds {0.L} is right-ideal proof let L be right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr); let p,x be Element of L; assume x in {0.L}; then A1: x = 0.L by TARSKI:def 1; reconsider p' = p as Element of L; x*p' = 0.L by A1,BINOM:1; hence x*p in {0.L} by TARSKI:def 1; end; theorem Th7: for L being add-associative right_zeroed right_complementable distributive (non empty doubleLoopStr) holds {0.L} is Ideal of L proof let L be add-associative right_zeroed right_complementable distributive (non empty doubleLoopStr); A1: {0.L} is add-closed proof let x,y be Element of L; assume x in {0.L} & y in {0.L}; then x = 0.L & y= 0.L by TARSKI:def 1; then x + y = 0.L by RLVECT_1:10; hence x + y in {0.L} by TARSKI:def 1; end; A2: {0.L} is left-ideal proof let p,x be Element of L; assume x in {0.L}; then x = 0.L by TARSKI:def 1; then p*x = 0.L by VECTSP_1:36; hence p*x in {0.L} by TARSKI:def 1; end; {0.L} is right-ideal proof let p,x be Element of L; assume x in {0.L}; then x = 0.L by TARSKI:def 1; then x*p = 0.L by VECTSP_1:39; hence x*p in {0.L} by TARSKI:def 1; end; hence thesis by A1,A2; end; theorem for L being add-associative right_zeroed right_complementable right-distributive (non empty doubleLoopStr) holds {0.L} is LeftIdeal of L proof let L be add-associative right_zeroed right_complementable right-distributive (non empty doubleLoopStr); A1: {0.L} is add-closed proof let x,y be Element of L; assume x in {0.L} & y in {0.L}; then x = 0.L & y= 0.L by TARSKI:def 1; then x+y = 0.L by RLVECT_1:10; hence x + y in {0.L} by TARSKI:def 1; end; {0.L} is left-ideal proof let p,x be Element of L; assume x in {0.L}; then x = 0.L by TARSKI:def 1; then p*x = 0.L by VECTSP_1:36; hence p*x in {0.L} by TARSKI:def 1; end; hence thesis by A1; end; theorem for L being add-associative right_zeroed right_complementable left-distributive (non empty doubleLoopStr) holds {0.L} is RightIdeal of L proof let L be add-associative right_zeroed right_complementable left-distributive (non empty doubleLoopStr); A1: {0.L} is add-closed proof let x,y be Element of L; assume x in {0.L} & y in {0.L}; then x = 0.L & y= 0.L by TARSKI:def 1; then x + y = 0.L by RLVECT_1:10; hence x + y in {0.L} by TARSKI:def 1; end; {0.L} is right-ideal proof let p,x be Element of L; assume x in {0.L}; then x = 0.L by TARSKI:def 1; then x*p = 0.L by VECTSP_1:39; hence x*p in {0.L} by TARSKI:def 1; end; hence thesis by A1; end; theorem Th10: for L being non empty doubleLoopStr holds the carrier of L is Ideal of L proof let L be non empty doubleLoopStr; the carrier of L c= the carrier of L; then reconsider cL = the carrier of L as Subset of L; A1: cL is add-closed proof let x, y be Element of L; thus thesis; end; A2: cL is left-ideal proof let x, y be Element of L; thus thesis; end; cL is right-ideal proof let x, y be Element of L; thus thesis; end; hence thesis by A1,A2; end; theorem Th11: for L being non empty doubleLoopStr holds the carrier of L is LeftIdeal of L proof let L be non empty doubleLoopStr; the carrier of L c= the carrier of L; then reconsider cL = the carrier of L as Subset of L; A1: cL is add-closed proof let x, y be Element of L; thus thesis; end; cL is left-ideal proof let x, y be Element of L; thus thesis; end; hence thesis by A1; end; theorem Th12: for L being non empty doubleLoopStr holds the carrier of L is RightIdeal of L proof let L be non empty doubleLoopStr; the carrier of L c= the carrier of L; then reconsider cL = the carrier of L as Subset of L; A1: cL is add-closed proof let x, y be Element of L; thus thesis; end; cL is right-ideal proof let x, y be Element of L; thus thesis; end; hence thesis by A1; end; definition let R be left_zeroed right_zeroed add-cancelable distributive (non empty doubleLoopStr), I be Ideal of R; redefine attr I is trivial means I = {0.R}; compatibility proof now assume I is trivial; then consider x being set such that A1: I = {x} by REALSET1:def 12; 0.R in {x} by A1,Th3; hence I = {0.R} by A1,TARSKI:def 1; end; hence thesis by REALSET1:def 12; end; end; definition let S be 1-sorted, T be Subset of S; attr T is proper means :Def5: T <> the carrier of S; end; definition let S be non empty 1-sorted; cluster proper Subset of S; existence proof for u being set holds u in {} implies u in the carrier of S; then reconsider e = {} as Subset of S by TARSKI:def 3; reconsider e as Subset of S; take e; thus thesis by Def5; end; end; definition let R be non trivial left_zeroed right_zeroed add-cancelable distributive (non empty doubleLoopStr); cluster proper Ideal of R; existence proof reconsider M = {0.R} as Ideal of R by Th4,Th5,Th6; ex a being Element of R st a <> 0.R proof assume A1: not(ex a being Element of R st a <> 0.R); A2: for u being set holds u in {0.R} implies u in the carrier of R; for u being set holds u in the carrier of R implies u in {0.R} proof let u be set; assume u in the carrier of R; then reconsider u as Element of R; u = 0.R by A1; hence thesis by TARSKI:def 1; end; then A3: the carrier of R = {0.R} by A2,TARSKI:2; the carrier of R is non trivial by REALSET1:def 13; hence thesis by A3,REALSET1:def 12; end; then {0.R} <> the carrier of R by TARSKI:def 1; then M is proper by Def5; hence thesis; end; end; theorem Th13: for L being add-associative right_zeroed right_complementable left-distributive left_unital (non empty doubleLoopStr), I being left-ideal (non empty Subset of L), x being Element of L st x in I holds -x in I proof let L be add-associative right_zeroed right_complementable left-distributive left_unital (non empty doubleLoopStr); let I be left-ideal (non empty Subset of L); let x being Element of L; assume x in I; then A1: (- 1_ L)*x in I by Def2; 0. L = 0.L*x by VECTSP_1:39 .= (1_ L + (- 1_ L))*x by RLVECT_1:def 10 .= 1_ L*x + (- 1_ L)*x by VECTSP_1:def 12 .= x + (- 1_ L)*x by VECTSP_1:def 19; hence - x in I by A1,RLVECT_1:def 10; end; theorem Th14: for L being add-associative right_zeroed right_complementable right-distributive right_unital (non empty doubleLoopStr), I being right-ideal (non empty Subset of L), x being Element of L st x in I holds -x in I proof let L be add-associative right_zeroed right_complementable right-distributive right_unital (non empty doubleLoopStr); let I be right-ideal (non empty Subset of L); let x being Element of L; assume x in I; then A1: x*(- 1_ L) in I by Def3; 0. L = x*0.L by VECTSP_1:36 .= x*(1_ L + (- 1_ L)) by RLVECT_1:def 10 .= x*1_ L + x*(- 1_ L) by VECTSP_1:def 11 .= x + x*(- 1_ L) by VECTSP_1:def 13; hence -x in I by A1,RLVECT_1:def 10; end; theorem for L being add-associative right_zeroed right_complementable left-distributive left_unital (non empty doubleLoopStr), I being LeftIdeal of L, x,y being Element of L st x in I & y in I holds x-y in I proof let L being add-associative right_zeroed right_complementable left-distributive left_unital (non empty doubleLoopStr); let I being LeftIdeal of L; let x, y being Element of L; assume A1: x in I & y in I; then - y in I by Th13; then x + (- y) in I by A1,Def1; hence x - y in I by RLVECT_1:def 11; end; theorem for L being add-associative right_zeroed right_complementable right-distributive right_unital (non empty doubleLoopStr), I being RightIdeal of L, x,y being Element of L st x in I & y in I holds x-y in I proof let L being add-associative right_zeroed right_complementable right-distributive right_unital (non empty doubleLoopStr); let I being RightIdeal of L; let x, y being Element of L; assume A1: x in I & y in I; then - y in I by Th14; then x + (- y) in I by A1,Def1 ; hence x - y in I by RLVECT_1:def 11; end; theorem Th17: for R being left_zeroed right_zeroed add-cancelable add-associative distributive (non empty doubleLoopStr), I being add-closed right-ideal (non empty Subset of R), a being Element of I, n being Nat holds n*a in I proof let R be left_zeroed right_zeroed add-cancelable add-associative distributive (non empty doubleLoopStr), I be add-closed right-ideal (non empty Subset of R), a be Element of I, n be Nat; defpred P[Nat] means $1*a in I; 0*a = 0.R by BINOM:13; then A1: P[0] by Th3; A2: for n being Nat holds P[n] implies P[n+1] proof let n be Nat; assume A3: n*a in I; (n+1)*a = 1*a + n*a by BINOM:16 .= a + n*a by BINOM:14; hence thesis by A3,Def1; end; for n being Nat holds P[n] from Ind(A1,A2); hence thesis; end; theorem for R being unital left_zeroed right_zeroed add-cancelable associative distributive (non empty doubleLoopStr), I being right-ideal (non empty Subset of R), a being Element of I, n being Nat st n <> 0 holds a|^n in I proof let R be unital left_zeroed right_zeroed add-cancelable associative distributive (non empty doubleLoopStr), I be right-ideal (non empty Subset of R), a be Element of I, n be Nat; defpred P[Nat] means a|^$1 in I; assume n <> 0; then A1: 1 <= n by RLVECT_1:99; a|^1 = a by BINOM:8; then A2: P[1]; A3: for n being Nat st 1 <= n holds P[n] implies P[n+1] proof let n be Nat; assume 1 <= n; assume A4: a|^n in I; a|^(n+1) = (a|^n)*(a|^1) by BINOM:11; hence thesis by A4,Def3; end; for n being Nat st 1 <= n holds P[n] from Ind2(A2,A3); hence thesis by A1; end; definition let R be non empty LoopStr, I be add-closed (non empty Subset of R); func add|(I,R) -> BinOp of I equals :Def6: (the add of R)|[:I,I:]; coherence proof reconsider f = (the add of R)|[:I,I:] as Function of [:I,I:],the carrier of R by FUNCT_2:38; A1: dom f = [:I,I:] by FUNCT_2:def 1; for x being set st x in [:I,I:] holds f.x in I proof let x be set; assume A2: x in [:I,I:]; then consider u,v being set such that A3: u in I & v in I & x = [u,v] by ZFMISC_1:def 2; reconsider u,v as Element of R by A3; reconsider u,v as Element of R; f.x = (the add of R).[u,v] by A1,A2,A3,FUNCT_1:70 .= u + v by RLVECT_1:def 3; hence thesis by A3,Def1; end; hence thesis by A1,FUNCT_2:5; end; end; definition let R be non empty HGrStr, I be right-ideal (non empty Subset of R); func mult|(I,R) -> BinOp of I equals (the mult of R)|[:I,I:]; coherence proof reconsider f = (the mult of R)|[:I,I:] as Function of [:I,I:],the carrier of R by FUNCT_2:38; A1: dom f = [:I,I:] by FUNCT_2:def 1; for x being set st x in [:I,I:] holds f.x in I proof let x be set; assume x in [:I,I:]; then consider u,v being set such that A2: u in I & v in I & x = [u,v] by ZFMISC_1:def 2; reconsider u,v as Element of I by A2; f.x = (the mult of R).[u,v] by A1,A2,FUNCT_1:70 .= (the mult of R).(u,v) by BINOP_1:def 1 .= u*v by VECTSP_1:def 10; hence thesis by Def3; end; hence thesis by A1,FUNCT_2:5; end; end; definition let R be non empty LoopStr, I be add-closed (non empty Subset of R); func Gr(I,R) -> non empty LoopStr equals :Def8: LoopStr (#I,add|(I,R),In (0.R,I)#); coherence; end; definition let R be left_zeroed right_zeroed add-cancelable add-associative distributive (non empty doubleLoopStr), I be add-closed (non empty Subset of R); cluster Gr(I,R) -> add-associative; coherence proof set M = LoopStr (#I,add|(I,R),In (0.R,I)#); A1: M = Gr(I,R) by Def8; reconsider M as non empty LoopStr; now let u be set; assume A2: u in [:I,I:]; dom(the add of R) = [:the carrier of R,the carrier of R:] by FUNCT_2:def 1 ; hence u in dom(the add of R) by A2; end; then [:I,I:] c= dom(the add of R) by TARSKI:def 3; then A3: dom((the add of R)|[:I,I:]) = [:I,I:] by RELAT_1:91; A4: for a,b being Element of M, a',b' being Element of I st a' = a & b' = b holds a + b = a' + b' proof let a,b be Element of M, a',b' be Element of I; assume a' = a & b' = b; hence a + b = (add|(I,R)).[a',b'] by RLVECT_1:def 3 .= ((the add of R)|[:I,I:]).[a',b'] by Def6 .= (the add of R).[a',b'] by A3,FUNCT_1:70 .= a' + b' by RLVECT_1:def 3; end; now let a,b,c be Element of M; reconsider a' = a, b' = b, c' = c as Element of I; a' + b' in I by Def1; then A5: [a'+b',c'] in dom((the add of R)|[:I,I:]) by A3,ZFMISC_1:def 2; b' + c' in I by Def1; then A6: [a',b'+c'] in dom((the add of R)|[:I,I:]) by A3,ZFMISC_1:def 2; thus (a + b) + c = (the add of M).[a+b,c] by RLVECT_1:def 3 .= (add|(I,R)).[a'+b',c'] by A4 .= ((the add of R)|[:I,I:]).[a'+b',c'] by Def6 .= (the add of R).[a'+b',c'] by A5,FUNCT_1:70 .= (a'+b') + c' by RLVECT_1:def 3 .= a' + (b' + c') by RLVECT_1:def 6 .= (the add of R).[a',b'+c'] by RLVECT_1:def 3 .= ((the add of R)|[:I,I:]).[a',b'+c'] by A6,FUNCT_1:70 .= (add|(I,R)).[a',b'+c'] by Def6 .= (the add of M).[a,b+c] by A4 .= a + (b + c) by RLVECT_1:def 3; end; hence thesis by A1,RLVECT_1:def 6; end; end; definition let R be left_zeroed right_zeroed add-cancelable add-associative distributive (non empty doubleLoopStr), I be add-closed right-ideal (non empty Subset of R); cluster Gr(I,R) -> right_zeroed; coherence proof set M = LoopStr (#I,add|(I,R),In (0.R,I)#); A1: M = Gr(I,R) by Def8; reconsider M as non empty LoopStr; now let u be set; assume A2: u in [:I,I:]; dom(the add of R) = [:the carrier of R,the carrier of R:] by FUNCT_2:def 1 ; hence u in dom(the add of R) by A2; end; then [:I,I:] c= dom(the add of R) by TARSKI:def 3; then A3: dom((the add of R)|[:I,I:]) = [:I,I:] by RELAT_1:91; now let a be Element of M; reconsider a' = a as Element of I; A4: 0.R in I by Th3; then A5: [a',0.R] in dom((the add of R)|[:I,I:]) by A3,ZFMISC_1:def 2; thus a + 0.M = a + (the Zero of M) by RLVECT_1:def 2 .= (the add of M).[a,(In (0.R,I))] by RLVECT_1:def 3 .= (the add of M).[a,0.R] by A4,FUNCT_7:def 1 .= ((the add of R)|[:I,I:]).[a',0.R] by Def6 .= (the add of R).[a',0.R] by A5,FUNCT_1:70 .= a' + 0.R by RLVECT_1:def 3 .= a by RLVECT_1:def 7; end; hence thesis by A1,RLVECT_1:def 7; end; end; definition let R be Abelian (non empty doubleLoopStr), I be add-closed (non empty Subset of R); cluster Gr(I,R) -> Abelian; coherence proof set M = LoopStr (#I,add|(I,R),In (0.R,I)#); A1: M = Gr(I,R) by Def8; reconsider M as non empty LoopStr; now let u be set; assume A2: u in [:I,I:]; dom(the add of R) = [:the carrier of R,the carrier of R:] by FUNCT_2:def 1 ; hence u in dom(the add of R) by A2; end; then [:I,I:] c= dom(the add of R) by TARSKI:def 3; then A3: dom((the add of R)|[:I,I:]) = [:I,I:] by RELAT_1:91; A4: for a,b being Element of M, a',b' being Element of I st a' = a & b' = b holds a + b = a' + b' proof let a,b be Element of M, a',b' be Element of I; assume a' = a & b' = b; hence a + b = (add|(I,R)).[a',b'] by RLVECT_1:def 3 .= ((the add of R)|[:I,I:]).[a',b'] by Def6 .= (the add of R).[a',b'] by A3,FUNCT_1:70 .= a' + b' by RLVECT_1:def 3; end; now let a,b be Element of M; reconsider a' = a, b' = b as Element of I; thus a + b = a' + b' by A4 .= b + a by A4; end; hence thesis by A1,RLVECT_1:def 5; end; end; definition let R be Abelian right_unital left_zeroed right_zeroed right_complementable add-associative distributive (non empty doubleLoopStr), I be add-closed right-ideal (non empty Subset of R); cluster Gr(I,R) -> right_complementable; coherence proof set M = LoopStr (#I,add|(I,R),In (0.R,I)#); A1: M = Gr(I,R) by Def8; reconsider M as non empty LoopStr; now let u be set; assume A2: u in [:I,I:]; dom(the add of R) = [:the carrier of R,the carrier of R:] by FUNCT_2:def 1 ; hence u in dom(the add of R) by A2; end; then [:I,I:] c= dom(the add of R) by TARSKI:def 3; then A3: dom((the add of R)|[:I,I:]) = [:I,I:] by RELAT_1:91; A4: for a,b being Element of M, a',b' being Element of I st a' = a & b' = b holds a + b = a' + b' proof let a,b be Element of M, a',b' be Element of I; assume a' = a & b' = b; hence a + b = (add|(I,R)).[a',b'] by RLVECT_1:def 3 .= ((the add of R)|[:I,I:]).[a',b'] by Def6 .= (the add of R).[a',b'] by A3,FUNCT_1:70 .= a' + b' by RLVECT_1:def 3; end; reconsider I as RightIdeal of R; now let a be Element of M; A5: 0.R in I by Th3; reconsider a' = a as Element of I; reconsider b = -a' as Element of M by Th14; a + b = a' + -a' by A4 .= 0.R by RLVECT_1:16 .= the Zero of M by A5,FUNCT_7:def 1 .= 0.M by RLVECT_1:def 2; hence ex b being Element of M st a + b = 0.M; end; hence thesis by A1,RLVECT_1:def 8; end; end; Lm1: for R being comRing, a being Element of R holds {a*r where r is Element of R : not contradiction} is Ideal of R proof let R be comRing, a be Element of R; set M = {a*r where r is Element of R : not contradiction}; A1: now let u be set; assume u in M; then consider r being Element of R such that A2: u = a*r; thus u in the carrier of R by A2; end; a*1_ R in M; then reconsider M as non empty Subset of R by A1,TARSKI:def 3; reconsider M as non empty Subset of R; A3: now let b,c be Element of R; assume A4: b in M & c in M; then consider r being Element of R such that A5: a*r = b; consider s being Element of R such that A6: a*s = c by A4; b + c = a*(r + s) by A5,A6,VECTSP_1:def 18; hence b + c in M; end; A7: now let s,b be Element of R; assume b in M; then consider r being Element of R such that A8: a*r = b; b*s = a*(r*s) by A8,VECTSP_1:def 16; hence b*s in M; end; now let s,b be Element of R; assume b in M; then consider r being Element of R such that A9: a*r = b; s*b = (s*r)*a by A9,VECTSP_1:def 16; hence s*b in M; end; hence thesis by A3,A7,Def1,Def2,Def3; end; theorem Th19: for R being right_unital (non empty doubleLoopStr), I being left-ideal (non empty Subset of R) holds I is proper iff not(1_ R in I) proof let R be right_unital (non empty doubleLoopStr), I be left-ideal (non empty Subset of R); A1: now assume A2: I is proper; thus not(1_ R in I) proof assume A3: 1_ R in I; A4: for u being set holds u in I implies u in the carrier of R; now let u be set; assume u in the carrier of R; then reconsider u' = u as Element of R; u'*1_ R = u' by VECTSP_1:def 13; hence u in I by A3,Def2; end; then I = the carrier of R by A4,TARSKI:2; hence thesis by A2,Def5; end; end; now assume not(1_ R in I); then I <> the carrier of R; hence I is proper by Def5; end; hence thesis by A1; end; theorem for R being left_unital right_unital (non empty doubleLoopStr), I being right-ideal (non empty Subset of R) holds I is proper iff for u being Element of R st u is unital holds not(u in I) proof let R be left_unital right_unital (non empty doubleLoopStr), I be right-ideal (non empty Subset of R); A1: now assume A2: I is proper; A3: not(1_ R in I) proof assume A4: 1_ R in I; A5: for u being set holds u in I implies u in the carrier of R; now let u be set; assume u in the carrier of R; then reconsider u' = u as Element of R; 1_ R*u' = u' by VECTSP_1:def 19; hence u in I by A4,Def3; end; then I = the carrier of R by A5,TARSKI:2; hence thesis by A2,Def5; end; thus for u being Element of R st u is unital holds not(u in I) proof let u be Element of R; assume u is unital; then u divides 1_ R by GCD_1:def 2; then ex b being Element of R st 1_ R = u*b by GCD_1:def 1; hence thesis by A3,Def3; end; end; now assume A6: for u being Element of R st u is unital holds not(u in I); 1_ R divides 1_ R; then 1_ R is unital by GCD_1:def 2; then I <> the carrier of R by A6; hence I is proper by Def5; end; hence thesis by A1; end; theorem for R being right_unital (non empty doubleLoopStr), I being left-ideal right-ideal (non empty Subset of R) holds I is proper iff for u being Element of R st u is unital holds not(u in I) proof let R be right_unital (non empty doubleLoopStr), I be left-ideal right-ideal (non empty Subset of R); A1: now assume A2: I is proper; A3: not(1_ R in I) proof assume A4: 1_ R in I; A5: for u being set holds u in I implies u in the carrier of R; now let u be set; assume u in the carrier of R; then reconsider u' = u as Element of R; u'*1_ R = u' by VECTSP_1:def 13; hence u in I by A4,Def2; end; then I = the carrier of R by A5,TARSKI:2; hence thesis by A2,Def5; end; thus for u being Element of R st u is unital holds not(u in I) proof let u be Element of R; assume u is unital; then u divides 1_ R by GCD_1:def 2; then ex b being Element of R st 1_ R = u*b by GCD_1:def 1; hence thesis by A3,Def3; end; end; now assume A6: for u being Element of R st u is unital holds not(u in I); 1_ R*1_ R = 1_ R by VECTSP_1:def 13; then 1_ R divides 1_ R by GCD_1:def 1; then 1_ R is unital by GCD_1:def 2; then I <> the carrier of R by A6; hence I is proper by Def5; end; hence thesis by A1; end; theorem for R being non degenerated comRing holds R is Field iff for I being Ideal of R holds (I = {0.R} or I = the carrier of R) proof let R be non degenerated comRing; A1: now assume A2: R is Field; thus for I being Ideal of R holds (I = {0.R} or I = the carrier of R) proof let I be Ideal of R; assume A3: I <> {0.R}; reconsider R as Field by A2; ex a being Element of R st a in I & a <> 0.R proof assume A4: not(ex a being Element of R st a in I & a <> 0.R); A5: now let u be set; assume u in I; then reconsider u' = u as Element of I; u' = 0.R by A4; hence u in {0.R} by TARSKI:def 1; end; now let u be set; assume A6: u in {0.R}; then reconsider u' = u as Element of R; u' = 0.R by A6,TARSKI:def 1; hence u in I by Th3; end; hence thesis by A3,A5,TARSKI:2; end; then consider a being Element of R such that A7: a in I & a <> 0.R; consider b being Element of R such that A8: a*b = 1_ R by A7,VECTSP_1:def 20; 1_ R in I by A7,A8,Def3; then I is non proper by Th19; hence thesis by Def5; end; end; now assume A9: for I being Ideal of R holds (I= {0.R} or I = the carrier of R ) ; now let a be Element of R; assume A10: a <> 0.R; reconsider a' = a as Element of R; reconsider M = {a'*r where r is Element of R : not contradiction} as Ideal of R by Lm1; a*1_ R = a by VECTSP_1:def 19; then a in M; then M <> {0.R} by A10,TARSKI:def 1; then M = the carrier of R by A9; then 1_ R in M; then consider b being Element of R such that A11: a*b = 1_ R; thus ex b being Element of R st a*b = 1_ R by A11; end; hence R is Field by VECTSP_1:def 20; end; hence thesis by A1; end; begin :: Linear combinations definition let R be non empty multLoopStr, A be non empty Subset of R; mode LinearCombination of A -> FinSequence of the carrier of R means :Def9: for i being set st i in dom it ex u,v being Element of R, a being Element of A st it/.i = u*a*v; existence proof set p = <*>(the carrier of R); take p; let i be set; assume i in dom p; hence thesis; end; mode LeftLinearCombination of A -> FinSequence of the carrier of R means :Def10: for i being set st i in dom it ex u being Element of R, a being Element of A st it/.i = u*a; existence proof consider a being Element of A; reconsider aR = a as Element of R; reconsider a' = a*a as Element of R; set p = <*a'*>; take p; let i be set; assume A1: i in dom p; dom p = {1} by FINSEQ_1:4,55; then A2: i = 1 by A1,TARSKI:def 1; take aR, a; thus p/.i = p.i by A1,FINSEQ_4:def 4 .= aR*a by A2,FINSEQ_1:57; end; mode RightLinearCombination of A -> FinSequence of the carrier of R means :Def11: for i being set st i in dom it ex u being Element of R, a being Element of A st it/.i = a*u; existence proof consider a being Element of A; reconsider aR = a as Element of R; reconsider a' = a*a as Element of R; set p = <*a'*>; take p; let i be set; assume A3: i in dom p; dom p = {1} by FINSEQ_1:4,55; then A4: i = 1 by A3,TARSKI:def 1; take aR, a; thus p/.i = p.i by A3,FINSEQ_4:def 4 .= a*aR by A4,FINSEQ_1:57; end; end; definition let R be non empty multLoopStr, A be non empty Subset of R; cluster non empty LinearCombination of A; existence proof consider u, v being Element of R; consider a being Element of A; reconsider p = <*u*a*v*> as FinSequence of the carrier of R; take p; now let i be set; assume i in dom p; then i in {1} by FINSEQ_1:4,55 ; then A1: i = 1 by TARSKI:def 1; take u,v, a; thus p/.i = u*a*v by A1,FINSEQ_4:25; end; hence thesis by Def9; end; cluster non empty LeftLinearCombination of A; existence proof consider u being Element of R; consider a being Element of A; reconsider p = <*u*a*> as FinSequence of the carrier of R; take p; now let i be set; assume i in dom p; then i in {1} by FINSEQ_1:4,55 ; then A2: i = 1 by TARSKI:def 1; take u, a; thus p/.i = u*a by A2,FINSEQ_4:25; end; hence thesis by Def10; end; cluster non empty RightLinearCombination of A; existence proof consider v being Element of R; consider a being Element of A; reconsider p = <*a*v*> as FinSequence of the carrier of R; take p; now let i be set; assume i in dom p; then i in {1} by FINSEQ_1:4,55 ; then A3: i = 1 by TARSKI:def 1; take v, a; thus p/.i = a*v by A3,FINSEQ_4:25; end; hence thesis by Def11; end; end; definition let R be non empty multLoopStr, A,B be non empty Subset of R, F be LinearCombination of A, G be LinearCombination of B; redefine func F^G -> LinearCombination of (A \/ B); coherence proof set H = F^G; thus H is LinearCombination of (A \/ B) proof let i be set; assume A1: i in dom H; then A2: H/.i = H.i by FINSEQ_4:def 4; per cases; suppose A3: i in dom F; then A4: F/.i = F.i by FINSEQ_4:def 4; consider u,v being Element of R, a being Element of A such that A5: F/.i = u*a*v by A3,Def9; A6: F.i = H.i by A3,FINSEQ_1:def 7; a in A \/ B by XBOOLE_0:def 2; hence thesis by A2,A4,A5,A6; suppose not i in dom F; then consider n being Nat such that A7: n in dom G & i=len F + n by A1,FINSEQ_1:38; consider u,v being Element of R, b being Element of B such that A8: G/.n = u*b*v by A7,Def9; A9: G/.n = G.n by A7,FINSEQ_4:def 4; A10: G.n = H.i by A7,FINSEQ_1:def 7; b in A \/ B by XBOOLE_0:def 2; hence thesis by A2,A8,A9,A10; end; end; end; theorem Th23: for R be associative (non empty multLoopStr), A be non empty Subset of R, a be Element of R, F be LinearCombination of A holds a*F is LinearCombination of A proof let R be associative (non empty multLoopStr), A be non empty Subset of R, a be Element of R, F be LinearCombination of A; let i be set; assume i in dom (a*F); then A1: i in dom F by POLYNOM1:def 2; then consider u, v being Element of R, b being Element of A such that A2: F/.i = u*b*v by Def9; take x = a*u, v, b; thus (a*F)/.i = a*F/.i by A1,POLYNOM1:def 2 .= a*(u*b)*v by A2,VECTSP_1:def 16 .= x*b*v by VECTSP_1:def 16; end; theorem Th24: for R be associative (non empty multLoopStr), A be non empty Subset of R, a be Element of R, F be LinearCombination of A holds F*a is LinearCombination of A proof let R be associative (non empty multLoopStr), A be non empty Subset of R, a be Element of R, F be LinearCombination of A; let i be set; assume i in dom (F*a); then A1: i in dom F by POLYNOM1:def 3; then consider u, v being Element of R, b being Element of A such that A2: F/.i = u*b*v by Def9; take u, x = v*a, b; thus (F*a)/.i = (F/.i)*a by A1,POLYNOM1:def 3 .= u*(b*v)*a by A2,VECTSP_1:def 16 .= u*(b*v*a) by VECTSP_1:def 16 .= u*(b*(v*a)) by VECTSP_1:def 16 .= u*b*x by VECTSP_1:def 16; end; theorem Th25: for R being right_unital (non empty multLoopStr), A being non empty Subset of R, f being LeftLinearCombination of A holds f is LinearCombination of A proof let R be right_unital (non empty multLoopStr), A be non empty Subset of R, f be LeftLinearCombination of A; let i be set; assume i in dom f; then consider r being Element of R, a being Element of A such that A1: f/.i = r*a by Def10; f/.i = r*a*1_ R by A1,VECTSP_1:def 13; hence thesis; end; definition let R be non empty multLoopStr, A,B be non empty Subset of R, F be LeftLinearCombination of A, G be LeftLinearCombination of B; redefine func F^G -> LeftLinearCombination of (A \/ B); coherence proof set H = F^G; thus H is LeftLinearCombination of (A \/ B) proof let i be set; assume A1: i in dom H; then A2: H/.i = H.i by FINSEQ_4:def 4; per cases; suppose A3: i in dom F; then A4: F/.i = F.i by FINSEQ_4:def 4; consider u being Element of R, a being Element of A such that A5: F/.i = u*a by A3,Def10; A6: F.i = H.i by A3,FINSEQ_1:def 7; a in A \/ B by XBOOLE_0:def 2; hence thesis by A2,A4,A5,A6; suppose not i in dom F; then consider n being Nat such that A7: n in dom G & i=len F + n by A1,FINSEQ_1:38; consider u being Element of R, b being Element of B such that A8: G/.n = u*b by A7,Def10; A9: G/.n = G.n by A7,FINSEQ_4:def 4; A10: G.n = H.i by A7,FINSEQ_1:def 7; b in A \/ B by XBOOLE_0:def 2; hence thesis by A2,A8,A9,A10; end; end; end; theorem Th26: for R be associative (non empty multLoopStr), A be non empty Subset of R, a be Element of R, F be LeftLinearCombination of A holds a*F is LeftLinearCombination of A proof let R be associative (non empty multLoopStr), A be non empty Subset of R, a be Element of R, F be LeftLinearCombination of A; let i be set; assume i in dom (a*F); then A1: i in dom F by POLYNOM1:def 2; then consider u being Element of R, b being Element of A such that A2: F/.i = u*b by Def10; take x = a*u, b; thus (a*F)/.i = a*F/.i by A1,POLYNOM1:def 2 .= x*b by A2,VECTSP_1:def 16; end; theorem for R be non empty multLoopStr, A be non empty Subset of R, a be Element of R, F be LeftLinearCombination of A holds F*a is LinearCombination of A proof let R be non empty multLoopStr, A be non empty Subset of R, a be Element of R, F be LeftLinearCombination of A; let i be set; assume i in dom (F*a); then A1: i in dom F by POLYNOM1:def 3; then consider u being Element of R, b being Element of A such that A2: F/.i = u*b by Def10; reconsider c = a as Element of R; take u, c, b; thus (F*a)/.i = u*b*c by A1,A2,POLYNOM1:def 3; end; theorem Th28: for R being left_unital (non empty multLoopStr), A being non empty Subset of R, f being RightLinearCombination of A holds f is LinearCombination of A proof let R be left_unital (non empty multLoopStr), A be non empty Subset of R, f be RightLinearCombination of A; let i be set; assume i in dom f; then consider r being Element of R, a being Element of A such that A1: f/.i = a*r by Def11; f/.i = 1_ R*a*r by A1,VECTSP_1:def 19; hence thesis; end; definition let R be non empty multLoopStr, A,B be non empty Subset of R, F be RightLinearCombination of A, G be RightLinearCombination of B; redefine func F^G -> RightLinearCombination of (A \/ B); coherence proof set H = F^G; thus H is RightLinearCombination of (A \/ B) proof let i be set; assume A1: i in dom H; then A2: H/.i = H.i by FINSEQ_4:def 4; per cases; suppose A3: i in dom F; then A4: F/.i = F.i by FINSEQ_4:def 4; consider u being Element of R, a being Element of A such that A5: F/.i = a*u by A3,Def11; A6: F.i = H.i by A3,FINSEQ_1:def 7; a in A \/ B by XBOOLE_0:def 2; hence thesis by A2,A4,A5,A6; suppose not i in dom F; then consider n being Nat such that A7: n in dom G & i=len F + n by A1,FINSEQ_1:38; consider u being Element of R, b being Element of B such that A8: G/.n = b*u by A7,Def11; A9: G/.n = G.n by A7,FINSEQ_4:def 4; A10: G.n = H.i by A7,FINSEQ_1:def 7; b in A \/ B by XBOOLE_0:def 2; hence thesis by A2,A8,A9,A10; end; end; end; theorem Th29: for R be associative (non empty multLoopStr), A be non empty Subset of R, a be Element of R, F be RightLinearCombination of A holds F*a is RightLinearCombination of A proof let R be associative (non empty multLoopStr), A be non empty Subset of R, a be Element of R, F be RightLinearCombination of A; let i be set; assume i in dom (F*a); then A1: i in dom F by POLYNOM1:def 3; then consider u being Element of R, b being Element of A such that A2: F/.i = b*u by Def11; take x = u*a, b; thus (F*a)/.i=(F/.i)*a by A1,POLYNOM1:def 3.= b*x by A2,VECTSP_1:def 16; end; theorem for R be associative (non empty multLoopStr), A be non empty Subset of R, a be Element of R, F be RightLinearCombination of A holds a*F is LinearCombination of A proof let R be associative (non empty multLoopStr), A be non empty Subset of R, a be Element of R, F be RightLinearCombination of A; let i be set; assume i in dom (a*F); then A1: i in dom F by POLYNOM1:def 2; then consider u being Element of R, b being Element of A such that A2: F/.i = b*u by Def11; reconsider c = a as Element of R; take c, u, b; thus (a*F)/.i=a*(F/.i) by A1,POLYNOM1:def 2.= c*b*u by A2,VECTSP_1:def 16; end; theorem Th31: for R being commutative associative (non empty multLoopStr), A being non empty Subset of R, f being LinearCombination of A holds f is LeftLinearCombination of A & f is RightLinearCombination of A proof let R being commutative associative (non empty multLoopStr), A being non empty Subset of R, f being LinearCombination of A; hereby let i be set; assume i in dom f; then consider r,s being Element of R, a being Element of A such that A1: f/.i = r*a*s by Def9; f/.i = (r*s)*a by A1,VECTSP_1:def 16; hence ex r being Element of R, a being Element of A st f/.i = r*a; end; let i be set; assume i in dom f; then consider r,s being Element of R, a being Element of A such that A2: f/.i = r*a*s by Def9; f/.i = a*(r*s) by A2,VECTSP_1:def 16; hence ex r being Element of R, a being Element of A st f/.i = a*r; end; theorem Th32: for S being non empty doubleLoopStr, F being non empty Subset of S, lc being non empty LinearCombination of F ex p being LinearCombination of F, e being Element of S st lc = p^<* e *> & <*e*> is LinearCombination of F proof let S be non empty doubleLoopStr, F be non empty Subset of S, lc be non empty LinearCombination of F; len lc <> 0 by FINSEQ_1:25; then consider p being FinSequence of the carrier of S, e being Element of S such that A1: lc = p^<*e*> by FINSEQ_2:22; now let i be set; assume A2: i in dom p; A3: dom p c= dom lc by A1,FINSEQ_1:39; then consider u, v being Element of S, a being Element of F such that A4: lc/.i = u*a*v by A2,Def9; take u, v, a; thus p/.i = p.i by A2,FINSEQ_4:def 4 .= lc.i by A1,A2,FINSEQ_1:def 7 .= u*a*v by A2,A3,A4,FINSEQ_4:def 4; end; then reconsider p as LinearCombination of F by Def9; take p; take e; thus lc = p^<* e *> by A1; A5: len lc = len p +1 by A1,FINSEQ_2:19; A6: len lc in dom lc by FINSEQ_5:6; then consider u, v being Element of S, a being Element of F such that A7: lc/.(len lc) = u*a*v by Def9; A8: lc/.(len lc) = lc.(len lc) by A6,FINSEQ_4:def 4; let i be set such that A9: i in dom <*e*>; dom <*e*> = {1} by FINSEQ_1:4,55; then A10: i = 1 by A9,TARSKI:def 1; take u, v, a; thus <*e*>/.i = <*e*>.i by A9,FINSEQ_4:def 4 .= e by A10,FINSEQ_1:57 .= u*a*v by A1,A5,A7,A8,JORDAN3:16; end; theorem Th33: for S being non empty doubleLoopStr, F being non empty Subset of S, lc being non empty LeftLinearCombination of F ex p being LeftLinearCombination of F, e being Element of S st lc = p^<* e *> & <*e*> is LeftLinearCombination of F proof let S be non empty doubleLoopStr, F be non empty Subset of S, lc be non empty LeftLinearCombination of F; len lc <> 0 by FINSEQ_1:25; then consider p being FinSequence of the carrier of S, e being Element of S such that A1: lc = p^<*e*> by FINSEQ_2:22; now let i be set; assume A2: i in dom p; A3: dom p c= dom lc by A1,FINSEQ_1:39; then consider u being Element of S, a being Element of F such that A4: lc/.i = u*a by A2,Def10; take u, a; thus p/.i = p.i by A2,FINSEQ_4:def 4 .= lc.i by A1,A2,FINSEQ_1:def 7 .= u*a by A2,A3,A4,FINSEQ_4:def 4; end; then reconsider p as LeftLinearCombination of F by Def10; take p; take e; thus lc = p^<* e *> by A1; A5: len lc = len p +1 by A1,FINSEQ_2:19; A6: len lc in dom lc by FINSEQ_5:6; then consider u being Element of S, a being Element of F such that A7: lc/.(len lc) = u*a by Def10; A8: lc/.(len lc) = lc.(len lc) by A6,FINSEQ_4:def 4; let i be set such that A9: i in dom <*e*>; dom <*e*> = {1} by FINSEQ_1:4,55; then A10: i = 1 by A9,TARSKI:def 1; take u, a; thus <*e*>/.i = <*e*>.i by A9,FINSEQ_4:def 4 .= e by A10,FINSEQ_1:57 .= u*a by A1,A5,A7,A8,JORDAN3:16; end; theorem Th34: for S being non empty doubleLoopStr, F being non empty Subset of S, lc being non empty RightLinearCombination of F ex p being RightLinearCombination of F, e being Element of S st lc = p^<* e *> & <*e*> is RightLinearCombination of F proof let S be non empty doubleLoopStr, F be non empty Subset of S, lc be non empty RightLinearCombination of F; len lc <> 0 by FINSEQ_1:25; then consider p being FinSequence of the carrier of S, e being Element of S such that A1: lc = p^<*e*> by FINSEQ_2:22; now let i be set; assume A2: i in dom p; A3: dom p c= dom lc by A1,FINSEQ_1:39; then consider u being Element of S, a being Element of F such that A4: lc/.i = a*u by A2,Def11; take u, a; thus p/.i = p.i by A2,FINSEQ_4:def 4 .= lc.i by A1,A2,FINSEQ_1:def 7 .= a*u by A2,A3,A4,FINSEQ_4:def 4; end; then reconsider p as RightLinearCombination of F by Def11; take p; take e; thus lc = p^<* e *> by A1; A5: len lc = len p +1 by A1,FINSEQ_2:19; A6: len lc in dom lc by FINSEQ_5:6; then consider u being Element of S, a being Element of F such that A7: lc/.(len lc) = a*u by Def11; A8: lc/.(len lc) = lc.(len lc) by A6,FINSEQ_4:def 4; let i be set such that A9: i in dom <*e*>; dom <*e*> = {1} by FINSEQ_1:4,55; then A10: i = 1 by A9,TARSKI:def 1; take u, a; thus <*e*>/.i = <*e*>.i by A9,FINSEQ_4:def 4 .= e by A10,FINSEQ_1:57 .= a*u by A1,A5,A7,A8,JORDAN3:16; end; definition let R be non empty multLoopStr, A be non empty Subset of R, L be LinearCombination of A, E be FinSequence of [:the carrier of R, the carrier of R, the carrier of R:]; pred E represents L means :Def12 : len E = len L & for i being set st i in dom L holds L.i = ((E/.i)`1)*((E/.i)`2)*((E/.i)`3) & ((E/.i)`2) in A; end; theorem for R being non empty multLoopStr, A being non empty Subset of R, L be LinearCombination of A ex E be FinSequence of [:the carrier of R, the carrier of R, the carrier of R:] st E represents L proof let R be non empty multLoopStr,A be non empty Subset of R, L be LinearCombination of A; set D = [:the carrier of R, the carrier of R, the carrier of R:]; defpred P[set,set] means ex x, y, z being Element of R st $2 = [x, y, z] & y in A & L/.$1 = x*y*z; A1: now let k be Nat; assume k in Seg len L; then k in dom L by FINSEQ_1:def 3; then consider u,v being Element of R, a being Element of A such that A2: L/.k = u*a*v by Def9; reconsider b = a as Element of R; reconsider d = [u, b, v] as Element of D; take d; thus P[k, d] by A2; end; consider E being FinSequence of D such that A3: dom E = Seg len L and A4: for k be Nat st k in Seg len L holds P[k,E/.k] from SeqExD(A1); take E; A5: dom L = Seg len L by FINSEQ_1:def 3; thus len E = len L by A3,FINSEQ_1:def 3; let i being set such that A6: i in dom L; reconsider k = i as Nat by A6; consider x, y, z being Element of R such that A7: E/.k = [x, y, z] & y in A & L/.k = x*y*z by A4,A5,A6; [x,y,z]`1 = x & [x,y,z]`2 = y & [x,y,z]`3 = z by MCART_1:def 5,def 6,def 7; hence L.i = ((E/.i)`1)*((E/.i)`2)*((E/.i)`3) by A6,A7,FINSEQ_4:def 4; thus ((E/.i)`2) in A by A7,MCART_1:def 6; end; theorem for R, S being non empty multLoopStr, F being non empty Subset of R, lc being LinearCombination of F, G being non empty Subset of S, P being Function of the carrier of R, the carrier of S, E being FinSequence of [:the carrier of R, the carrier of R, the carrier of R:] st P.:F c= G & E represents lc holds ex LC being LinearCombination of G st len lc = len LC & for i being set st i in dom LC holds LC.i = (P.(E/.i)`1)*(P.(E/.i)`2)*(P.(E/.i)`3) proof let R, S be non empty multLoopStr, F be non empty Subset of R, lc be LinearCombination of F, G be non empty Subset of S, P be Function of the carrier of R, the carrier of S, E being FinSequence of [:the carrier of R, the carrier of R, the carrier of R:]; assume A1: P.:F c= G; assume A2: E represents lc; deffunc F(Nat)=(P.(E/.$1)`1)*(P.(E/.$1)`2)*(P.(E/.$1)`3); consider LC being FinSequence of the carrier of S such that A3: len LC = len lc and A4: for k being Nat st k in Seg len lc holds LC.k = F(k) from SeqLambdaD; now let i be set such that A5: i in dom LC; A6: dom LC = Seg len lc by A3,FINSEQ_1:def 3; reconsider u = P.(E/.i)`1, v = P.(E/.i)`3 as Element of S; A7: dom P = the carrier of R by FUNCT_2:def 1; dom lc = dom LC by A3,FINSEQ_3:31; then (E/.i)`2 in F by A2,A5,Def12 ; then P.((E/.i)`2) in P.:F by A7,FUNCT_1:def 12; then reconsider a = P.(E/.i)`2 as Element of G by A1; take u, v, a; LC.i = LC/.i by A5,FINSEQ_4:def 4; hence LC/.i = u*a*v by A4,A5,A6; end; then reconsider LC as LinearCombination of G by Def9; take LC; thus len lc = len LC by A3; let i be set such that A8: i in dom LC; dom LC = Seg len lc by A3,FINSEQ_1:def 3; hence thesis by A4,A8; end; definition let R be non empty multLoopStr, A be non empty Subset of R, L be LeftLinearCombination of A, E be FinSequence of [:the carrier of R, the carrier of R:]; pred E represents L means :Def13: len E = len L & for i being set st i in dom L holds L.i = ((E/.i)`1)*((E/.i)`2) & ((E/.i)`2) in A; end; theorem for R being non empty multLoopStr, A being non empty Subset of R, L be LeftLinearCombination of A ex E be FinSequence of [:the carrier of R, the carrier of R:] st E represents L proof let R be non empty multLoopStr,A be non empty Subset of R, L be LeftLinearCombination of A; set D = [:the carrier of R, the carrier of R:]; defpred P[set,set] means ex x, y being Element of R st $2 = [x, y] & y in A & L/.$1 = x*y; A1: now let k be Nat; assume k in Seg len L; then k in dom L by FINSEQ_1:def 3; then consider u being Element of R, a being Element of A such that A2: L/.k = u*a by Def10; reconsider b = a as Element of R; reconsider d = [u, b] as Element of D; take d; thus P[k, d] by A2; end; consider E being FinSequence of D such that A3: dom E = Seg len L and A4: for k be Nat st k in Seg len L holds P[k,E/.k] from SeqExD(A1); take E; A5: dom L = Seg len L by FINSEQ_1:def 3; thus len E = len L by A3,FINSEQ_1:def 3; let i being set such that A6: i in dom L; reconsider k = i as Nat by A6; consider x, y being Element of R such that A7: E/.k = [x, y] & y in A & L/.k = x*y by A4,A5,A6; [x,y]`1 = x & [x,y]`2 = y by MCART_1:def 1,def 2; hence L.i = ((E/.i)`1)*((E/.i)`2) by A6,A7,FINSEQ_4:def 4; thus ((E/.i)`2) in A by A7,MCART_1:def 2; end; theorem for R, S being non empty multLoopStr, F being non empty Subset of R, lc being LeftLinearCombination of F, G being non empty Subset of S, P being Function of the carrier of R, the carrier of S, E being FinSequence of [:the carrier of R, the carrier of R:] st P.:F c= G & E represents lc holds ex LC being LeftLinearCombination of G st len lc = len LC & for i being set st i in dom LC holds LC.i = (P.(E/.i)`1)*(P.(E/.i)`2) proof let R, S be non empty multLoopStr, F be non empty Subset of R, lc be LeftLinearCombination of F, G be non empty Subset of S, P be Function of the carrier of R, the carrier of S, E being FinSequence of [:the carrier of R, the carrier of R:]; assume A1: P.:F c= G; assume A2: E represents lc; deffunc F(Nat)=(P.(E/.$1)`1)*(P.(E/.$1)`2); consider LC being FinSequence of the carrier of S such that A3: len LC = len lc and A4: for k being Nat st k in Seg len lc holds LC.k = F(k) from SeqLambdaD; now let i be set such that A5: i in dom LC; A6: dom LC = Seg len lc by A3,FINSEQ_1:def 3; reconsider u = P.(E/.i)`1 as Element of S; A7: dom P = the carrier of R by FUNCT_2:def 1; dom lc = dom LC by A3,FINSEQ_3:31; then (E/.i)`2 in F by A2,A5,Def13 ; then P.((E/.i)`2) in P.:F by A7,FUNCT_1:def 12; then reconsider a = P.(E/.i)`2 as Element of G by A1; take u, a; LC.i = LC/.i by A5,FINSEQ_4:def 4; hence LC/.i = u*a by A4,A5,A6; end; then reconsider LC as LeftLinearCombination of G by Def10; take LC; thus len lc = len LC by A3; let i be set such that A8: i in dom LC; dom LC = Seg len lc by A3,FINSEQ_1:def 3; hence thesis by A4,A8; end; definition let R be non empty multLoopStr, A be non empty Subset of R, L be RightLinearCombination of A, E be FinSequence of [:the carrier of R, the carrier of R:]; pred E represents L means :Def14: len E = len L & for i being set st i in dom L holds L.i = ((E/.i)`1)*((E/.i)`2) & ((E/.i)`1) in A; end; theorem for R being non empty multLoopStr, A being non empty Subset of R, L be RightLinearCombination of A ex E be FinSequence of [:the carrier of R, the carrier of R:] st E represents L proof let R be non empty multLoopStr,A be non empty Subset of R, L be RightLinearCombination of A; set D = [:the carrier of R, the carrier of R:]; defpred P[set,set] means ex x, y being Element of R st $2 = [x, y] & x in A & L/.$1 = x*y; A1: now let k be Nat; assume k in Seg len L; then k in dom L by FINSEQ_1:def 3; then consider v being Element of R, a being Element of A such that A2: L/.k = a*v by Def11; reconsider b = a as Element of R; reconsider d = [b, v] as Element of D; take d; thus P[k, d] by A2; end; consider E being FinSequence of D such that A3: dom E = Seg len L and A4: for k be Nat st k in Seg len L holds P[k,E/.k] from SeqExD(A1); take E; A5: dom L = Seg len L by FINSEQ_1:def 3; thus len E = len L by A3,FINSEQ_1:def 3; let i being set such that A6: i in dom L; reconsider k = i as Nat by A6; consider x, y being Element of R such that A7: E/.k = [x, y] & x in A & L/.k = x*y by A4,A5,A6; [x,y]`1 = x & [x,y]`2 = y by MCART_1:def 1,def 2; hence L.i = ((E/.i)`1)*((E/.i)`2) by A6,A7,FINSEQ_4:def 4; thus ((E/.i)`1) in A by A7,MCART_1:def 1; end; theorem for R, S being non empty multLoopStr, F being non empty Subset of R, lc being RightLinearCombination of F, G being non empty Subset of S, P being Function of the carrier of R, the carrier of S, E being FinSequence of [:the carrier of R, the carrier of R:] st P.:F c= G & E represents lc holds ex LC being RightLinearCombination of G st len lc = len LC & for i being set st i in dom LC holds LC.i = (P.(E/.i)`1)*(P.(E/.i)`2) proof let R, S be non empty multLoopStr, F be non empty Subset of R, lc be RightLinearCombination of F, G be non empty Subset of S, P be Function of the carrier of R, the carrier of S, E being FinSequence of [:the carrier of R, the carrier of R:]; assume A1: P.:F c= G; assume A2: E represents lc; deffunc F(Nat)=(P.(E/.$1)`1)*(P.(E/.$1)`2); consider LC being FinSequence of the carrier of S such that A3: len LC = len lc and A4: for k being Nat st k in Seg len lc holds LC.k = F(k) from SeqLambdaD; now let i be set such that A5: i in dom LC; A6: dom LC = Seg len lc by A3,FINSEQ_1:def 3; reconsider v = P.(E/.i)`2 as Element of S; A7: dom P = the carrier of R by FUNCT_2:def 1; dom lc = dom LC by A3,FINSEQ_3:31; then (E/.i)`1 in F by A2,A5,Def14 ; then P.((E/.i)`1) in P.:F by A7,FUNCT_1:def 12; then reconsider a = P.(E/.i)`1 as Element of G by A1; take v, a; LC.i = LC/.i by A5,FINSEQ_4:def 4; hence LC/.i = a*v by A4,A5,A6; end; then reconsider LC as RightLinearCombination of G by Def11; take LC; thus len lc = len LC by A3; let i be set such that A8: i in dom LC; dom LC = Seg len lc by A3,FINSEQ_1:def 3; hence thesis by A4,A8; end; theorem for R being non empty multLoopStr, A being non empty Subset of R, l being LinearCombination of A, n being Nat holds l|Seg n is LinearCombination of A proof let R be non empty multLoopStr, A be non empty Subset of R, l be LinearCombination of A, n being Nat; reconsider ln = l|Seg n as FinSequence of the carrier of R by FINSEQ_1:23; now let i being set such that A1: i in dom ln; A2: dom ln c= dom l by RELAT_1:89; then consider u,v being Element of R, a being Element of A such that A3: l/.i = u*a*v by A1,Def9; take u, v, a; thus ln/.i = ln.i by A1,FINSEQ_4:def 4 .= l.i by A1,FUNCT_1:70 .= u*a*v by A1,A2,A3,FINSEQ_4:def 4; end; hence l|Seg n is LinearCombination of A by Def9; end; theorem for R being non empty multLoopStr, A being non empty Subset of R, l being LeftLinearCombination of A, n being Nat holds l|Seg n is LeftLinearCombination of A proof let R be non empty multLoopStr, A be non empty Subset of R, l be LeftLinearCombination of A, n being Nat; reconsider ln = l|Seg n as FinSequence of the carrier of R by FINSEQ_1:23; now let i being set such that A1: i in dom ln; A2: dom ln c= dom l by RELAT_1:89; then consider u being Element of R, a being Element of A such that A3: l/.i = u*a by A1,Def10; take u, a; thus ln/.i = ln.i by A1,FINSEQ_4:def 4 .= l.i by A1,FUNCT_1:70 .= u*a by A1,A2,A3,FINSEQ_4:def 4; end; hence l|Seg n is LeftLinearCombination of A by Def10; end; theorem for R being non empty multLoopStr, A being non empty Subset of R, l being RightLinearCombination of A, n being Nat holds l|Seg n is RightLinearCombination of A proof let R be non empty multLoopStr, A be non empty Subset of R, l be RightLinearCombination of A, n being Nat; reconsider ln = l|Seg n as FinSequence of the carrier of R by FINSEQ_1:23; now let i being set such that A1: i in dom ln; A2: dom ln c= dom l by RELAT_1:89; then consider v being Element of R, a being Element of A such that A3: l/.i = a*v by A1,Def11; take v, a; thus ln/.i = ln.i by A1,FINSEQ_4:def 4 .= l.i by A1,FUNCT_1:70 .= a*v by A1,A2,A3,FINSEQ_4:def 4; end; hence l|Seg n is RightLinearCombination of A by Def11; end; begin :: Generated ideals definition let L be non empty doubleLoopStr, F be Subset of L; assume A1: F is non empty; func F-Ideal -> Ideal of L means :Def15: F c= it & for I being Ideal of L st F c= I holds it c= I; existence proof set Id = { I where I is Element of bool the carrier of L: F c= I & I is Ideal of L }; set I = meet Id; the carrier of L is Ideal of L by Th10; then A2: the carrier of L in Id; then A3: I c= the carrier of L by SETFAM_1:4; A4: now let X be set; assume X in Id; then ex X' being Element of bool the carrier of L st X'=X & F c= X' & X' is Ideal of L; hence F c= X; end; then A5: F c= I by A2,SETFAM_1:6; consider x being set such that A6: x in F by A1,XBOOLE_0:def 1; reconsider I as non empty Subset of L by A3,A5,A6; A7: I is add-closed proof let x, y being Element of L; assume A8: x in I & y in I; now let X be set; assume A9: X in Id; then A10: x in X by A8,SETFAM_1:def 1; A11: y in X by A8,A9,SETFAM_1:def 1; consider X' being Element of bool the carrier of L such that A12: X'=X & F c= X' & X' is Ideal of L by A9; x+y in X' by A10,A11,A12,Def1; hence {x+y} c= X by A12,ZFMISC_1:37; end; then {x+y} c= I by A2,SETFAM_1:6; hence x+y in I by ZFMISC_1:37; end; A13: I is left-ideal proof let p, x being Element of L; assume A14: x in I; now let X be set; assume A15: X in Id; then A16: x in X by A14,SETFAM_1:def 1; consider X' being Element of bool the carrier of L such that A17: X'=X & F c= X' & X' is Ideal of L by A15; p*x in X' by A16,A17,Def2; hence {p*x} c= X by A17,ZFMISC_1:37; end; then {p*x} c= I by A2,SETFAM_1:6; hence p*x in I by ZFMISC_1:37; end; I is right-ideal proof let p, x being Element of L; assume A18: x in I; now let X be set; assume A19: X in Id; then A20: x in X by A18,SETFAM_1:def 1; consider X' being Element of bool the carrier of L such that A21: X'=X & F c= X' & X' is Ideal of L by A19; x*p in X' by A20,A21,Def3; hence {x*p} c= X by A21,ZFMISC_1:37; end; then {x*p} c= I by A2,SETFAM_1:6; hence x*p in I by ZFMISC_1:37; end; then reconsider I as Ideal of L by A7,A13; take I; now let X be Ideal of L; assume F c= X; then X in Id; hence I c= X by SETFAM_1:4; end; hence thesis by A2,A4,SETFAM_1:6; end; uniqueness proof let X,Y be Ideal of L such that A22: F c= X & for I being Ideal of L st F c= I holds X c= I and A23: F c= Y & for I being Ideal of L st F c= I holds Y c= I; A24: X c= Y by A22,A23; Y c= X by A22,A23; hence X=Y by A24,XBOOLE_0:def 10; end; func F-LeftIdeal -> LeftIdeal of L means :Def16 : F c= it & for I being LeftIdeal of L st F c= I holds it c= I; existence proof set Id = { I where I is Element of bool the carrier of L: F c= I & I is LeftIdeal of L }; set I = meet Id; the carrier of L is LeftIdeal of L by Th11; then A25: the carrier of L in Id; then A26: I c= the carrier of L by SETFAM_1:4; A27: now let X be set; assume X in Id; then consider X' being Element of bool the carrier of L such that A28: X'=X & F c= X' & X' is LeftIdeal of L; thus F c= X by A28; end; then A29: F c= I by A25,SETFAM_1:6; consider x being set such that A30: x in F by A1,XBOOLE_0:def 1; reconsider I as non empty Subset of L by A26,A29,A30; A31: I is add-closed proof let x, y being Element of L; assume A32: x in I & y in I; now let X be set; assume A33: X in Id; then A34: x in X by A32,SETFAM_1:def 1; A35: y in X by A32,A33,SETFAM_1:def 1; consider X' being Element of bool the carrier of L such that A36: X'=X & F c= X' & X' is LeftIdeal of L by A33; x+y in X' by A34,A35,A36,Def1; hence {x+y} c= X by A36,ZFMISC_1:37; end; then {x+y} c= I by A25,SETFAM_1:6; hence x+y in I by ZFMISC_1:37; end; I is left-ideal proof let p, x being Element of L; assume A37: x in I; now let X be set; assume A38: X in Id; then A39: x in X by A37,SETFAM_1:def 1; consider X' being Element of bool the carrier of L such that A40: X'=X & F c= X' & X' is LeftIdeal of L by A38; p*x in X' by A39,A40,Def2; hence {p*x} c= X by A40,ZFMISC_1:37; end; then {p*x} c= I by A25,SETFAM_1:6; hence p*x in I by ZFMISC_1:37; end; then reconsider I as LeftIdeal of L by A31; take I; now let X be LeftIdeal of L; assume F c= X; then X in Id; hence I c= X by SETFAM_1:4; end; hence thesis by A25,A27,SETFAM_1:6; end; uniqueness proof let X,Y be LeftIdeal of L such that A41: F c= X & for I being LeftIdeal of L st F c= I holds X c= I and A42: F c= Y & for I being LeftIdeal of L st F c= I holds Y c= I; A43: X c= Y by A41,A42; Y c= X by A41,A42; hence X=Y by A43,XBOOLE_0:def 10; end; func F-RightIdeal -> RightIdeal of L means :Def17 : F c= it & for I being RightIdeal of L st F c= I holds it c= I; existence proof set Id = { I where I is Element of bool the carrier of L: F c= I & I is RightIdeal of L }; set I = meet Id; the carrier of L is RightIdeal of L by Th12; then A44: the carrier of L in Id; then A45: I c= the carrier of L by SETFAM_1:4; A46: now let X be set; assume X in Id; then consider X' being Element of bool the carrier of L such that A47: X'=X & F c= X' & X' is RightIdeal of L; thus F c= X by A47; end; then A48: F c= I by A44,SETFAM_1:6; consider x being set such that A49: x in F by A1,XBOOLE_0:def 1; reconsider I as non empty Subset of L by A45,A48,A49; A50: I is add-closed proof let x, y being Element of L; assume A51: x in I & y in I; now let X be set; assume A52: X in Id; then A53: x in X by A51,SETFAM_1:def 1; A54: y in X by A51,A52,SETFAM_1:def 1; consider X' being Element of bool the carrier of L such that A55: X'=X & F c= X' & X' is RightIdeal of L by A52; x+y in X' by A53,A54,A55,Def1; hence {x+y} c= X by A55,ZFMISC_1:37; end; then {x+y} c= I by A44,SETFAM_1:6; hence x+y in I by ZFMISC_1:37; end; I is right-ideal proof let p, x being Element of L; assume A56: x in I; now let X be set; assume A57: X in Id; then A58: x in X by A56,SETFAM_1:def 1; consider X' being Element of bool the carrier of L such that A59: X'=X & F c= X' & X' is RightIdeal of L by A57; x*p in X' by A58,A59,Def3; hence {x*p} c= X by A59,ZFMISC_1:37; end; then {x*p} c= I by A44,SETFAM_1:6; hence x*p in I by ZFMISC_1:37; end; then reconsider I as RightIdeal of L by A50; take I; now let X be RightIdeal of L; assume F c= X; then X in Id; hence I c= X by SETFAM_1:4; end; hence thesis by A44,A46,SETFAM_1:6; end; uniqueness proof let X,Y be RightIdeal of L such that A60: F c= X & for I being RightIdeal of L st F c= I holds X c= I and A61: F c= Y & for I being RightIdeal of L st F c= I holds Y c= I; A62: X c= Y by A60,A61; Y c= X by A60,A61; hence X=Y by A62,XBOOLE_0:def 10; end; end; theorem Th44: for L being non empty doubleLoopStr, I being Ideal of L holds I-Ideal = I proof let L be non empty doubleLoopStr, I be Ideal of L; I c= I-Ideal & I-Ideal c= I by Def15; hence thesis by XBOOLE_0:def 10; end; theorem Th45: for L being non empty doubleLoopStr, I being LeftIdeal of L holds I-LeftIdeal = I proof let L be non empty doubleLoopStr, I be LeftIdeal of L; I c= I-LeftIdeal & I-LeftIdeal c= I by Def16; hence thesis by XBOOLE_0:def 10 ; end; theorem Th46: for L being non empty doubleLoopStr, I being RightIdeal of L holds I-RightIdeal = I proof let L be non empty doubleLoopStr, I be RightIdeal of L; I c= I-RightIdeal & I-RightIdeal c= I by Def17; hence thesis by XBOOLE_0: def 10; end; definition let L be non empty doubleLoopStr, I be Ideal of L; mode Basis of I -> non empty Subset of L means it-Ideal = I; existence proof take I; thus thesis by Th44; end; end; theorem Th47: for L being add-associative right_zeroed right_complementable distributive (non empty doubleLoopStr) holds {0.L}-Ideal = {0.L} proof let L be add-associative right_zeroed right_complementable distributive (non empty doubleLoopStr); {0.L} is Ideal of L by Th7; hence thesis by Th44; end; theorem for L being left_zeroed right_zeroed add-cancelable distributive (non empty doubleLoopStr) holds {0.L}-Ideal = {0.L} proof let L be left_zeroed right_zeroed add-cancelable distributive (non empty doubleLoopStr); {0.L} is Ideal of L by Th4,Th5,Th6; hence thesis by Th44; end; theorem for L being left_zeroed right_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr) holds {0.L}-LeftIdeal = {0.L} proof let L be left_zeroed right_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr); {0.L} is LeftIdeal of L by Th4,Th5; hence thesis by Th45; end; theorem for L being right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr) holds {0.L}-RightIdeal = {0.L} proof let L be right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr); {0.L} is RightIdeal of L by Th4,Th6; hence thesis by Th46; end; theorem for L being well-unital (non empty doubleLoopStr) holds {1_ L}-Ideal = the carrier of L proof let L be well-unital (non empty doubleLoopStr); the carrier of L c= {1_ L}-Ideal proof let x be set; assume x in the carrier of L; then reconsider x'=x as Element of L; A1: 1_ L in {1_ L} by TARSKI:def 1; {1_ L} c= {1_ L}-Ideal by Def15; then x'*1_ L in {1_ L}-Ideal by A1,Def2; hence thesis by VECTSP_2:def 2; end; hence thesis by XBOOLE_0:def 10; end; theorem for L being right_unital (non empty doubleLoopStr) holds {1_ L}-LeftIdeal = the carrier of L proof let L be right_unital (non empty doubleLoopStr); the carrier of L c= {1_ L}-LeftIdeal proof let x be set; assume x in the carrier of L; then reconsider x'=x as Element of L; A1: 1_ L in {1_ L} by TARSKI:def 1; {1_ L} c= {1_ L}-LeftIdeal by Def16; then x'*1_ L in {1_ L}-LeftIdeal by A1,Def2; hence thesis by VECTSP_1:def 13; end; hence thesis by XBOOLE_0:def 10; end; theorem for L being left_unital (non empty doubleLoopStr) holds {1_ L}-RightIdeal = the carrier of L proof let L be left_unital (non empty doubleLoopStr); the carrier of L c= {1_ L}-RightIdeal proof let x be set; assume x in the carrier of L; then reconsider x'=x as Element of L; A1: 1_ L in {1_ L} by TARSKI:def 1; {1_ L} c= {1_ L}-RightIdeal by Def17; then (1_ L)*x' in {1_ L}-RightIdeal by A1,Def3; hence thesis by VECTSP_1:def 19; end; hence thesis by XBOOLE_0:def 10; end; theorem for L being non empty doubleLoopStr holds ([#] L)-Ideal = the carrier of L proof let L be non empty doubleLoopStr; A1: [#] L = the carrier of L by PRE_TOPC:def 3; [#] L c= ([#] L)-Ideal by Def15; hence thesis by A1,XBOOLE_0:def 10; end; theorem for L being non empty doubleLoopStr holds ([#] L)-LeftIdeal = the carrier of L proof let L be non empty doubleLoopStr; A1: [#] L = the carrier of L by PRE_TOPC:def 3; [#] L c= ([#] L)-LeftIdeal by Def16; hence thesis by A1,XBOOLE_0:def 10; end; theorem for L being non empty doubleLoopStr holds ([#] L)-RightIdeal = the carrier of L proof let L be non empty doubleLoopStr; A1: [#] L = the carrier of L by PRE_TOPC:def 3; [#] L c= ([#] L)-RightIdeal by Def17; hence thesis by A1,XBOOLE_0:def 10; end; theorem Th57: for L being non empty doubleLoopStr, A, B being non empty Subset of L st A c= B holds A-Ideal c= B-Ideal proof let L be non empty doubleLoopStr, A,B be non empty Subset of L; assume A1: A c= B; B c= B-Ideal by Def15; then A c= B-Ideal by A1,XBOOLE_1:1 ; hence thesis by Def15; end; theorem for L being non empty doubleLoopStr, A, B being non empty Subset of L st A c= B holds A-LeftIdeal c= B-LeftIdeal proof let L be non empty doubleLoopStr, A,B be non empty Subset of L; assume A1: A c= B; B c= B-LeftIdeal by Def16; then A c= B-LeftIdeal by A1,XBOOLE_1:1; hence thesis by Def16; end; theorem for L being non empty doubleLoopStr, A, B being non empty Subset of L st A c= B holds A-RightIdeal c= B-RightIdeal proof let L be non empty doubleLoopStr, A,B be non empty Subset of L; assume A1: A c= B; B c= B-RightIdeal by Def17; then A c= B-RightIdeal by A1,XBOOLE_1:1; hence thesis by Def17; end; theorem Th60: for L being add-associative left_zeroed right_zeroed add-cancelable associative distributive well-unital (non empty doubleLoopStr), F being non empty Subset of L, x being set holds x in F-Ideal iff ex f being LinearCombination of F st x = Sum f proof let L be add-associative right_zeroed add-cancelable left_zeroed associative distributive well-unital (non empty doubleLoopStr), F be non empty Subset of L; set I = { x where x is Element of L: ex lc being LinearCombination of F st Sum lc = x }; A1: I c= F-Ideal proof let x be set; defpred P[Nat] means for lc being LinearCombination of F st len lc <= $1 holds Sum lc in F-Ideal; A2: P[0] proof let lc be LinearCombination of F; assume len lc <= 0; then len lc = 0 by NAT_1:18; then lc = <*>(the carrier of L) by FINSEQ_1:25; then A3: Sum lc = 0.L by RLVECT_1:60; consider y being Element of F; F c= F-Ideal by Def15; then A4: y in F-Ideal by TARSKI:def 3; 0.L*y = 0.L by BINOM:1; hence thesis by A3,A4,Def2; end; A5: for k being Nat st P[k] holds P[k+1] proof let k be Nat; assume A6: P[k]; thus P[k+1] proof let lc be LinearCombination of F; assume A7: len lc <= k+1; per cases by A7,NAT_1:26; suppose len lc <= k; hence thesis by A6; suppose A8: len lc = k+1; then lc is non empty by FINSEQ_1:25; then consider q being LinearCombination of F, r being Element of L such that A9: lc=q^<*r*> & <*r*> is LinearCombination of F by Th32; k+1 = len q + len <*r*> by A8,A9,FINSEQ_1:35 .= len q + 1 by FINSEQ_1:56; then len q = k by XCMPLX_1:2; then A10: Sum q in F-Ideal by A6; dom <*r*> = {1} by FINSEQ_1:4,55; then A11: 1 in dom <*r*> by TARSKI:def 1; then consider u,v being Element of L, a being Element of F such that A12: <* r *>/.1 = u*a*v by A9,Def9; A13: <*r*>/.1 = <*r*>.1 by A11,FINSEQ_4:def 4; A14: Sum <* r *> = r by BINOM:3 .= u*a*v by A12,A13,FINSEQ_1:57; F c= F-Ideal by Def15 ; then a in F-Ideal by TARSKI:def 3; then u*a in F-Ideal by Def2; then A15: Sum <* r *> in F-Ideal by A14,Def3; Sum lc = Sum q + Sum <* r *> by A9,RLVECT_1:58; hence thesis by A10,A15,Def1; end; end; A16: for k being Nat holds P[k] from Ind(A2,A5); assume x in I; then consider x' being Element of L such that A17: x'=x & ex lc being LinearCombination of F st Sum lc = x'; consider lc being LinearCombination of F such that A18: Sum lc = x' by A17; P[len lc] by A16; hence x in F-Ideal by A17,A18; end; A19: F c= I proof let x be set; assume A20: x in F; then reconsider x as Element of L; set lc = <* x *>; now let i be set; assume A21: i in dom lc; then A22: lc/.i = lc.i by FINSEQ_4:def 4; dom lc = {1} by FINSEQ_1:4,55; then i = 1 by A21,TARSKI:def 1; then lc.i = x by FINSEQ_1:57 .= 1_ L*x by VECTSP_2:def 2 .= 1_ L*x*1_ L by VECTSP_2:def 2; hence ex u,v being Element of L, a being Element of F st lc/.i = u*a*v by A20,A22; end; then reconsider lc as LinearCombination of F by Def9; Sum lc = x by BINOM:3; hence thesis; end; A23: I c= the carrier of L proof let x be set; assume x in I; then consider x' being Element of L such that A24: x'=x & ex lc being LinearCombination of F st Sum lc = x'; thus thesis by A24; end; consider x being set such that A25: x in F by XBOOLE_0:def 1; reconsider I as non empty Subset of L by A19,A23,A25; reconsider I'=I as non empty Subset of L; A26: I' is add-closed proof let x, y be Element of L; assume A27: x in I' & y in I'; then consider x' being Element of L such that A28: x'=x & ex lc being LinearCombination of F st Sum lc = x'; consider y' being Element of L such that A29: y'=y & ex lc being LinearCombination of F st Sum lc = y' by A27; consider lcx being LinearCombination of F such that A30: Sum lcx = x' by A28; consider lcy being LinearCombination of F such that A31: Sum lcy = y' by A29; Sum (lcx^lcy) = x' + y' by A30,A31,RLVECT_1:58; hence x+y in I' by A28,A29; end; A32: I' is left-ideal proof let p, x be Element of L; assume x in I'; then consider x' being Element of L such that A33: x'=x & ex lc being LinearCombination of F st Sum lc = x'; consider lcx being LinearCombination of F such that A34: Sum lcx = x' by A33; reconsider plcx = p*lcx as LinearCombination of F by Th23; p*x = Sum plcx by A33,A34,BINOM:4; hence p*x in I'; end; I' is right-ideal proof let p, x be Element of L; assume x in I'; then consider x' being Element of L such that A35: x'=x & ex lc being LinearCombination of F st Sum lc = x'; consider lcx being LinearCombination of F such that A36: Sum lcx = x' by A35; reconsider lcxp = lcx*p as LinearCombination of F by Th24; x*p = Sum lcxp by A35,A36,BINOM:5; hence x*p in I'; end; then F-Ideal c= I by A19,A26,A32,Def15; then A37: I = F-Ideal by A1,XBOOLE_0:def 10; let x be set; hereby assume x in F-Ideal; then consider x' being Element of L such that A38: x'=x & ex lc being LinearCombination of F st Sum lc = x' by A37; thus ex f being LinearCombination of F st x = Sum f by A38; end; assume ex f being LinearCombination of F st x = Sum f; hence x in F-Ideal by A37; end; theorem Th61: for L being add-associative left_zeroed right_zeroed add-cancelable associative distributive well-unital (non empty doubleLoopStr), F being non empty Subset of L, x being set holds x in F-LeftIdeal iff ex f being LeftLinearCombination of F st x = Sum f proof let L be add-associative right_zeroed add-cancelable left_zeroed associative distributive well-unital (non empty doubleLoopStr), F be non empty Subset of L; set I = { x where x is Element of L: ex lc being LeftLinearCombination of F st Sum lc = x }; A1: I c= F-LeftIdeal proof let x be set; defpred P[Nat] means for lc being LeftLinearCombination of F st len lc <= $1 holds Sum lc in F-LeftIdeal; A2: P[0] proof let lc be LeftLinearCombination of F; assume len lc <= 0; then len lc = 0 by NAT_1:18; then lc = <*>(the carrier of L) by FINSEQ_1:25; then A3: Sum lc = 0.L by RLVECT_1:60; consider y being Element of F; F c= F-LeftIdeal by Def16; then A4: y in F-LeftIdeal by TARSKI:def 3; 0.L*y = 0.L by BINOM:1; hence thesis by A3,A4,Def2; end; A5: for k being Nat st P[k] holds P[k+1] proof let k be Nat; assume A6: P[k]; thus P[k+1] proof let lc be LeftLinearCombination of F; assume A7: len lc <= k+1; per cases by A7,NAT_1:26; suppose len lc <= k; hence thesis by A6; suppose A8: len lc = k+1; then lc is non empty by FINSEQ_1:25; then consider q being LeftLinearCombination of F, r being Element of L such that A9: lc=q^<*r*> & <*r*> is LeftLinearCombination of F by Th33; k+1 = len q + len <*r*> by A8,A9,FINSEQ_1:35 .= len q + 1 by FINSEQ_1:56; then len q = k by XCMPLX_1:2; then A10: Sum q in F-LeftIdeal by A6; dom <*r*> = {1} by FINSEQ_1:4,55; then A11: 1 in dom <*r*> by TARSKI:def 1; then consider u being Element of L, a being Element of F such that A12: <* r *>/.1 = u*a by A9,Def10; A13: <*r*>/.1 = <*r*>.1 by A11,FINSEQ_4:def 4; A14: Sum <* r *> = r by BINOM:3 .= u*a by A12,A13,FINSEQ_1:57; F c= F-LeftIdeal by Def16; then a in F-LeftIdeal by TARSKI:def 3; then A15: Sum <* r *> in F-LeftIdeal by A14,Def2; Sum lc = Sum q + Sum <* r *> by A9,RLVECT_1:58; hence thesis by A10,A15,Def1; end; end; A16: for k being Nat holds P[k] from Ind(A2,A5); assume x in I; then consider x' being Element of L such that A17: x'=x & ex lc being LeftLinearCombination of F st Sum lc = x'; consider lc being LeftLinearCombination of F such that A18: Sum lc = x' by A17; P[len lc] by A16; hence x in F-LeftIdeal by A17,A18; end; A19: F c= I proof let x be set; assume A20: x in F; then reconsider x as Element of L; set lc = <* x *>; now let i be set; assume A21: i in dom lc; then A22: lc/.i = lc.i by FINSEQ_4:def 4; dom lc = {1} by FINSEQ_1:4,55; then i = 1 by A21,TARSKI:def 1; then lc.i = x by FINSEQ_1:57 .= 1_ L*x by VECTSP_2:def 2; hence ex u being Element of L, a being Element of F st lc/.i = u*a by A20,A22; end; then reconsider lc as LeftLinearCombination of F by Def10; Sum lc = x by BINOM:3; hence thesis; end; A23: I c= the carrier of L proof let x be set; assume x in I; then consider x' being Element of L such that A24: x'=x & ex lc being LeftLinearCombination of F st Sum lc = x'; thus thesis by A24; end; consider x being set such that A25: x in F by XBOOLE_0:def 1; reconsider I as non empty Subset of L by A19,A23,A25; reconsider I'=I as non empty Subset of L; A26: I' is add-closed proof let x, y be Element of L; assume A27: x in I' & y in I'; then consider x' being Element of L such that A28: x'=x & ex lc being LeftLinearCombination of F st Sum lc = x'; consider y' being Element of L such that A29: y'=y & ex lc being LeftLinearCombination of F st Sum lc = y' by A27; consider lcx being LeftLinearCombination of F such that A30: Sum lcx = x' by A28; consider lcy being LeftLinearCombination of F such that A31: Sum lcy = y' by A29; Sum (lcx^lcy) = x' + y' by A30,A31,RLVECT_1:58; hence x+y in I' by A28,A29; end; I' is left-ideal proof let p, x be Element of L; assume x in I'; then consider x' being Element of L such that A32: x'=x & ex lc being LeftLinearCombination of F st Sum lc = x'; consider lcx being LeftLinearCombination of F such that A33: Sum lcx = x' by A32; reconsider plcx = p*lcx as LeftLinearCombination of F by Th26; p*x = Sum plcx by A32,A33,BINOM:4; hence p*x in I'; end; then F-LeftIdeal c= I by A19,A26,Def16; then A34: I = F-LeftIdeal by A1,XBOOLE_0:def 10; let x be set; hereby assume x in F-LeftIdeal; then consider x' being Element of L such that A35: x'=x & ex lc being LeftLinearCombination of F st Sum lc = x' by A34; thus ex f being LeftLinearCombination of F st x = Sum f by A35; end; assume ex f being LeftLinearCombination of F st x = Sum f; hence x in F-LeftIdeal by A34; end; theorem Th62: for L being add-associative left_zeroed right_zeroed add-cancelable associative distributive well-unital (non empty doubleLoopStr), F being non empty Subset of L, x being set holds x in F-RightIdeal iff ex f being RightLinearCombination of F st x = Sum f proof let L be add-associative left_zeroed right_zeroed add-cancelable associative distributive well-unital (non empty doubleLoopStr), F be non empty Subset of L; set I = { x where x is Element of L: ex lc being RightLinearCombination of F st Sum lc = x }; A1: I c= F-RightIdeal proof let x be set; defpred P[Nat] means for lc being RightLinearCombination of F st len lc <= $1 holds Sum lc in F-RightIdeal; A2: P[0] proof let lc be RightLinearCombination of F; assume len lc <= 0; then len lc = 0 by NAT_1:18; then lc = <*>(the carrier of L) by FINSEQ_1:25; then A3: Sum lc = 0.L by RLVECT_1:60; consider y being Element of F; F c= F-RightIdeal by Def17; then A4: y in F-RightIdeal by TARSKI:def 3; y*0.L = 0.L by BINOM:2; hence thesis by A3,A4,Def3; end; A5: for k being Nat st P[k] holds P[k+1] proof let k be Nat; assume A6: P[k]; thus P[k+1] proof let lc be RightLinearCombination of F; assume A7: len lc <= k+1; per cases by A7,NAT_1:26; suppose len lc <= k; hence thesis by A6; suppose A8: len lc = k+1; then lc is non empty by FINSEQ_1:25; then consider q being RightLinearCombination of F, r being Element of L such that A9: lc=q^<*r*> & <*r*> is RightLinearCombination of F by Th34; k+1 = len q + len <*r*> by A8,A9,FINSEQ_1:35 .= len q + 1 by FINSEQ_1:56; then len q = k by XCMPLX_1:2; then A10: Sum q in F-RightIdeal by A6; dom <*r*> = {1} by FINSEQ_1:4,55; then A11: 1 in dom <*r*> by TARSKI:def 1; then consider v being Element of L, a being Element of F such that A12: <* r *>/.1 = a*v by A9,Def11; A13: <*r*>/.1 = <*r*>.1 by A11,FINSEQ_4:def 4; A14: Sum <* r *> = r by BINOM:3 .= a*v by A12,A13,FINSEQ_1:57; F c= F-RightIdeal by Def17; then a in F-RightIdeal by TARSKI:def 3; then A15: Sum <* r *> in F-RightIdeal by A14,Def3; Sum lc = Sum q + Sum <* r *> by A9,RLVECT_1:58; hence thesis by A10,A15,Def1; end; end; A16: for k being Nat holds P[k] from Ind(A2,A5); assume x in I; then consider x' being Element of L such that A17: x'=x & ex lc being RightLinearCombination of F st Sum lc = x'; consider lc being RightLinearCombination of F such that A18: Sum lc = x' by A17; P[len lc] by A16; hence x in F-RightIdeal by A17,A18; end; A19: F c= I proof let x be set; assume A20: x in F; then reconsider x as Element of L; set lc = <* x *>; now let i be set; assume A21: i in dom lc; then A22: lc/.i = lc.i by FINSEQ_4:def 4; dom lc = {1} by FINSEQ_1:4,55; then i = 1 by A21,TARSKI:def 1; then lc.i = x by FINSEQ_1:57 .= x*1_ L by VECTSP_2:def 2; hence ex v being Element of L, a being Element of F st lc/.i = a*v by A20,A22; end; then reconsider lc as RightLinearCombination of F by Def11; Sum lc = x by BINOM:3; hence thesis; end; A23: I c= the carrier of L proof let x be set; assume x in I; then consider x' being Element of L such that A24: x'=x & ex lc being RightLinearCombination of F st Sum lc = x'; thus thesis by A24; end; consider x being set such that A25: x in F by XBOOLE_0:def 1; reconsider I as non empty Subset of L by A19,A23,A25; reconsider I'=I as non empty Subset of L; A26: I' is add-closed proof let x, y be Element of L; assume A27: x in I' & y in I'; then consider x' being Element of L such that A28: x'=x & ex lc being RightLinearCombination of F st Sum lc = x'; consider y' being Element of L such that A29: y'=y & ex lc being RightLinearCombination of F st Sum lc = y' by A27; consider lcx being RightLinearCombination of F such that A30: Sum lcx = x' by A28; consider lcy being RightLinearCombination of F such that A31: Sum lcy = y' by A29; Sum (lcx^lcy) = x' + y' by A30,A31,RLVECT_1:58; hence x+y in I' by A28,A29; end; I' is right-ideal proof let p, x be Element of L; assume x in I'; then consider x' being Element of L such that A32: x'=x & ex lc being RightLinearCombination of F st Sum lc = x'; consider lcx being RightLinearCombination of F such that A33: Sum lcx = x' by A32; reconsider lcxp = lcx*p as RightLinearCombination of F by Th29; x*p = Sum lcxp by A32,A33,BINOM:5; hence x*p in I'; end; then F-RightIdeal c= I by A19,A26,Def17; then A34: I = F-RightIdeal by A1,XBOOLE_0:def 10; let x be set; hereby assume x in F-RightIdeal; then consider x' being Element of L such that A35: x'=x & ex lc being RightLinearCombination of F st Sum lc = x' by A34; thus ex f being RightLinearCombination of F st x = Sum f by A35; end; assume ex f being RightLinearCombination of F st x = Sum f; hence x in F-RightIdeal by A34; end; theorem Th63: for R being add-associative left_zeroed right_zeroed add-cancelable well-unital associative commutative distributive (non empty doubleLoopStr), F being non empty Subset of R holds F-Ideal = F-LeftIdeal & F-Ideal = F-RightIdeal proof let R be add-associative left_zeroed right_zeroed add-cancelable well-unital associative commutative distributive (non empty doubleLoopStr), F be non empty Subset of R; now let x be set; hereby assume x in F-Ideal; then consider lc being LinearCombination of F such that A1: x = Sum lc by Th60; lc is LeftLinearCombination of F by Th31; hence x in F-LeftIdeal by A1,Th61; end; assume x in F-LeftIdeal; then consider lc being LeftLinearCombination of F such that A2: x = Sum lc by Th61; lc is LinearCombination of F by Th25; hence x in F-Ideal by A2,Th60; end; hence F-Ideal = F-LeftIdeal by TARSKI:2; now let x be set; hereby assume x in F-Ideal; then consider lc being LinearCombination of F such that A3: x = Sum lc by Th60; lc is RightLinearCombination of F by Th31; hence x in F-RightIdeal by A3,Th62; end; assume x in F-RightIdeal; then consider lc being RightLinearCombination of F such that A4: x = Sum lc by Th62; lc is LinearCombination of F by Th28; hence x in F-Ideal by A4,Th60; end; hence F-Ideal = F-RightIdeal by TARSKI:2; end; theorem Th64: for R being add-associative left_zeroed right_zeroed add-cancelable well-unital associative commutative distributive (non empty doubleLoopStr), a being Element of R holds {a}-Ideal = {a*r where r is Element of R : not contradiction} proof let R be add-associative left_zeroed right_zeroed add-cancelable well-unital commutative associative distributive (non empty doubleLoopStr), a be Element of R; set A = {a}; reconsider a as Element of A by TARSKI:def 1; set M = {Sum s where s is LinearCombination of A : not contradiction}; set N = {a*r where r is Element of R : not contradiction}; A1: for u being set holds u in M implies u in N proof let u be set; assume u in M; then consider s being LinearCombination of A such that A2: u = Sum s; A3: 0 <= len s by NAT_1:18; consider f being Function of NAT,the carrier of R such that A4: Sum s = f.(len s) and A5: f.0 = 0.R and A6: for j being Nat,v being Element of R st j < len s & v = s.(j + 1) holds f.(j + 1) = f.j + v by RLVECT_1:def 12; defpred P[Nat] means ex r being Element of R st f.($1) = a*r; f.0 = a*0.R by A5,BINOM:2; then A7: P[0]; A8: now let j be Nat; assume A9: 0 <= j & j < len s; thus P[j] implies P[j+1] proof assume ex r being Element of R st f.j = a*r; then consider r1 being Element of R such that A10: f.j = a*r1; A11: 0 + 1 <= j + 1 by A9,AXIOMS:24; j + 1 <= len s by A9,NAT_1:38; then j + 1 in Seg(len s) by A11,FINSEQ_1:3; then A12: j + 1 in dom s by FINSEQ_1:def 3; then s.(j+1) = s/.(j+1) by FINSEQ_4:def 4; then A13: f.(j+1) = f.j + s/.(j+1) by A6,A9; consider r2,r3 being Element of R, a' being Element of A such that A14: s/.(j+1) = r2*a'*r3 by A12,Def9; f.(j+1) = a*r1 + r2*a*r3 by A10,A13,A14,TARSKI:def 1 .= a*r1 + a*(r2*r3) by VECTSP_1:def 16 .= a*(r1 + r2*r3) by VECTSP_1:def 18; hence thesis; end; end; for k being Nat st 0 <= k & k <= len s holds P[k] from FinInd(A7,A8); then ex r being Element of R st Sum s = a*r by A3,A4; hence thesis by A2; end; for u being set holds u in N implies u in M proof let u be set; assume u in N; then consider r being Element of R such that A15: u = a*r; set s = <* a*r *>; for i being set st i in dom s ex r,t being Element of R, a being Element of A st s/.i = r*a*t proof let i be set; assume A16: i in dom s; len s = 1 by FINSEQ_1:57; then i in {1} by A16,FINSEQ_1:4,def 3; then i = 1 by TARSKI:def 1; then s/.i = a*r by FINSEQ_4:25 .= (r*a)*1_ R by VECTSP_1:def 13; hence thesis; end; then reconsider s as LinearCombination of A by Def9; Sum s = a*r by BINOM:3; hence u in M by A15; end; then A17: M = N by A1,TARSKI:2; now let x be set; hereby assume x in {a}-Ideal; then x in {a}-RightIdeal by Th63; then consider f being RightLinearCombination of A such that A18: x = Sum f by Th62; f is LinearCombination of A by Th28; hence x in M by A18; end; assume x in M; then consider s being LinearCombination of A such that A19: x = Sum s; thus x in {a}-Ideal by A19,Th60; end; hence thesis by A17,TARSKI:2; end; theorem Th65: for R being Abelian left_zeroed right_zeroed add-cancelable well-unital add-associative associative commutative distributive (non empty doubleLoopStr), a,b being Element of R holds {a,b}-Ideal = {a*r + b*s where r,s is Element of R : not contradiction} proof let R be Abelian left_zeroed right_zeroed add-cancelable associative well-unital add-associative commutative distributive (non empty doubleLoopStr), a,b be Element of R; set A = {a,b}; reconsider a,b as Element of A by TARSKI:def 2; set M = {Sum s where s is LinearCombination of A : not contradiction}; set N = {a*r + b*s where r,s is Element of R : not contradiction}; A1: for u being set holds u in M implies u in N proof let u be set; assume u in M; then consider s being LinearCombination of A such that A2: u = Sum s; A3: 0 <= len s by NAT_1:18; consider f being Function of NAT,the carrier of R such that A4: Sum s = f.(len s) and A5: f.0 = 0.R and A6: for j being Nat,v being Element of R st j < len s & v = s.(j + 1) holds f.(j + 1) = f.j + v by RLVECT_1:def 12; defpred P[Nat] means ex r,s being Element of R st f.($1) = a*r + b*s; f.0 = a*0.R by A5,BINOM:2 .= a*0.R + 0.R by RLVECT_1:def 7 .= a*0.R + b*0.R by BINOM:2; then A7: P[0]; A8: now let j be Nat; assume A9: 0 <= j & j < len s; thus P[j] implies P[j+1] proof assume ex r,s being Element of R st f.j = a*r + b*s; then consider r1,s1 being Element of R such that A10: f.j = a*r1 + b*s1; A11: 0 + 1 <= j + 1 by A9,AXIOMS:24; j + 1 <= len s by A9,NAT_1:38; then j + 1 in Seg(len s) by A11,FINSEQ_1:3; then A12: j + 1 in dom s by FINSEQ_1:def 3; then consider r2,r3 being Element of R, a' being Element of A such that A13: s/.(j+1) = r2*a'*r3 by Def9; A14: s/.(j+1) = s.(j+1) by A12,FINSEQ_4:def 4; per cases by TARSKI:def 2; suppose a' = a; then f.(j+1) = (a*r1 + b*s1) + r2*a*r3 by A6,A9,A10,A13,A14 .= (a*r1 + a*r2*r3) + b*s1 by RLVECT_1:def 6 .= (a*r1 + a*(r2*r3)) + b*s1 by VECTSP_1:def 16 .= a*(r1 + r2*r3) + b*s1 by VECTSP_1:def 18; hence thesis; suppose a' = b; then f.(j+1) = (a*r1 + b*s1) + r2*b*r3 by A6,A9,A10,A13,A14 .= a*r1 + (b*s1 + b*r2*r3) by RLVECT_1:def 6 .= a*r1 + (b*s1 + b*(r2*r3)) by VECTSP_1:def 16 .= a*r1 + b*(s1 + r2*r3) by VECTSP_1:def 18; hence thesis; end; end; for k being Nat st 0 <= k & k <= len s holds P[k] from FinInd(A7,A8); then ex r,t being Element of R st Sum s = a*r + b*t by A3,A4; hence thesis by A2; end; for u being set holds u in N implies u in M proof let u be set; assume u in N; then consider r,t being Element of R such that A15: u = a*r + b*t; set s = <* a*r, b*t *>; for i being set st i in dom s ex r,t being Element of R, a being Element of A st s/.i = r*a*t proof let i be set; assume A16: i in dom s; then i in Seg(len s) by FINSEQ_1:def 3; then A17: i in {1,2} by FINSEQ_1:4,61; per cases by A17,TARSKI:def 2; suppose i = 1; then s/.i = s.1 by A16,FINSEQ_4:def 4 .= a*r by FINSEQ_1:61 .= 1_ R*a*r by VECTSP_1:def 19; hence thesis; suppose i = 2; then s/.i = s.2 by A16,FINSEQ_4:def 4 .= b*t by FINSEQ_1:61 .= 1_ R*b*t by VECTSP_1:def 19; hence thesis; end; then reconsider s as LinearCombination of A by Def9; Sum s = a*r + b*t by Th1; hence u in M by A15; end; then A18: M = N by A1,TARSKI:2; now let x be set; hereby assume x in {a,b}-Ideal; then x in {a,b}-RightIdeal by Th63; then consider f being RightLinearCombination of A such that A19: x = Sum f by Th62; f is LinearCombination of A by Th28; hence x in M by A19; end; assume x in M; then consider s being LinearCombination of A such that A20: x = Sum s; thus x in {a,b}-Ideal by A20,Th60; end; hence thesis by A18,TARSKI:2; end; theorem Th66: for R being non empty doubleLoopStr, a being Element of R holds a in {a}-Ideal proof let R be non empty doubleLoopStr, a be Element of R; a in {a} & {a} c= {a}-Ideal by Def15,TARSKI:def 1; hence thesis; end; theorem for R being Abelian left_zeroed right_zeroed right_complementable add-associative associative commutative distributive well-unital (non empty doubleLoopStr), A being non empty Subset of R, a being Element of R holds a in A-Ideal implies {a}-Ideal c= A-Ideal proof let R be left_zeroed right_zeroed right_complementable add-associative associative distributive well-unital commutative Abelian (non empty doubleLoopStr), A be non empty Subset of R, a be Element of R; assume a in A-Ideal; then consider s being LinearCombination of A such that A1: a = Sum s by Th60; now let u be set; assume u in {a}-Ideal; then u in {a*r where r is Element of R : not contradiction} by Th64; then consider r being Element of R such that A2: u = a*r; set t = s*r; A3: dom s = dom t by POLYNOM1:def 3; for i being set st i in dom t ex u,v being Element of R, a being Element of A st t/.i = u*a*v proof let i be set; assume A4: i in dom t; then consider u,v being Element of R, b being Element of A such that A5: s/.i = u*b*v by A3,Def9; t/.i = (u*b*v)*r by A3,A4,A5,POLYNOM1:def 3 .= u*b*(v*r) by VECTSP_1:def 16; hence thesis; end; then A6: t is LinearCombination of A by Def9; Sum t = u by A1,A2,BINOM:5; hence u in A-Ideal by A6,Th60; end; hence thesis by TARSKI:def 3; end; Lm2: for a,b being set holds {a} c= {a,b} proof let a,b be set; now let u be set; assume u in {a}; then u = a by TARSKI:def 1; hence u in {a,b} by TARSKI:def 2; end; hence thesis by TARSKI:def 3; end; theorem for R being non empty doubleLoopStr, a,b being Element of R holds a in {a,b}-Ideal & b in {a,b}-Ideal proof let R be non empty doubleLoopStr, a,b be Element of R; {a} c= {a,b} by Lm2; then A1: {a}-Ideal c= {a,b}-Ideal by Th57; a in {a}-Ideal by Th66; hence a in {a,b}-Ideal by A1; {b} c= {a,b} by Lm2; then A2: {b}-Ideal c= {a,b}-Ideal by Th57; b in {b}-Ideal by Th66; hence b in {a,b}-Ideal by A2; end; theorem for R being non empty doubleLoopStr, a,b being Element of R holds {a}-Ideal c= {a,b}-Ideal & {b}-Ideal c= {a,b}-Ideal proof let R be non empty doubleLoopStr, a,b be Element of R; {a} c= {a,b} & {b} c= {a,b} by Lm2; hence thesis by Th57; end; begin :: Some Operations on Ideals definition let R be non empty HGrStr, I be Subset of R, a be Element of R; func a*I -> Subset of R equals :Def19: {a*i where i is Element of R : i in I}; coherence proof set M = {a*i where i is Element of R : i in I}; M is Subset of R proof per cases; suppose A1: I is empty; M is empty proof assume M is non empty; then reconsider M as non empty set; consider b being Element of M; b in {a*i where i is Element of R : i in I}; then consider i being Element of R such that A2: b = a*i & i in I; thus thesis by A1,A2; end; then for u being set holds u in M implies u in the carrier of R; hence thesis by TARSKI:def 3; suppose I is non empty; then reconsider I as non empty set; consider j' being Element of I; j' in I; then reconsider j = j' as Element of R; a*j in {a*i where i is Element of R : i in I}; then reconsider M as non empty set; for x being set holds x in M implies x in the carrier of R proof let x be set; assume x in M; then consider i being Element of R such that A3: x = a*i & i in I; thus thesis by A3; end; hence thesis by TARSKI:def 3; end; hence thesis; end; end; definition let R be non empty multLoopStr, I be non empty Subset of R, a be Element of R; cluster a*I -> non empty; coherence proof consider j being Element of I; a*j in {a*i where i is Element of R : i in I} ; hence thesis by Def19; end; end; definition let R be distributive (non empty doubleLoopStr), I be add-closed Subset of R, a be Element of R; cluster a*I -> add-closed; coherence proof set M = {a*i where i is Element of R : i in I}; A1: M = a*I by Def19; for x,y being Element of R st x in M & y in M holds x+y in M proof let x,y be Element of R; assume A2: x in M & y in M; then consider i being Element of R such that A3: x = a*i & i in I; consider j being Element of R such that A4: y = a*j & j in I by A2; reconsider k = i + j as Element of R; A5: k in I by A3,A4,Def1; x + y = a*k by A3,A4,VECTSP_1:def 18; hence thesis by A5; end; hence thesis by A1,Def1; end; end; definition let R be associative (non empty doubleLoopStr), I be right-ideal Subset of R, a be Element of R; cluster a*I -> right-ideal; coherence proof set M = {a*i where i is Element of R : i in I}; A1: M = a*I by Def19; for y,x being Element of R st x in M holds x*y in M proof let y,x be Element of R; assume x in M; then consider i being Element of R such that A2: x = a*i & i in I; A3: x*y = a*(i*y) by A2,VECTSP_1:def 16; i*y in I by A2,Def3; hence thesis by A3; end; hence thesis by A1,Def3; end; end; theorem Th70: for R being right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), I being non empty Subset of R holds 0.R*I = {0.R} proof let R be right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), I be non empty Subset of R; A1: now let u be set; assume u in {0.R}; then A2: u = 0.R by TARSKI:def 1; consider j being Element of I; 0.R*j = 0.R by BINOM:1; then 0.R in {0.R*i where i is Element of R : i in I}; hence u in 0.R*I by A2,Def19; end; now let u be set; assume u in 0.R*I; then u in {0.R*i where i is Element of R : i in I} by Def19; then consider i being Element of R such that A3: u = 0.R*i & i in I; u = 0.R by A3,BINOM:1; hence u in {0.R} by TARSKI:def 1; end; hence 0.R*I = {0.R} by A1,TARSKI:2; end; theorem for R being left_unital (non empty doubleLoopStr), I being Subset of R holds 1_ R*I = I proof let R be left_unital (non empty doubleLoopStr), I be Subset of R; A1: now let u be set; assume A2: u in I; then reconsider u' = u as Element of R; 1_ R*u' = u' by VECTSP_1:def 19; then u' in {1_ R*i where i is Element of R : i in I} by A2; hence u in 1_ R*I by Def19; end; now let u be set; assume u in 1_ R*I; then u in {1_ R*i where i is Element of R : i in I } by Def19; then ex i being Element of R st u = 1_ R*i & i in I; hence u in I by VECTSP_1:def 19; end; hence thesis by A1,TARSKI:2; end; definition let R be non empty LoopStr, I,J be Subset of R; func I + J -> Subset of R equals :Def20: {a + b where a,b is Element of R : a in I & b in J}; coherence proof set M = {a + b where a,b is Element of R : a in I & b in J}; M is Subset of R proof per cases; suppose A1: (I is empty) or (J is empty); now per cases by A1; case A2: I is empty; M is empty proof assume M is non empty; then reconsider M as non empty set; consider x being Element of M; x in {a + b where a,b is Element of R : a in I & b in J}; then consider a,b being Element of R such that A3: x = a + b & a in I & b in J; thus thesis by A2,A3; end; then for u being set holds u in M implies u in the carrier of R; hence thesis by TARSKI:def 3; case A4: J is empty; M is empty proof assume M is non empty; then reconsider M as non empty set; consider x being Element of M; x in {a + b where a,b is Element of R : a in I & b in J}; then consider a,b being Element of R such that A5: x = a + b & a in I & b in J; thus thesis by A4,A5; end; then for u being set holds u in M implies u in the carrier of R; hence thesis by TARSKI:def 3; end; hence thesis; suppose A6: I is non empty & J is non empty; then reconsider I as non empty set; reconsider J as non empty set by A6; consider x' being Element of I; consider y' being Element of J; x' in I & y' in J; then reconsider x = x',y = y' as Element of R; x+y in {a + b where a,b is Element of R : a in I & b in J}; then reconsider M as non empty set; for x being set holds x in M implies x in the carrier of R proof let x be set; assume x in M; then consider a,b being Element of R such that A7: x = a + b & a in I & b in J; thus thesis by A7; end; hence thesis by TARSKI:def 3; end; hence thesis; end; end; definition let R be non empty LoopStr, I,J be non empty Subset of R; cluster I + J -> non empty; coherence proof {x + y where x,y is Element of R : x in I & y in J} is non empty proof consider x being Element of I; consider y being Element of J; x+y in {a + b where a,b is Element of R : a in I & b in J}; hence thesis; end; hence thesis by Def20; end; end; definition let R be Abelian (non empty LoopStr), I,J be Subset of R; redefine func I + J; commutativity proof now let I,J be Subset of R; A1: I+J = {a + b where a,b is Element of R : a in I & b in J} by Def20; A2: J+I = {a + b where a,b is Element of R : a in J & b in I} by Def20; A3: now let u be set; assume u in I+J; then consider a,b being Element of R such that A4: u = a + b & a in I & b in J by A1; thus u in J+I by A2,A4; end; now let u be set; assume u in J+I; then consider a,b being Element of R such that A5: u = a + b & a in J & b in I by A2; thus u in I+J by A1,A5; end; hence I + J = J + I by A3,TARSKI:2; end; hence thesis; end; end; definition let R be Abelian add-associative (non empty LoopStr), I,J be add-closed Subset of R; cluster I + J -> add-closed; coherence proof set M = {a + b where a,b is Element of R : a in I & b in J}; A1: M = I + J by Def20; for x,y being Element of R st x in M & y in M holds x+y in M proof let x,y be Element of R; assume A2: x in M & y in M; then consider a',b' being Element of R such that A3: x = a' + b' & a' in I & b' in J; consider c,d being Element of R such that A4: y = c + d & c in I & d in J by A2; A5: a' + c in I & b' + d in J by A3,A4,Def1; (a' + c) + (b' + d) = ((a' + c) + b') + d by RLVECT_1:def 6 .= (c + x) + d by A3,RLVECT_1:def 6 .= x + y by A4,RLVECT_1:def 6; hence thesis by A5; end; hence thesis by A1,Def1; end; end; definition let R be left-distributive (non empty doubleLoopStr), I,J be right-ideal Subset of R; cluster I + J -> right-ideal; coherence proof set M = {a + b where a,b is Element of R : a in I & b in J}; A1: M = I + J by Def20; for y,x being Element of R st x in M holds x*y in M proof let y,x be Element of R; assume x in M; then consider a',b' being Element of R such that A2: x = a' + b' & a' in I & b' in J; A3: a'*y in I & b'*y in J by A2,Def3; (a'*y) + (b'*y) = x*y by A2,VECTSP_1:def 12; hence thesis by A3; end; hence thesis by A1,Def3; end; end; definition let R be right-distributive (non empty doubleLoopStr), I,J be left-ideal Subset of R; cluster I + J -> left-ideal; coherence proof set M = {a + b where a,b is Element of R : a in I & b in J}; A1: M = I + J by Def20; for y,x being Element of R st x in M holds y*x in M proof let y,x be Element of R; assume x in M; then consider a',b' being Element of R such that A2: x = a' + b' & a' in I & b' in J; A3: y*a' in I & y*b' in J by A2,Def2; (y*a') + (y*b') = y*x by A2,VECTSP_1:def 11; hence thesis by A3; end; hence thesis by A1,Def2; end; end; theorem for R being add-associative (non empty LoopStr), I,J,K being Subset of R holds I + (J + K) = (I + J) + K proof let R be add-associative (non empty LoopStr), I,J,K be Subset of R; A1: now let u be set; assume u in I + (J + K); then u in {a + b where a,b is Element of R : a in I & b in J+K} by Def20; then consider a,b being Element of R such that A2: u = a + b & a in I & b in J+K; b in {a' + b' where a',b' is Element of R : a' in J & b' in K} by A2,Def20 ; then consider c,d being Element of R such that A3: b = c + d & c in J & d in K; a + c in {a' + b' where a',b' is Element of R : a' in I & b' in J} by A2, A3 ; then a + c in I + J by Def20; then (a+ c) + d in {a' + b' where a',b' is Element of R : a' in I+J & b' in K}by A3; then (a + c) + d in (I + J) + K by Def20; hence u in (I + J) + K by A2,A3,RLVECT_1:def 6; end; now let u be set; assume u in (I + J) + K; then u in {a + b where a,b is Element of R : a in I+J & b in K} by Def20; then consider a,b being Element of R such that A4: u = a + b & a in I+J & b in K; a in {a' + b' where a',b' is Element of R : a' in I & b' in J} by A4,Def20 ; then consider c,d being Element of R such that A5: a = c + d & c in I & d in J; d + b in {a' + b' where a',b' is Element of R : a' in J & b' in K} by A4, A5 ; then d + b in J + K by Def20; then c+(d + b) in {a' + b' where a',b' is Element of R : a' in I & b' in J+K} by A5; then c + (d + b) in I + (J + K) by Def20; hence u in I + (J + K) by A4,A5,RLVECT_1:def 6; end; hence thesis by A1,TARSKI:2; end; theorem Th73: for R being left_zeroed right_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr), I,J being right-ideal (non empty Subset of R) holds I c= I + J proof let R be left_zeroed add-right-cancelable right_zeroed right-distributive (non empty doubleLoopStr), I,J be right-ideal (non empty Subset of R); now let u be set; assume u in I; then reconsider u' = u as Element of I; 0.R is Element of J by Th3; then A1: u' + 0.R in {a + b where a,b is Element of R : a in I & b in J}; u' + 0.R = u' by RLVECT_1:def 7; hence u in I + J by A1,Def20; end; hence I c= I + J by TARSKI:def 3; end; theorem Th74: for R being left_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr), I,J being right-ideal (non empty Subset of R) holds J c= I + J proof let R be left_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr), I,J be right-ideal (non empty Subset of R); now let u be set; assume u in J; then reconsider u' = u as Element of J; 0.R is Element of I by Th3; then A1: 0.R + u' in {a + b where a,b is Element of R : a in I & b in J}; 0.R + u' = u' by ALGSTR_1:def 5; hence u in I + J by A1,Def20; end; hence thesis by TARSKI:def 3; end; theorem for R being non empty LoopStr, I,J being Subset of R, K being add-closed Subset of R st I c= K & J c= K holds I + J c= K proof let R be (non empty LoopStr), I,J be Subset of R, K being add-closed ( Subset of R); assume A1: I c= K & J c= K; now let u be set; assume u in I + J; then u in {x + y where x,y is Element of R : x in I & y in J} by Def20; then consider i,j being Element of R such that A2: u = i + j & i in I & j in J; thus u in K by A1,A2,Def1; end; hence thesis by TARSKI:def 3; end; theorem for R being Abelian left_zeroed right_zeroed add-cancelable well-unital add-associative associative commutative distributive (non empty doubleLoopStr), a,b being Element of R holds {a,b}-Ideal = {a}-Ideal + {b}-Ideal proof let R be Abelian left_zeroed right_zeroed add-cancelable well-unital add-associative associative commutative distributive (non empty doubleLoopStr), a,b be Element of R; A1: now let u be set; assume u in {a,b}-Ideal; then u in {a*r + b*s where r,s is Element of R : not contradiction} by Th65 ; then consider r,s being Element of R such that A2: u = a*r + b*s; a*r in {a*v where v is Element of R : not contradiction } ; then reconsider a' = a*r as Element of {a}-Ideal by Th64; b*s in {b*v where v is Element of R : not contradiction}; then reconsider b' = b*s as Element of {b}-Ideal by Th64; a' + b' in {x + y where x,y is Element of R : x in {a}-Ideal & y in {b}-Ideal}; hence u in {a}-Ideal + {b}-Ideal by A2,Def20; end; now let u be set; assume u in {a}-Ideal + {b}-Ideal; then u in {x + y where x,y is Element of R : x in {a}-Ideal & y in {b}-Ideal} by Def20; then consider x,y being Element of R such that A3: u = x + y & x in {a}-Ideal & y in {b}-Ideal; x in {a*v where v is Element of R : not contradiction} by A3,Th64; then consider r being Element of R such that A4: x = a*r; y in {b*v where v is Element of R : not contradiction} by A3,Th64; then consider s being Element of R such that A5: y = b*s; u in {a*v + b*d where v,d is Element of R : not contradiction} by A3,A4,A5 ; hence u in {a,b}-Ideal by Th65; end; hence thesis by A1,TARSKI:2; end; definition let R be non empty 1-sorted, I,J be Subset of R; func I /\ J -> Subset of R equals :Def21: { x where x is Element of R : x in I & x in J }; coherence proof set M = { x where x is Element of R : x in I & x in J }; now let u be set; assume u in M; then consider x being Element of R such that A1: u = x & x in I & x in J; thus u in the carrier of R by A1; end; then reconsider M as Subset of R by TARSKI:def 3; M is Subset of R; hence thesis; end; end; definition let R be right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), I,J be left-ideal (non empty Subset of R); cluster I /\ J -> non empty; coherence proof 0.R in I & 0.R in J by Th2; then 0.R in { x where x is Element of R : x in I & x in J }; hence thesis by Def21; end; end; definition let R be non empty LoopStr, I,J be add-closed Subset of R; cluster I /\ J -> add-closed; coherence proof set M = { x where x is Element of R : x in I & x in J }; A1: M = I /\ J by Def21; then reconsider M as Subset of R; for x,y being Element of R st x in M & y in M holds x+y in M proof let x,y be Element of R; assume A2: x in M & y in M; then consider a being Element of R such that A3: x = a & a in I & a in J; consider c being Element of R such that A4: c = y & c in I & c in J by A2; a + c in I & a + c in J by A3,A4,Def1; hence thesis by A3,A4; end; hence thesis by A1,Def1; end; end; definition let R be non empty multLoopStr, I,J be left-ideal Subset of R; cluster I /\ J -> left-ideal; coherence proof set M = { x where x is Element of R : x in I & x in J }; A1: M = I /\ J by Def21; then reconsider M as Subset of R; for y,x being Element of R st x in M holds y*x in M proof let y,x be Element of R; assume x in M; then consider a being Element of R such that A2: x = a & a in I & a in J; y*a in I & y*a in J by A2,Def2; hence thesis by A2; end; hence thesis by A1,Def2; end; end; definition let R be non empty multLoopStr, I,J be right-ideal Subset of R; cluster I /\ J -> right-ideal; coherence proof set M = { x where x is Element of R : x in I & x in J }; A1: M = I /\ J by Def21; then reconsider M as Subset of R; for y,x being Element of R st x in M holds x*y in M proof let y,x be Element of R; assume x in M; then consider a being Element of R such that A2: x = a & a in I & a in J; a*y in I & a*y in J by A2,Def3; hence thesis by A2; end; hence thesis by A1,Def3; end; end; theorem Th77: for R being non empty 1-sorted, I,J being Subset of R holds I /\ J c= I & I /\ J c= J proof let R be non empty 1-sorted, I,J be Subset of R; now let u be set; assume u in I /\ J; then u in {x where x is Element of R : x in I & x in J} by Def21; then consider a being Element of R such that A1: u = a & a in I & a in J; thus u in I by A1; end; hence I /\ J c= I by TARSKI:def 3; now let u be set; assume u in I /\ J; then u in {x where x is Element of R : x in I & x in J} by Def21; then consider a being Element of R such that A2: u = a & a in I & a in J; thus u in J by A2; end; hence I /\ J c= J by TARSKI:def 3; end; theorem for R being non empty 1-sorted, I,J,K being Subset of R holds I /\ (J /\ K) = (I /\ J) /\ K proof let R be non empty 1-sorted, I,J,K be Subset of R; A1: now let u be set; assume u in I /\ (J /\ K); then u in {x where x is Element of R : x in I & x in (J /\ K)} by Def21; then consider a being Element of R such that A2: u = a & a in I & a in J/\K; a in {x where x is Element of R : x in J & x in K} by A2,Def21; then consider b being Element of R such that A3: b = a & b in J & b in K; a in {x where x is Element of R : x in I & x in J} by A2,A3; then a in (I /\ J) by Def21; then a in {x where x is Element of R : x in (I /\ J) & x in K} by A3; hence u in (I /\ J) /\ K by A2,Def21; end; now let u be set; assume u in (I /\ J) /\ K; then u in {x where x is Element of R : x in (I /\ J) & x in K} by Def21; then consider a being Element of R such that A4: u = a & a in (I /\ J) & a in K; a in {x where x is Element of R : x in I & x in J} by A4,Def21; then consider b being Element of R such that A5: b = a & b in I & b in J; a in {x where x is Element of R : x in J & x in K} by A4,A5; then a in (J /\ K) by Def21; then a in {x where x is Element of R : x in I & x in (J /\ K)} by A5; hence u in I /\ (J /\ K) by A4,Def21; end; hence thesis by A1,TARSKI:2; end; theorem for R being non empty 1-sorted, I,J,K being Subset of R st K c= I & K c= J holds K c= I /\ J proof let R be non empty 1-sorted, I,J,K be Subset of R; assume A1: K c= I & K c= J; now let u be set; assume A2: u in K; then reconsider u' = u as Element of R; u' in {x where x is Element of R : x in I & x in J} by A1,A2; hence u in (I /\ J) by Def21; end; hence thesis by TARSKI:def 3; end; theorem for R being Abelian left_zeroed right_zeroed right_complementable left_unital add-associative left-distributive (non empty doubleLoopStr), I being add-closed left-ideal (non empty Subset of R), J being Subset of R, K being non empty Subset of R st J c= I holds I /\ (J + K) = J + (I /\ K) proof let R be Abelian left_zeroed right_zeroed right_complementable left_unital add-associative left-distributive (non empty doubleLoopStr), I be add-closed left-ideal (non empty Subset of R), J be Subset of R, K be non empty Subset of R; assume A1: J c= I; A2: now let u be set; assume u in I /\ (J + K); then u in {x where x is Element of R : x in I & x in J+K} by Def21; then consider v being Element of R such that A3: u = v & v in I & v in J + K; v in {x + y where x,y is Element of R : x in J & y in K} by A3,Def20; then consider j,k being Element of R such that A4: v = j + k & j in J & k in K; reconsider j' = j as Element of I by A1,A4; -j' in I by Th13; then (j' + k) + -j' in I by A3,A4,Def1; then (j' + -j') + k in I by RLVECT_1:def 6; then 0.R + k in I by RLVECT_1:16; then k in I by ALGSTR_1:def 5; then k in {x where x is Element of R : x in I & x in K} by A4; then k in (I /\ K) by Def21; then u in {x+y where x,y is Element of R : x in J & y in (I /\ K)} by A3,A4 ; hence u in J + (I /\ K) by Def20; end; now let u be set; assume u in J + (I /\ K); then u in {x + y where x,y is Element of R: x in J & y in (I /\ K)} by Def20; then consider j,ik being Element of R such that A5: u = j + ik & j in J & ik in (I /\ K); ik in {x where x is Element of R : x in I & x in K} by A5,Def21; then consider z being Element of R such that A6: z = ik & z in I & z in K; reconsider i' = ik as Element of I by A6; reconsider k' = ik as Element of K by A6; reconsider j' = j as Element of I by A1,A5; u = j' + i' by A5; then A7: u in I by Def1; u = j + k' by A5; then u in {x + y where x,y is Element of R: x in J & y in K} by A5; then u in J + K by Def20; then u in {x where x is Element of R : x in I & x in J+K} by A7; hence u in I /\ (J + K) by Def21; end; hence thesis by A2,TARSKI:2; end; definition let R be non empty doubleLoopStr, I,J be Subset of R; func I *' J -> Subset of R equals :Def22: { Sum s where s is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I & b in J}; coherence proof set M = {Sum s where s is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I & b in J}; now let u be set; assume u in M; then consider s being FinSequence of the carrier of R such that A1: u = Sum s & for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I & b in J; thus u in the carrier of R by A1; end; then reconsider M as Subset of R by TARSKI:def 3; M is Subset of R; hence thesis; end; end; definition let R be non empty doubleLoopStr, I,J be Subset of R; cluster I *' J -> non empty; coherence proof set M = {Sum s where s is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I & b in J}; M is non empty proof set p = <*>(the carrier of R); for i being Nat st 1 <= i & i <= len p ex a,b being Element of R st p.i = a*b & a in I & b in J proof let i be Nat; assume A1: 1 <= i & i <= len p; len p = 0 by FINSEQ_1:32; hence thesis by A1,AXIOMS:22; end; then Sum p in M; hence thesis; end; hence thesis by Def22; end; end; definition let R be commutative (non empty doubleLoopStr), I,J be Subset of R; redefine func I *' J; commutativity proof now let I,J be Subset of R; A1: I*'J = {Sum s where s is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I & b in J} by Def22; A2: J*'I = {Sum s where s is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in J & b in I} by Def22; A3: now let u be set; assume u in I*'J; then consider s being FinSequence of the carrier of R such that A4: u = Sum s & for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I & b in J by A1; for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in J & b in I proof let i be Nat; assume 1 <= i & i <= len s; then consider a,b being Element of R such that A5: s.i = a*b & a in I & b in J by A4; thus thesis by A5; end; hence u in J*'I by A2,A4; end; now let u be set; assume u in J*'I; then consider s being FinSequence of the carrier of R such that A6: u = Sum s & for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in J & b in I by A2; for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I & b in J proof let i be Nat; assume 1 <= i & i <= len s; then consider a,b being Element of R such that A7: s.i = a*b & a in J & b in I by A6; thus thesis by A7; end; hence u in I*'J by A1,A6; end; hence I *' J = J *' I by A3,TARSKI:2; end; hence thesis; end; end; definition let R be right_zeroed add-associative (non empty doubleLoopStr), I,J be Subset of R; cluster I *' J -> add-closed; coherence proof set M = {Sum s where s is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I & b in J}; A1: M = I *' J by Def22; then reconsider M as non empty Subset of R; for x,y being Element of R st x in M & y in M holds x+y in M proof let x,y be Element of R; assume A2: x in M & y in M; then consider s being FinSequence of the carrier of R such that A3: x = Sum s & for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I & b in J; consider t being FinSequence of the carrier of R such that A4: y = Sum t & for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in I & b in J by A2; set q = s^t; A5: Sum q = x + y by A3,A4,RLVECT_1:58; now let i be Nat; assume A6: 1 <= i & i <= len q; thus ex a,r being Element of R st q.i = a*r & a in I & r in J proof per cases; suppose A7: i <= len s; then i in Seg(len s) by A6,FINSEQ_1:3; then i in dom s by FINSEQ_1:def 3; then q.i = s.i by FINSEQ_1:def 7; hence thesis by A3,A6,A7; suppose A8: len s < i; then reconsider j = i - len s as Nat by INT_1:18; len s - len s < j by A8,REAL_1:54 ; then 0 < j by XCMPLX_1:14; then A9: 1 <= j by RLVECT_1:99; i <= len s + len t by A6,FINSEQ_1:35; then j <= (len s + len t) - len s by REAL_1:49; then j <= (len s + len t) + -len s by XCMPLX_0:def 8; then j <= len t + (len s + -len s) by XCMPLX_1:1; then j <= len t + (len s - len s) by XCMPLX_0:def 8; then A10: j <= len t + 0 by XCMPLX_1:14; t.j = q.i by A6,A8,FINSEQ_1:37; hence thesis by A4,A9,A10; end; end; hence thesis by A5; end; hence thesis by A1,Def1; end; end; definition let R be right_zeroed add-left-cancelable associative left-distributive (non empty doubleLoopStr), I,J be right-ideal Subset of R; cluster I *' J -> right-ideal; coherence proof set M = {Sum s where s is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I & b in J}; A1: M = I *' J by Def22; then reconsider M as non empty Subset of R; for y,x being Element of R st x in M holds x*y in M proof let y,x be Element of R; assume x in M; then consider s being FinSequence of the carrier of R such that A2: x = Sum s & for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I & b in J; set q = s*y; A3: Sum q = Sum s*y by BINOM:5; A4: Seg(len q) = dom q by FINSEQ_1:def 3 .= dom s by POLYNOM1:def 3 .= Seg(len s) by FINSEQ_1:def 3; then A5: len q = len s by FINSEQ_1:8; now let i be Nat; assume A6: 1 <= i & i <= len q; then consider c,r' being Element of R such that A7: s.i = c*r' & c in I & r' in J by A2,A5; i in Seg(len s) & i in Seg(len q) by A4,A6,FINSEQ_1:3; then A8: i in dom s & i in dom q by FINSEQ_1:def 3; then A9: s/.i = c*r' by A7,FINSEQ_4:def 4; A10: q.i = q/.i by A8,FINSEQ_4:def 4 .= (c*r')*y by A8,A9,POLYNOM1:def 3 .= c*(r'*y) by VECTSP_1:def 16; thus ex b,r being Element of R st q.i = b*r & b in I & r in J proof take c,r'*y; thus thesis by A7,A10,Def3; end; end; hence thesis by A2,A3; end; hence thesis by A1,Def3; end; end; definition let R be left_zeroed add-right-cancelable associative right-distributive (non empty doubleLoopStr), I,J be left-ideal Subset of R; cluster I *' J -> left-ideal; coherence proof set M = {Sum s where s is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I & b in J}; A1: M = I *' J by Def22; then reconsider M as non empty Subset of R; for y,x being Element of R st x in M holds y*x in M proof let y,x be Element of R; assume x in M; then consider s being FinSequence of the carrier of R such that A2: x = Sum s & for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I & b in J; set q = y*s; A3: Sum q = y*Sum s by BINOM:4; A4: Seg(len q) = dom q by FINSEQ_1:def 3 .= dom s by POLYNOM1:def 2 .= Seg(len s) by FINSEQ_1:def 3; then A5: len q = len s by FINSEQ_1:8; now let i be Nat; assume A6: 1 <= i & i <= len q; then consider c,r' being Element of R such that A7: s.i = c*r' & c in I & r' in J by A2,A5; i in Seg(len s) & i in Seg(len q) by A4,A6,FINSEQ_1:3; then A8: i in dom s & i in dom q by FINSEQ_1:def 3; then A9: s/.i = c*r' by A7,FINSEQ_4:def 4; A10: q.i = q/.i by A8,FINSEQ_4:def 4 .= y*(c*r') by A8,A9,POLYNOM1:def 2 .= (y*c)*r' by VECTSP_1:def 16; thus ex b,r being Element of R st q.i = b*r & b in I & r in J proof take y*c,r'; thus thesis by A7,A10,Def2; end; end; hence thesis by A2,A3; end; hence thesis by A1,Def2; end; end; theorem for R being left_zeroed right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), I being non empty Subset of R holds {0.R} *' I = {0.R} proof let R be left_zeroed right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), I be non empty Subset of R; A1: now let u be set; assume A2: u in {0.R}; consider a being Element of I; set q = <* 0.R*a *>; A3: Sum q = 0.R*a by BINOM:3 .= 0.R by BINOM:1 .= u by A2,TARSKI:def 1; A4: len q = 1 & q.1 = 0.R*a by FINSEQ_1:57; reconsider o = 0.R as Element of {0.R} by TARSKI:def 1; for i being Nat st 1 <= i & i <= len q holds ex b,r being Element of R st q.i = b*r & b in {0.R} & r in I proof let i be Nat; assume 1 <= i & i <= len q; then q.i = o*a by A4,AXIOMS:21; hence thesis; end; then u in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in {0.R} & b in I} by A3; hence u in {0.R} *' I by Def22; end; now let u be set; assume u in ({0.R}) *' I; then u in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in {0.R} & b in I} by Def22; then consider s being FinSequence of the carrier of R such that A5: Sum s = u & for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in {0.R} & b in I; now per cases; case len s = 0; then s = <*>(the carrier of R) by FINSEQ_1:32; hence Sum s = 0.R by RLVECT_1:60; case len s <> 0; then 1 <= len s by RLVECT_1:99; then 1 in Seg(len s) by FINSEQ_1:3; then A6: 1 in dom s by FINSEQ_1:def 3; A7: for i being Nat st i in dom s holds s/.i = 0.R proof let i be Nat; assume A8: i in dom s; then i in Seg(len s) by FINSEQ_1:def 3; then 1 <= i & i <= len s by FINSEQ_1:3; then consider a ,b being Element of R such that A9: s.i = a*b & a in {0.R} & b in I by A5; A10: s/.i = a*b by A8,A9,FINSEQ_4:def 4; a = 0.R by A9,TARSKI:def 1; hence thesis by A10,BINOM:1; end; then for i being Nat st i in dom s & i <> 1 holds s/.i = 0.R; hence Sum s = s/.1 by A6,POLYNOM2:5 .= 0.R by A6,A7; end; hence u in {0.R} by A5,TARSKI:def 1; end; hence ({0.R}) *' I = {0.R} by A1,TARSKI:2; end; theorem Th82: for R being left_zeroed right_zeroed add-cancelable distributive (non empty doubleLoopStr), I being add-closed right-ideal (non empty Subset of R), J being add-closed left-ideal (non empty Subset of R) holds I *' J c= I /\ J proof let R be left_zeroed right_zeroed add-cancelable distributive (non empty doubleLoopStr), I be add-closed right-ideal (non empty Subset of R), J be add-closed left-ideal (non empty Subset of R); now let u be set; assume u in I *' J; then u in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in I & b in J} by Def22; then consider s being FinSequence of the carrier of R such that A1: Sum s = u & for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I & b in J; consider f being Function of NAT,the carrier of R such that A2: Sum s = f.(len s) & f.0 = 0.R & for j being Nat, v being Element of R st j < len s & v = s.(j + 1) holds f.(j + 1) = f.j + v by RLVECT_1:def 12; defpred P[Nat] means f.$1 in I & f.$1 in J; A3: P[0] by A2,Th2,Th3; A4: now let j be Nat; assume A5: 0 <= j & j < len s; thus P[j] implies P[j+1] proof assume A6: f.j in I & f.j in J; A7: j + 1 <= len s by A5,NAT_1:38; A8: 0 + 1 <= j + 1 by A5,AXIOMS:24; then j + 1 in Seg(len s) by A7,FINSEQ_1:3; then j + 1 in dom s by FINSEQ_1:def 3; then A9: s.(j+1) = s/.(j+1) by FINSEQ_4:def 4; then A10: f.(j+1) = f.j + s/.(j+1) by A2,A5; consider a,b being Element of R such that A11: s.(j+1) = a*b & a in I & b in J by A1,A7,A8; s/.(j+1) in I & s/.(j+1) in J by A9,A11,Def2,Def3; hence thesis by A6,A10,Def1; end; end; A12: for j being Nat st 0 <= j & j <= len s holds P[j] from FinInd(A3,A4); 0 <= len s by NAT_1:18; then Sum s in I & Sum s in J by A2,A12; then Sum s in {z where z is Element of R : z in I & z in J}; hence u in I /\ J by A1,Def21; end; hence thesis by TARSKI:def 3; end; theorem Th83: for R being Abelian left_zeroed right_zeroed add-cancelable add-associative associative distributive (non empty doubleLoopStr), I,J,K being right-ideal (non empty Subset of R) holds I *' (J + K) = (I *' J) + (I *' K) proof let R be Abelian left_zeroed right_zeroed add-cancelable add-associative associative distributive (non empty doubleLoopStr), I,J,K be right-ideal (non empty Subset of R); A1: now let u be set; assume u in I *' (J + K); then u in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in I & b in J+K} by Def22; then consider s being FinSequence of the carrier of R such that A2: Sum s = u & for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I & b in J+K; consider f being Function of NAT,the carrier of R such that A3: Sum s = f.(len s) & f.0 = 0.R & for j being Nat,v being Element of R st j < len s & v = s.(j + 1) holds f.(j + 1) = f.j + v by RLVECT_1:def 12; defpred P[Nat] means ex x,y being Element of R st f.$1=x+y & x in I*'J & y in I*'K; A4: P[0] proof take 0.R,0.R; thus thesis by A3,Th3,RLVECT_1:def 7; end; A5: now let n be Nat; assume A6: 0 <= n & n < len s; thus P[n] implies P[n+1] proof assume ex x,y being Element of R st f.n = x+y & x in I*'J & y in I*'K; then consider x,y being Element of R such that A7: f.n = x+y & x in I*'J & y in I*'K; x in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in I & b in J} by A7,Def22 ; then consider q being FinSequence of the carrier of R such that A8: Sum q = x & for i being Nat st 1 <= i & i <= len q ex a,b being Element of R st q.i = a*b & a in I & b in J; y in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in I & b in K} by A7, Def22; then consider p being FinSequence of the carrier of R such that A9: Sum p = y & for i being Nat st 1 <= i & i <= len p ex a,b being Element of R st p.i = a*b & a in I & b in K; A10: 0 + 1 <= n + 1 by A6,AXIOMS:24; A11: n + 1 <= len s by A6,NAT_1:38; then n + 1 in Seg(len s) by A10,FINSEQ_1:3; then A12: n + 1 in dom s by FINSEQ_1:def 3; then A13: s.(n+1) = s/.(n+1) by FINSEQ_4:def 4; consider a,b being Element of R such that A14: s.(n+1) = a*b & a in I & b in J+K by A2,A10,A11; b in {a'+b' where a',b' is Element of R : a' in J & b' in K} by A14,Def20; then consider c,d being Element of R such that A15: b = c + d & c in J & d in K; A16: s/.(n+1) = a*(c + d) by A12,A14,A15,FINSEQ_4:def 4 .= a*c + a*d by VECTSP_1:def 18; set q1 = q^<*a*c*>, p1 = p^<*a*d*>; A17: len q1 = len q + len(<*a*c*>) by FINSEQ_1:35 .= len q + 1 by FINSEQ_1:57; for i being Nat st 1 <= i & i <= len q1 ex a,b being Element of R st q1.i = a*b & a in I & b in J proof let i be Nat; assume A18: 1 <= i & i <= len q1; per cases; suppose i = len q1; then q1.i = a*c by A17,FINSEQ_1:59; hence thesis by A14,A15; suppose i <> len q1; then i < len q1 by A18,REAL_1:def 5; then A19: i <= len q by A17,NAT_1:38; then i in Seg(len q) by A18,FINSEQ_1:3; then A20: i in dom q by FINSEQ_1:def 3; consider a,b being Element of R such that A21: q.i = a*b & a in I & b in J by A8,A18,A19; q1.i = a*b by A20,A21,FINSEQ_1:def 7; hence thesis by A21; end; then Sum q1 in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in I & b in J}; then A22: Sum q1 in I *' J by Def22; A23: len p1 = len p + len(<*a*d*>) by FINSEQ_1:35 .= len p + 1 by FINSEQ_1:57; for i being Nat st 1 <= i & i <= len p1 ex a,b being Element of R st p1.i = a*b & a in I & b in K proof let i be Nat; assume A24: 1 <= i & i <= len p1; per cases; suppose i = len p1; then p1.i = a*d by A23,FINSEQ_1:59; hence thesis by A14,A15; suppose i <> len p1; then i < len p1 by A24,REAL_1:def 5; then A25: i <= len p by A23,NAT_1:38; then i in Seg(len p) by A24,FINSEQ_1:3; then A26: i in dom p by FINSEQ_1:def 3; consider a,b being Element of R such that A27: p.i = a*b & a in I & b in K by A9,A24,A25; p1.i = a*b by A26,A27,FINSEQ_1:def 7; hence thesis by A27; end; then Sum p1 in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in I & b in K}; then A28: Sum p1 in I *' K by Def22; Sum q1 + Sum p1 = (Sum q + Sum <*a*c*>) + Sum p1 by RLVECT_1:58 .= (Sum q + a*c) + Sum p1 by BINOM:3 .= (Sum q + a*c) + (Sum p + Sum <*a*d*>) by RLVECT_1:58 .= (Sum q + a*c) + (Sum p + a*d) by BINOM:3 .= ((Sum q + a*c) + Sum p) + a*d by RLVECT_1:def 6 .= (a*c + (Sum q + Sum p)) + a*d by RLVECT_1:def 6 .= f.n + (a*c + a*d) by A7,A8,A9,RLVECT_1:def 6 .= f.(n+1) by A3,A6,A13,A16; hence thesis by A22,A28; end; end; A29: for n being Nat st 0 <= n &n <= len s holds P[n] from FinInd(A4,A5); 0 <= len s by NAT_1:18; then ex x,y being Element of R st Sum s = x+y & x in I*'J & y in I*'K by A3, A29; then u in {a+b where a,b is Element of R : a in I*'J & b in I*'K} by A2; hence u in (I *' J) + (I *' K) by Def20; end; now let u be set; assume u in (I *' J) + (I *' K); then u in {a+b where a,b is Element of R: a in I*'J & b in I*'K} by Def20; then consider a,b being Element of R such that A30: u = a + b & a in I*'J & b in I*'K; a in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in I & b in J} by A30,Def22; then consider q being FinSequence of the carrier of R such that A31: a = Sum q & for i being Nat st 1 <= i & i <= len q ex a,b being Element of R st q.i = a*b & a in I & b in J; b in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in I & b in K} by A30,Def22; then consider p being FinSequence of the carrier of R such that A32: b = Sum p & for i being Nat st 1 <= i & i <= len p ex a,b being Element of R st p.i = a*b & a in I & b in K; set s = p^q; A33: for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I & b in J+K proof let i be Nat; assume A34: 1 <= i & i <= len s; then i in Seg(len s) by FINSEQ_1:3; then A35: i in dom s by FINSEQ_1:def 3 ; now per cases; case A36: i <= len p; then i in Seg(len p) by A34,FINSEQ_1:3; then A37: i in dom p by FINSEQ_1:def 3; consider a,b being Element of R such that A38: p.i = a*b & a in I & b in K by A32,A34,A36; 0.R in J by Th3; then 0.R + b in {a'+b' where a',b' is Element of R:a' in J & b' in K} by A38 ; then A39: 0.R + b in J + K by Def20; s.i=a*b by A37,A38,FINSEQ_1:def 7 .=a*(0.R + b) by ALGSTR_1:def 5; hence thesis by A38,A39; case i > len p; then not(i in Seg(len p)) by FINSEQ_1:3; then not(i in dom p) by FINSEQ_1:def 3; then consider n being Nat such that A40: n in dom q & i = len p + n by A35,FINSEQ_1:38; n in Seg(len q) by A40,FINSEQ_1:def 3; then 1 <= n & n <= len q by FINSEQ_1:3; then consider a,b being Element of R such that A41: q.n = a*b & a in I & b in J by A31; 0.R in K by Th3; then b + 0.R in {a'+b' where a',b' is Element of R:a' in J & b' in K} by A41 ; then A42: b + 0.R in J + K by Def20; s.i = q.n by A40,FINSEQ_1:def 7 .= a*(b + 0.R) by A41,RLVECT_1:def 7; hence thesis by A41,A42; end; hence thesis; end; Sum s = u by A30,A31,A32,RLVECT_1:58; then u in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in I & b in J+K} by A33; hence u in I *' (J + K) by Def22; end; hence thesis by A1,TARSKI:2; end; theorem Th84: for R being Abelian left_zeroed right_zeroed add-cancelable add-associative commutative associative distributive (non empty doubleLoopStr), I,J being right-ideal (non empty Subset of R) holds (I + J) *' (I /\ J) c= I *' J proof let R be Abelian left_zeroed right_zeroed add-cancelable add-associative commutative associative distributive (non empty doubleLoopStr), I,J be right-ideal (non empty Subset of R); A1: (I + J) *' (I /\ J) = (I *' (I /\ J)) + (J *' (I /\ J)) by Th83; now let u be set; assume u in (I *' (I /\ J)) + (J *' (I /\ J)); then u in {a + b where a,b is Element of R : a in (I *' (I /\ J)) & b in (J *' (I /\ J))} by Def20; then consider a,b being Element of R such that A2: u = a + b & a in (I *' (I /\ J)) & b in (J *' (I /\ J)); a in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in I & b in (I /\ J)} by A2, Def22; then consider q being FinSequence of the carrier of R such that A3: a = Sum q & for i being Nat st 1 <= i & i <= len q ex a,b being Element of R st q.i = a*b & a in I & b in (I /\ J); for i being Nat st 1 <= i & i <= len q ex x,y being Element of R st q.i = x*y & x in I & y in J proof let i be Nat; assume 1 <= i & i <= len q; then consider x,y being Element of R such that A4: q.i = x*y & x in I & y in (I /\ J) by A3; I /\ J c= J by Th77; hence thesis by A4; end; then A5: Sum q in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in I & b in J}; b in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in J & b in (I /\ J)} by A2, Def22; then consider s being FinSequence of the carrier of R such that A6: b = Sum s & for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in J & b in (I /\ J); for i being Nat st 1 <= i & i <= len s ex x,y being Element of R st s.i = x*y & x in I & y in J proof let i be Nat; assume 1 <= i & i <= len s; then consider x,y being Element of R such that A7: s.i = x*y & x in J & y in (I /\ J) by A6; I /\ J c= I by Th77; hence thesis by A7; end; then Sum s in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in I & b in J}; then a in I *' J & b in I *' J by A3,A5,A6,Def22; hence u in I *' J by A2,Def1; end; hence thesis by A1,TARSKI:def 3; end; theorem for R being right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), I,J being add-closed left-ideal (non empty Subset of R) holds (I + J) *' (I /\ J) c= I /\ J proof let R be right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), I,J be add-closed left-ideal (non empty Subset of R); now let u be set; assume u in (I + J) *' (I /\ J); then u in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in I+J & b in I/\J} by Def22; then consider s being FinSequence of the carrier of R such that A1: u = Sum s & for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I+J & b in I/\J; consider f being Function of NAT,the carrier of R such that A2: Sum s = f.(len s) & f.0 = 0.R & for j being Nat,v being Element of R st j < len s & v = s.(j + 1) holds f.(j + 1) = f.j + v by RLVECT_1:def 12; defpred P[Nat] means f.$1 in I/\J; f.0 in I & f.0 in J by A2,Th2; then f.0 in {t where t is Element of R : t in I & t in J}; then A3: P[0] by Def21; A4: now let n be Nat; assume A5: 0 <= n & n < len s; thus P[n] implies P[n+1] proof assume A6: f.n in I /\ J; A7: 0 + 1 <= n + 1 by A5,AXIOMS:24; A8: n + 1 <= len s by A5,NAT_1:38; then n + 1 in Seg(len s) by A7,FINSEQ_1:3; then A9: n + 1 in dom s by FINSEQ_1:def 3; consider x,y being Element of R such that A10: s.(n+1) = x*y & x in I+J & y in I/\J by A1,A7,A8; A11: s.(n+1) = s/.(n+1) by A9,FINSEQ_4:def 4; then s/.(n+1) in I /\ J by A10,Def2; then f.n + s/.(n+1) in I /\ J by A6,Def1; hence thesis by A2,A5,A11; end; end; A12: for n being Nat st 0 <= n &n <= len s holds P[n] from FinInd(A3,A4); 0 <= len s by NAT_1:18; hence u in I /\ J by A1,A2,A12; end; hence thesis by TARSKI:def 3; end; definition let R be non empty LoopStr, I,J be Subset of R; pred I,J are_co-prime means :Def23: I + J = the carrier of R; end; theorem Th86: for R being left_zeroed left_unital (non empty doubleLoopStr), I,J being non empty Subset of R st I,J are_co-prime holds I /\ J c= (I + J) *' (I /\ J) proof let R be left_zeroed left_unital (non empty doubleLoopStr), I,J be non empty Subset of R; assume I,J are_co-prime; then A1: I + J = the carrier of R by Def23; now let u be set; assume A2: u in I /\ J; then u in {g where g is Element of R : g in I & g in J} by Def21; then consider g being Element of R such that A3: u = g & g in I & g in J; reconsider u' = u as Element of R by A3; set q = <*1_ R*u'*>; A4: len q = 1 by FINSEQ_1:56; A5: for i being Nat st 1 <= i & i <= len q ex x,y being Element of R st q.i = x*y & x in I+J & y in I/\J proof let i be Nat; assume 1 <= i & i <= len q; then A6: i = 1 by A4,AXIOMS:21; take 1_ R,u'; thus thesis by A1,A2,A6,FINSEQ_1:57; end; Sum q = 1_ R*u' by BINOM:3 .= u' by VECTSP_1:def 19; then u' in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in I+J & b in I/\J} by A5; hence u in (I + J) *' (I /\ J) by Def22; end; hence thesis by TARSKI:def 3; end; theorem for R being Abelian left_zeroed right_zeroed add-cancelable add-associative left_unital commutative associative distributive (non empty doubleLoopStr), I being add-closed left-ideal right-ideal (non empty Subset of R), J being add-closed left-ideal (non empty Subset of R) st I,J are_co-prime holds I *' J = I /\ J proof let R be left_zeroed right_zeroed add-cancelable add-associative Abelian commutative associative distributive left_unital (non empty doubleLoopStr), I be add-closed left-ideal right-ideal (non empty Subset of R), J be add-closed left-ideal (non empty Subset of R); assume A1: I,J are_co-prime; A2: (I + J) *' (I /\ J) c= I *' J by Th84; I /\ J c= (I + J) *' (I /\ J) by A1,Th86; then A3: I /\ J c= I *' J by A2,XBOOLE_1:1; I *' J c= I /\ J by Th82; hence thesis by A3,XBOOLE_0:def 10; end; definition let R be non empty HGrStr, I,J be Subset of R; func I % J -> Subset of R equals :Def24: {a where a is Element of R: a*J c= I}; coherence proof set M = {a where a is Element of R: a*J c= I}; M is Subset of R proof for x being set holds x in M implies x in the carrier of R proof let x be set; assume x in M; then consider a being Element of R such that A1: x = a & a*J c= I; thus thesis by A1; end; hence thesis by TARSKI:def 3; end; hence thesis; end; end; definition let R be right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), I,J be left-ideal (non empty Subset of R); cluster I % J -> non empty; coherence proof set M = {a where a is Element of R: a*J c= I}; M is non empty Subset of R proof A1: 0.R*J = {0.R} by Th70; 0.R in I & 0.R in J by Th2; then for u being set holds u in {0.R} implies u in I by TARSKI:def 1; then {0.R} c= I by TARSKI:def 3; then A2: 0.R in M by A1; for x being set holds x in M implies x in the carrier of R proof let x be set; assume x in M; then consider a being Element of R such that A3: x = a & a*J c= I; thus thesis by A3; end; hence thesis by A2,TARSKI:def 3; end; hence thesis by Def24; end; end; definition let R be right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), I,J be add-closed left-ideal (non empty Subset of R); cluster I % J -> add-closed; coherence proof set M = {a where a is Element of R: a*J c= I}; A1: M = I % J by Def24; then reconsider M as non empty Subset of R; for x,y being Element of R st x in M & y in M holds x+y in M proof let x,y be Element of R; assume A2: x in M & y in M; then consider a being Element of R such that A3: x = a & a*J c= I; consider b being Element of R such that A4: y = b & b*J c= I by A2; now let u be set; assume u in (a+b)*J; then u in {(a+b)*i where i is Element of R : i in J} by Def19; then consider c being Element of R such that A5: u = (a + b)*c & c in J; A6: u = a*c + b*c by A5,VECTSP_1:def 12; a*c in {a*i where i is Element of R : i in J} by A5; then A7: a*c in a*J by Def19; b*c in {b*i where i is Element of R : i in J} by A5; then b*c in b*J by Def19; hence u in I by A3,A4,A6,A7,Def1; end; then (a+b)*J c= I by TARSKI:def 3; hence thesis by A3,A4; end; hence thesis by A1,Def1; end; end; definition let R be right_zeroed add-left-cancelable left-distributive associative commutative (non empty doubleLoopStr), I,J be left-ideal (non empty Subset of R); cluster I % J -> left-ideal; coherence proof set M = {a where a is Element of R: a*J c= I}; A1: M = I % J by Def24; then reconsider M as non empty Subset of R; for y,x being Element of R st x in M holds y*x in M proof let y,x be Element of R; assume x in M; then consider a being Element of R such that A2: x = a & a*J c= I; now let u be set; assume u in (y*a)*J; then u in {(y*a)*i where i is Element of R : i in J} by Def19; then consider c being Element of R such that A3: u = (y*a)*c & c in J; A4: u = a*(y*c) by A3,VECTSP_1:def 16; y*c in J by A3,Def2 ; then a*(y*c) in {a*i where i is Element of R : i in J}; then u in a*J by A4,Def19; hence u in I by A2; end; then (y*a)*J c= I by TARSKI:def 3; hence thesis by A2; end; hence thesis by A1,Def2; end; cluster I % J -> right-ideal; coherence proof set M = {a where a is Element of R: a*J c= I}; A5: M = I % J by Def24; then reconsider M as non empty Subset of R; for y,x being Element of R st x in M holds x*y in M proof let y,x be Element of R; assume x in M; then consider a being Element of R such that A6: x = a & a*J c= I; now let u be set; assume u in (a*y)*J; then u in {(a*y)*i where i is Element of R : i in J} by Def19; then consider c being Element of R such that A7: u = (a*y)*c & c in J; A8: u = y*(a*c) by A7,VECTSP_1:def 16; a*c in {a*i where i is Element of R : i in J} by A7; then a*c in a*J by Def19; hence u in I by A6,A8,Def2; end; then (a*y)*J c= I by TARSKI:def 3; hence thesis by A6; end; hence thesis by A5,Def3; end; end; theorem for R being (non empty multLoopStr), I being right-ideal (non empty Subset of R), J being Subset of R holds I c= I % J proof let R be (non empty multLoopStr), I be right-ideal (non empty Subset of R), J be Subset of R; now let u be set; assume A1: u in I; then reconsider u' = u as Element of R; now let v be set; assume v in u'*J; then v in {u'*j where j is Element of R : j in J} by Def19; then consider j being Element of R such that A2: v = u'*j & j in J; thus v in I by A1,A2,Def3; end; then u'*J c= I by TARSKI:def 3; then u' in {a where a is Element of R: a*J c= I}; hence u in (I % J) by Def24; end; hence thesis by TARSKI:def 3; end; theorem for R being right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), I being add-closed left-ideal (non empty Subset of R), J being Subset of R holds (I % J) *' J c= I proof let R be right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), I be add-closed left-ideal (non empty Subset of R), J be Subset of R; now let u be set; assume u in (I % J) *' J; then u in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in I%J & b in J} by Def22; then consider s being FinSequence of the carrier of R such that A1: Sum s = u & for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I%J & b in J; consider f being Function of NAT,the carrier of R such that A2: Sum s = f.(len s) & f.0 = 0.R & for j being Nat, v being Element of R st j < len s & v = s.(j + 1) holds f.(j + 1) = f.j + v by RLVECT_1:def 12; defpred P[Nat] means f.$1 in I; A3: P[0] by A2,Th2; A4: now let j be Nat; assume A5: 0 <= j & j < len s; thus P[j] implies P[j+1] proof assume A6: f.j in I; A7: j + 1 <= len s by A5,NAT_1:38; A8: 0 + 1 <= j + 1 by A5,AXIOMS:24; then j + 1 in Seg(len s) by A7,FINSEQ_1:3; then j + 1 in dom s by FINSEQ_1:def 3; then A9: s.(j+1) = s/.(j+1) by FINSEQ_4:def 4; then A10: f.(j+1) = f.j + s/.(j+1) by A2,A5; consider a,b being Element of R such that A11: s.(j+1) = a*b & a in I%J & b in J by A1,A7,A8; a in {d where d is Element of R: d*J c= I} by A11,Def24; then consider d being Element of R such that A12: a = d & d*J c= I; a*b in {d*i where i is Element of R : i in J} by A11,A12; then s.(j+1) in d*J by A11,Def19; hence thesis by A6,A9,A10,A12,Def1; end; end; A13: for j being Nat st 0 <= j & j <= len s holds P[j] from FinInd(A3,A4); 0 <= len s by NAT_1:18; hence u in I by A1,A2,A13; end; hence thesis by TARSKI:def 3; end; theorem Th90: for R being left_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr), I being add-closed right-ideal (non empty Subset of R), J being Subset of R holds (I % J) *' J c= I proof let R be left_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr), I be add-closed right-ideal (non empty Subset of R), J be Subset of R; now let u be set; assume u in (I % J) *' J; then u in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in I%J & b in J} by Def22; then consider s being FinSequence of the carrier of R such that A1: Sum s = u & for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I%J & b in J; consider f being Function of NAT,the carrier of R such that A2: Sum s = f.(len s) & f.0 = 0.R & for j being Nat, v being Element of R st j < len s & v = s.(j + 1) holds f.(j + 1) = f.j + v by RLVECT_1:def 12; defpred P[Nat] means f.$1 in I; A3: P[0] by A2,Th3; A4: now let j be Nat; assume A5: 0 <= j & j < len s; thus P[j] implies P[j+1] proof assume A6: f.j in I; A7: j + 1 <= len s by A5,NAT_1:38; A8: 0 + 1 <= j + 1 by A5,AXIOMS:24; then j + 1 in Seg(len s) by A7,FINSEQ_1:3; then j + 1 in dom s by FINSEQ_1:def 3; then A9: s.(j+1) = s/.(j+1) by FINSEQ_4:def 4; then A10: f.(j+1) = f.j + s/.(j+1) by A2,A5; consider a,b being Element of R such that A11: s.(j+1) = a*b & a in I%J & b in J by A1,A7,A8; a in {d where d is Element of R: d*J c= I} by A11,Def24; then consider d being Element of R such that A12: a = d & d*J c= I; a*b in {d*i where i is Element of R : i in J} by A11,A12; then s.(j+1) in d*J by A11,Def19; hence thesis by A6,A9,A10,A12,Def1; end; end; A13: for j being Nat st 0 <= j & j <= len s holds P[j] from FinInd(A3,A4); 0 <= len s by NAT_1:18; hence u in I by A1,A2,A13; end; hence thesis by TARSKI:def 3; end; theorem for R being left_zeroed add-right-cancelable right-distributive commutative associative (non empty doubleLoopStr), I being add-closed right-ideal (non empty Subset of R), J,K being Subset of R holds (I % J) % K = I % (J *' K) proof let R be left_zeroed add-right-cancelable right-distributive commutative associative (non empty doubleLoopStr), I be add-closed right-ideal (non empty Subset of R), J,K be Subset of R; A1: now let u be set; assume u in (I % J) % K; then u in {a where a is Element of R: a*K c= I%J} by Def24; then consider a being Element of R such that A2: u = a & a*K c= I % J; now let v be set; assume v in a*(J *' K); then v in {a*b where b is Element of R : b in J*'K} by Def19; then consider b being Element of R such that A3: v = a*b & b in J *' K; b in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in J & b in K} by A3,Def22; then consider s being FinSequence of the carrier of R such that A4: Sum s = b & for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in J & b in K; set q = a*s; A5: dom q = dom s by POLYNOM1:def 2; A6: Seg(len q) = dom q by FINSEQ_1:def 3 .= dom s by POLYNOM1:def 2 .= Seg(len s) by FINSEQ_1:def 3; then A7: len q = len s by FINSEQ_1:8; A8: for j being Nat st 1 <= j & j <= len q holds ex c,d being Element of R st q.j = c*d & c in I%J & d in J proof let j be Nat; assume A9: 1 <= j & j <= len q; then consider c,d being Element of R such that A10: s.j = c*d & c in J & d in K by A4,A7; j in Seg(len s) by A6,A9,FINSEQ_1:3; then A11: j in dom s by FINSEQ_1:def 3; then A12: s/.j = c*d by A10,FINSEQ_4:def 4; A13: q.j = q/.j by A5,A11,FINSEQ_4:def 4 .= a*(c*d) by A11,A12,POLYNOM1:def 2 .= (a*d)*c by VECTSP_1:def 16; a*d in {a*b' where b' is Element of R : b' in K} by A10; then a*d in a*K by Def19; hence thesis by A2,A10,A13; end; A14: Sum q = v by A3,A4,BINOM:4; Sum q in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in I%J & b in J} by A8; then A15: v in (I % J) *' J by A14,Def22; (I % J) *' J c= I by Th90; hence v in I by A15; end; then a*(J*'K) c= I by TARSKI:def 3; then u in {f where f is Element of R: f*(J*'K) c= I} by A2; hence u in I % (J *' K) by Def24; end; now let u be set; assume u in I % (J *' K); then u in {f where f is Element of R: f*(J*'K) c= I} by Def24; then consider a being Element of R such that A16: u = a & a*(J *' K) c= I; now let v be set; assume v in a*K; then v in {a*b where b is Element of R : b in K} by Def19; then consider b being Element of R such that A17: v = a*b & b in K; now let z be set; assume z in (a*b)*J; then z in {(a*b)*c where c is Element of R : c in J} by Def19; then consider c being Element of R such that A18: z = (a*b)*c & c in J; A19: z = a*(c*b) by A18,VECTSP_1:def 16; set q = <*c*b*>; A20: len q = 1 by FINSEQ_1:57; A21: for i being Nat st 1 <= i & i <= len q ex x,y being Element of R st q.i = x*y & x in J & y in K proof let i be Nat; assume 1 <= i & i <= len q; then q.i = q.1 by A20,AXIOMS:21 .= c*b by FINSEQ_1:57; hence thesis by A17,A18; end; Sum q = c*b by BINOM:3; then c*b in {Sum t where t is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len t ex a,b being Element of R st t.i = a*b & a in J & b in K} by A21; then c*b in J *' K by Def22; then z in {a*f where f is Element of R : f in J*'K} by A19; then z in a*(J *' K) by Def19; hence z in I by A16; end; then (a*b)*J c= I by TARSKI:def 3; then v in {f where f is Element of R: f*J c= I} by A17; hence v in I % J by Def24; end; then a*K c= I % J by TARSKI:def 3; then u in {f where f is Element of R: f*K c= I % J} by A16; hence u in (I % J) % K by Def24; end; hence thesis by A1,TARSKI:2; end; theorem for R being non empty multLoopStr, I,J,K being Subset of R holds (J /\ K) % I = (J % I) /\ (K % I) proof let R be (non empty multLoopStr), I,J,K be Subset of R; A1: now let u be set; assume u in (J /\ K) % I; then u in {a where a is Element of R: a*I c= (J /\ K)} by Def24; then consider a being Element of R such that A2: u = a & a*I c= (J /\ K); now let v be set; assume v in a*I; then v in J /\ K by A2; then v in { x where x is Element of R : x in J & x in K} by Def21; then consider x being Element of R such that A3: v = x & x in J & x in K; thus v in J by A3; end; then a*I c= J by TARSKI:def 3; then u in {a' where a' is Element of R: a'*I c= J} by A2; then A4: u in (J % I) by Def24; now let v be set; assume v in a*I; then v in J /\ K by A2; then v in { x where x is Element of R : x in J & x in K} by Def21; then consider x being Element of R such that A5: v = x & x in J & x in K; thus v in K by A5; end; then a*I c= K by TARSKI:def 3; then u in {a' where a' is Element of R: a'*I c= K} by A2; then u in (K % I) by Def24; then u in { x where x is Element of R : x in J%I & x in K%I } by A4; hence u in (J % I) /\ (K % I) by Def21; end; now let u be set; assume u in (J % I) /\ (K % I); then u in { x where x is Element of R : x in J%I & x in K%I} by Def21; then consider x being Element of R such that A6: x = u & x in (J % I) & x in (K % I); A7: u in {a' where a' is Element of R: a'*I c= J} & u in {a' where a' is Element of R: a'*I c= K} by A6,Def24; then consider a being Element of R such that A8: u = a & a*I c= J; consider b being Element of R such that A9: u = b & b*I c= K by A7; now let v be set; assume A10: v in a*I; v in { x' where x' is Element of R : x' in J & x' in K} by A8,A9,A10; hence v in J /\ K by Def21; end; then a*I c= J /\ K by TARSKI:def 3; then u in {a' where a' is Element of R: a'*I c= (J /\ K)} by A8; hence u in (J /\ K) % I by Def24; end; hence thesis by A1,TARSKI:2; end; theorem for R being left_zeroed right_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr), I being add-closed Subset of R, J,K being right-ideal (non empty Subset of R) holds I % (J + K) = (I % J) /\ (I % K) proof let R be left_zeroed right_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr), I be add-closed Subset of R, J,K be right-ideal (non empty Subset of R); A1: now let u be set; assume u in I % (J + K); then u in {a' where a' is Element of R: a'*(J+K) c= I} by Def24; then consider a being Element of R such that A2: u = a & a*(J+K) c= I; now let u be set; assume u in a*K; then u in {a*j where j is Element of R : j in K} by Def19; then consider j being Element of R such that A3: u = a*j & j in K; K c= J+K by Th74; then u in {a*j' where j' is Element of R : j' in J+K} by A3; then u in a*(J+K) by Def19; hence u in I by A2; end; then a*K c= I by TARSKI:def 3; then u in {a' where a' is Element of R: a'*K c= I} by A2; then A4: u in (I % K) by Def24; now let u be set; assume u in a*J; then u in {a*j where j is Element of R : j in J} by Def19; then consider j being Element of R such that A5: u = a*j & j in J; J c= J+K by Th73; then u in {a*j' where j' is Element of R : j' in J+K} by A5; then u in a*(J+K) by Def19; hence u in I by A2; end; then a*J c= I by TARSKI:def 3; then u in {a' where a' is Element of R: a'*J c= I} by A2; then u in (I % J) by Def24; then u in {x where x is Element of R: x in (I % J) & x in (I % K)} by A4 ; hence u in (I % J) /\ (I % K) by Def21; end; now let u be set; assume u in (I % J) /\ (I % K); then u in {x where x is Element of R: x in (I % J) & x in (I % K)} by Def21 ; then consider x being Element of R such that A6: u = x & x in (I % J) & x in (I % K); A7: u in {a where a is Element of R: a*J c= I} & u in {a where a is Element of R: a*K c= I} by A6,Def24; then consider a being Element of R such that A8: u = a & a*J c= I; consider b being Element of R such that A9: u = b & b*K c= I by A7; now let v be set; assume v in a*(J+K); then v in {a*j where j is Element of R : j in J+K} by Def19; then consider j being Element of R such that A10: v = a*j & j in J+K; j in {x'+y where x',y is Element of R : x' in J & y in K} by A10,Def20; then consider x',y being Element of R such that A11: j = x' + y & x' in J & y in K; A12: v = a*x' + b*y by A8,A9,A10,A11,VECTSP_1:def 11; a*x' in {a*j' where j' is Element of R : j' in J} by A11; then A13: a*x' in a*J by Def19; b*y in {b*j' where j' is Element of R : j' in K} by A11; then b*y in b*K by Def19; hence v in I by A8,A9,A12,A13,Def1; end; then a*(J+K) c= I by TARSKI:def 3; then u in {a' where a' is Element of R: a'*(J+K) c= I} by A8; hence u in I % (J + K) by Def24; end; hence thesis by A1,TARSKI:2; end; definition let R be unital (non empty doubleLoopStr), I be Subset of R; func sqrt I -> Subset of R equals :Def25: {a where a is Element of R: ex n being Nat st a|^n in I}; coherence proof set M = {a where a is Element of R:ex n being Nat st a|^n in I}; M is Subset of R proof for x being set holds x in M implies x in the carrier of R proof let x be set; assume x in M; then consider a being Element of R such that A1: a = x & ex n being Nat st a|^n in I; thus thesis by A1; end; hence thesis by TARSKI:def 3; end; hence thesis; end; end; definition let R be unital (non empty doubleLoopStr), I be non empty Subset of R; cluster sqrt I -> non empty; coherence proof set M ={a where a is Element of R: ex n being Nat st a|^n in I}; M is non empty proof consider a being Element of I; a|^1 = a by BINOM:8; then a in M; hence thesis; end; hence thesis by Def25; end; end; definition let R be Abelian add-associative left_zeroed right_zeroed commutative associative add-cancelable distributive unital (non empty doubleLoopStr), I be add-closed right-ideal (non empty Subset of R); cluster sqrt I -> add-closed; coherence proof set M ={a where a is Element of R: ex n being Nat st a|^n in I}; A1: M = sqrt I by Def25; then reconsider M as non empty Subset of R; for x,y being Element of R st x in M & y in M holds x+y in M proof let x,y be Element of R; assume A2: x in M & y in M; then consider a being Element of R such that A3: x = a & ex n being Nat st a|^n in I; consider n being Nat such that A4: a|^n in I by A3; consider b being Element of R such that A5: y = b & ex m being Nat st b|^m in I by A2; consider m being Nat such that A6: b|^m in I by A5; A7: (a+b)|^(n+m) = Sum((a,b) In_Power (n+m)) by BINOM:26; set p = ((a,b) In_Power (n+m)); consider f being Function of NAT,the carrier of R such that A8: Sum p = f.(len p) & f.0 = 0.R & for j being Nat, v being Element of R st j < len p & v = p.(j + 1) holds f.(j + 1) = f.j + v by RLVECT_1:def 12; A9: 0 <= len p by NAT_1:18; A10: for i being Nat st 1 <= i & i <= len p holds p.i in I proof let i be Nat; assume A11: 1 <= i & i <= len p; then i in Seg(len p) by FINSEQ_1:3; then A12: i in dom p by FINSEQ_1:def 3; set r = i - 1; set l = (n+m) - r; 1 - 1 <= i - 1 by A11,REAL_1:49; then reconsider r as Nat by INT_1:16; i <= (n+m) + 1 by A11,BINOM:def 10; then r <= ((n+m) + 1) - 1 by REAL_1:49; then r <= ((n+m) + 1) + -1 by XCMPLX_0:def 8; then r <= (n+m) + (1 + -1) by XCMPLX_1:1; then r - r <= (n+m) - r by REAL_1:49; then 0 <= l by XCMPLX_1:14; then reconsider l as Nat by INT_1:16; A13: p.i = p/.i by A12,FINSEQ_4:def 4 .= ((n+m) choose r)*a|^l*b|^r by A12,BINOM:def 10; per cases; suppose n <= l; then consider k being Nat such that A14: l = n + k by NAT_1:28; a|^l = (a|^n)*(a|^k) by A14,BINOM:11; then a|^l in I by A4,Def3; then ((n+m) choose r)*a|^l in I by Th17; hence thesis by A13,Def3; suppose l < n; then ((n+m) + -r) < n by XCMPLX_0:def 8; then ((n+m) + -r) + r < n + r by REAL_1:53; then (n+m) + (-r + r) < n + r by XCMPLX_1:1; then (n+m) + (r - r) < n + r by XCMPLX_0:def 8; then (n+m) + 0 < n + r by XCMPLX_1:14; then -n + (n+m) < -n + (n + r) by REAL_1:53; then (-n + n) + m < -n + (n + r) by XCMPLX_1:1; then (-n + n) + m < (-n + n) + r by XCMPLX_1:1; then (-n + n) + m < (n - n) + r by XCMPLX_0:def 8; then (-n + n) + m < 0 + r by XCMPLX_1:14; then (n - n) + m < 0 + r by XCMPLX_0:def 8; then 0 + m < r by XCMPLX_1:14; then consider k being Nat such that A15: r = m + k by NAT_1:28; b|^r = (b|^m)*(b|^k) by A15,BINOM:11; then b|^r in I by A6,Def3; hence thesis by A13,Def3; end; defpred P[Nat] means f.$1 in I; A16: P[0] by A8,Th2; A17: now let j be Nat; assume A18: 0 <= j & j < len p; thus P[j] implies P[j+1] proof assume A19: f.j in I; A20: 1 <= j + 1 by NAT_1:29; A21: j + 1 <= len p by A18,NAT_1:38; then j + 1 in Seg(len p) by A20,FINSEQ_1:3; then j + 1 in dom p by FINSEQ_1:def 3; then A22: p/.(j+1) = p.(j+1) by FINSEQ_4:def 4; then A23: f.(j + 1) = f.j + p/.(j+1) by A8,A18; p/.(j+1) in I by A10,A20,A21,A22; hence thesis by A19,A23,Def1; end; end; for i being Nat st 0 <= i & i <= len p holds P[i] from FinInd(A16,A17); then (a+b)|^(n+m) in I by A7,A8,A9; hence thesis by A3,A5; end; hence thesis by A1,Def1; end; end; definition let R be unital commutative associative (non empty doubleLoopStr), I be left-ideal (non empty Subset of R); cluster sqrt I -> left-ideal; coherence proof set M ={a where a is Element of R: ex n being Nat st a|^n in I}; A1: M = sqrt I by Def25; then reconsider M as non empty Subset of R; for y,x being Element of R st x in M holds y*x in M proof let y',x' be Element of R; assume A2: x' in M; reconsider x = x',y = y' as Element of R; consider a being Element of R such that A3: x = a & ex n being Nat st a|^n in I by A2; consider n being Nat such that A4: a|^n in I by A3; A5: (y|^n)*(a|^n) in I by A4,Def2; (y*a)|^n = (y|^n)*(a|^n) by BINOM:10 ; hence thesis by A3,A5; end; hence thesis by A1,Def2; end; cluster sqrt I -> right-ideal; coherence proof set M ={a where a is Element of R: ex n being Nat st a|^n in I}; A6: M = sqrt I by Def25; then reconsider M as non empty Subset of R; for y,x being Element of R st x in M holds x*y in M proof let y',x' be Element of R; assume A7: x' in M; reconsider x = x',y = y' as Element of R; consider a being Element of R such that A8: x = a & ex n being Nat st a|^n in I by A7; consider n being Nat such that A9: a|^n in I by A8; A10: (y|^n)*(a|^n) in I by A9,Def2; (y*a)|^n = (y|^n)*(a|^n) by BINOM:10; hence thesis by A8,A10; end; hence thesis by A6,Def3; end; end; theorem for R being unital associative (non empty doubleLoopStr), I being non empty Subset of R, a being Element of R holds a in sqrt I iff ex n being Nat st a|^n in sqrt I proof let R be unital associative (non empty doubleLoopStr), I be non empty Subset of R, a be Element of R; A1: now assume A2: a in sqrt I; a|^1 = a by BINOM:8; hence ex n being Nat st a|^n in sqrt I by A2; end; now assume ex n being Nat st a|^n in sqrt I; then consider n being Nat such that A3: a|^n in sqrt I; a|^n in {d where d is Element of R: ex m being Nat st d|^m in I} by A3,Def25; then consider d being Element of R such that A4: a|^n = d & ex m being Nat st d|^m in I; consider m being Nat such that A5: d|^m in I by A4; a|^(n*m) = d|^m by A4,BINOM:12; then a in {g where g is Element of R: ex l being Nat st g|^l in I} by A5; hence a in sqrt I by Def25; end; hence thesis by A1; end; theorem for R being left_zeroed right_zeroed add-cancelable distributive unital associative (non empty doubleLoopStr), I being add-closed right-ideal (non empty Subset of R), J being add-closed left-ideal (non empty Subset of R) holds sqrt (I *' J) = sqrt (I /\ J) proof let R be left_zeroed right_zeroed associative add-cancelable distributive unital (non empty doubleLoopStr), I be add-closed right-ideal (non empty Subset of R), J be add-closed left-ideal(non empty Subset of R); A1: now let u be set; assume u in sqrt (I *' J); then u in {d where d is Element of R: ex m being Nat st d|^m in I*'J} by Def25; then consider d being Element of R such that A2: u = d & ex m being Nat st d|^m in I *' J; consider m being Nat such that A3: d|^m in I *' J by A2; d|^m in {Sum s where s is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I & b in J} by A3,Def22; then consider s being FinSequence of the carrier of R such that A4: d|^m = Sum s & for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I & b in J; consider f being Function of NAT,the carrier of R such that A5: Sum s = f.(len s) & f.0 = 0.R & for j being Nat, v being Element of R st j < len s & v = s.(j + 1) holds f.(j + 1) = f.j + v by RLVECT_1:def 12; defpred P[Nat] means f.$1 in I /\ J; f.0 in I & f.0 in J by A5,Th2,Th3; then f.0 in {g where g is Element of R: g in I & g in J}; then A6: P[0] by Def21; A7: now let j be Nat; assume A8: 0 <= j & j < len s; thus P[j] implies P[j+1] proof assume f.j in I /\ J; then f.j in {g where g is Element of R: g in I & g in J} by Def21; then consider g being Element of R such that A9: g = f.j & g in I & g in J; A10: j + 1 <= len s by A8,NAT_1:38; A11: 0 + 1 <= j + 1 by A8,AXIOMS:24; then j + 1 in Seg(len s) by A10,FINSEQ_1:3; then j + 1 in dom s by FINSEQ_1:def 3; then A12: s.(j+1) = s/.(j+1) by FINSEQ_4:def 4; then A13: f.(j+1) = f.j + s/.(j+1) by A5,A8; consider a,b being Element of R such that A14: s.(j+1) = a*b & a in I & b in J by A4,A10,A11; s/.(j+1) in I & s/.(j+1) in J by A12,A14,Def2,Def3; then A15: f.(j+1) in I & f.(j+1) in J by A9,A13,Def1; f.(j+1) in {g' where g' is Element of R: g' in I & g' in J} by A15; hence thesis by Def21; end; end; A16: for j being Nat st 0 <= j&j<=len s holds P[j] from FinInd(A6,A7); 0 <= len s by NAT_1:18; then Sum s in I /\ J by A5,A16; then u in {d' where d' is Element of R: ex m being Nat st d'|^m in I /\ J} by A2,A4; hence u in sqrt (I /\ J) by Def25; end; now let u be set; assume u in sqrt (I /\ J); then u in {d where d is Element of R: ex m being Nat st d|^m in I/\J} by Def25; then consider d being Element of R such that A17: u = d & ex m being Nat st d|^m in I /\ J; consider m being Nat such that A18: d|^m in I /\ J by A17; d|^m in {g where g is Element of R: g in I & g in J} by A18,Def21; then consider g being Element of R such that A19: d|^m = g & g in I & g in J; set q = <*d|^m*d|^m*>; A20: len q = 1 by FINSEQ_1:57; A21: for i being Nat st 1 <= i & i <= len q ex x,y being Element of R st q.i = x*y & x in I & y in J proof let i be Nat; assume A22: 1 <= i & i <= len q; then i in Seg(len q) by FINSEQ_1:3; then i in dom q by FINSEQ_1:def 3; then A23: q.i = q/.i by FINSEQ_4:def 4; then q/.i = q.1 by A20,A22,AXIOMS:21 .= d|^m*d|^m by FINSEQ_1:57; hence thesis by A19,A23; end; d|^(m+m) = d|^m*d|^m by BINOM:11 .= Sum q by BINOM:3; then d|^(m+m) in {Sum s where s is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I & b in J} by A21; then d|^(m+m) in I *' J by Def22; then u in {d' where d' is Element of R: ex m being Nat st d'|^m in I *' J} by A17; hence u in sqrt (I *' J) by Def25; end; hence thesis by A1,TARSKI:2; end; begin :: Noetherian ideals definition let L be non empty doubleLoopStr, I be Ideal of L; attr I is finitely_generated means : Def26: ex F being non empty finite Subset of L st I = F-Ideal; end; definition let L be non empty doubleLoopStr; cluster finitely_generated Ideal of L; existence proof consider x being set such that A1: x in the carrier of L by XBOOLE_0:def 1; reconsider x as Element of L by A1; take {x}-Ideal; thus thesis by Def26; end; end; definition let L be non empty doubleLoopStr, F be non empty finite Subset of L; cluster F-Ideal -> finitely_generated; coherence by Def26; end; definition let L be non empty doubleLoopStr; attr L is Noetherian means :Def27: for I being Ideal of L holds I is finitely_generated; end; definition cluster Euclidian Abelian add-associative right_zeroed right_complementable well-unital distributive commutative associative non degenerated (non empty doubleLoopStr); existence proof take INT.Ring; thus thesis; end; end; definition let L be non empty doubleLoopStr; let I be Ideal of L; attr I is principal means :Def28: ex e being Element of L st I = {e}-Ideal; end; definition let L be non empty doubleLoopStr; attr L is PID means :Def29 : for I being Ideal of L holds I is principal; end; theorem Th96: for L being non empty doubleLoopStr, F being non empty Subset of L st F <> {0.L} ex x being Element of L st x <> 0.L & x in F proof let L be non empty doubleLoopStr, F be non empty Subset of L; assume A1: F <> {0.L}; now assume A2: for x being set st x in F holds x = 0.L; for x being set holds x in F iff x = 0.L proof let e be set; thus e in F implies e = 0.L by A2; assume A3: e = 0.L; ex a being set st a in F by XBOOLE_0:def 1; hence e in F by A2,A3; end; hence contradiction by A1,TARSKI:def 1; end; hence thesis; end; theorem Th97: for R being add-associative left_zeroed right_zeroed right_complementable distributive left_unital Euclidian (non empty doubleLoopStr) holds R is PID proof let R be add-associative left_zeroed right_zeroed right_complementable distributive left_unital Euclidian (non empty doubleLoopStr); let I be Ideal of R; per cases; suppose A1: I = {0.R}; set e = 0.R; take e; thus I = {e}-Ideal by A1,Th47; suppose I <> {0.R}; then consider x being Element of R such that A2: x <> 0.R & x in I by Th96; set I' = { y where y is Element of I : y <> 0.R }; A3: x in I' by A2; I' c= the carrier of R proof let x be set; assume x in I'; then consider y being Element of I such that A4: x=y & y <> 0.R; thus x in the carrier of R by A4; end; then reconsider I' as non empty Subset of R by A3; consider f being Function of the carrier of R, NAT such that A5: for a,b being Element of R st b <> 0.R holds (ex q,r being Element of R st (a = q*b + r & (r = 0.R or (f.r qua Nat) < (f.b qua Nat)))) by INT_3:def 9; set K = { f.i where i is Element of I' : not contradiction }; consider i being Element of I'; A6: f.i in K; K c= NAT proof let x be set; assume x in K; then consider e being Element of I' such that A7: f.e = x; thus x in NAT by A7; end; then reconsider K as non empty Subset of NAT by A6; set k=min K; k in K by CQC_SIM1:def 8; then consider e being Element of I' such that A8: f.e = k; e in I'; then consider e' being Element of I such that A9: e'=e & e' <> 0.R; take e; now let x be set; hereby assume A10: x in I; then reconsider x'=x as Element of R; consider q,r being Element of R such that A11: x' = q*e + r and A12: r = 0.R or (f.r qua Nat) < k by A5,A8,A9; now A13: -(q*e) + x' = -(q*e) + q*e + r by A11,RLVECT_1:def 6 .= 0.R + r by RLVECT_1:16 .= r by ALGSTR_1:def 5; q*e in I by A9,Def2; then -(q*e) in I by Th13; then A14: r in I by A10,A13,Def1; assume A15: r <> 0.R; then r in I' by A14; then f.r in K; hence contradiction by A12,A15,CQC_SIM1:def 8; end; then A16: x' = q*e by A11,RLVECT_1:def 7; A17: e in {e} by TARSKI:def 1; {e} c= {e}-Ideal by Def15; hence x in {e}-Ideal by A16,A17,Def2; end; assume A18: x in {e}-Ideal; {e} c= I by A9,ZFMISC_1:37; then {e}-Ideal c= I by Def15; hence x in I by A18; end; hence I={e}-Ideal by TARSKI:2; end; theorem Th98: for L being non empty doubleLoopStr st L is PID holds L is Noetherian proof let L be non empty doubleLoopStr; assume A1: L is PID; let I being Ideal of L; I is principal by A1,Def29; then consider e being Element of L such that A2: I = {e}-Ideal by Def28; take F={e}; thus F-Ideal=I by A2; end; definition cluster INT.Ring -> Noetherian; coherence proof INT.Ring is PID by Th97; hence thesis by Th98; end; end; definition cluster Noetherian Abelian add-associative right_zeroed right_complementable well-unital distributive commutative associative non degenerated (non empty doubleLoopStr); existence proof take INT.Ring; thus thesis; end; end; theorem :: Lemma_4_5_i_ii: for R being Noetherian add-associative left_zeroed right_zeroed add-cancelable associative distributive well-unital (non empty doubleLoopStr) for B being non empty Subset of R ex C being non empty finite Subset of R st C c= B & C-Ideal = B-Ideal proof let R be Noetherian add-associative left_zeroed right_zeroed add-cancelable associative distributive well-unital (non empty doubleLoopStr); let B be non empty Subset of R; B-Ideal is finitely_generated by Def27; then consider D being non empty finite Subset of R such that A1: D-Ideal = B-Ideal by Def26; A2: D c= B-Ideal by A1,Def15; defpred P[set,set] means ex cL being non empty LinearCombination of B st $1 = Sum cL & ex fsB being FinSequence of B st dom fsB = dom cL & $2 = rng fsB & for i being Nat st i in dom cL ex u, v being Element of R st cL/.i = u*((fsB/.i qua Element of B) qua Element of R)*v; A3: for e being set st e in D ex u being set st u in bool B & P[e,u] proof let e be set; assume e in D; then consider cL being LinearCombination of B such that A4: e = Sum cL by A2,Th60; per cases; suppose A5: cL is non empty; defpred P1[set,Element of B] means ex u, v being Element of R st cL/.$1 = u*(($2 qua Element of B) qua Element of R)*v; A6: now let k be Nat; assume k in Seg len cL; then k in dom cL by FINSEQ_1:def 3; then consider u, v being Element of R, a being Element of B such that A7: cL/.k = u*a*v by Def9; take d = a; thus P1[k,d] by A7; end; consider fsB being FinSequence of B such that A8: dom fsB = Seg len cL and A9: for k being Nat st k in Seg len cL holds P1[k,fsB/.k] from SeqExD(A6); take u = rng fsB; thus u in bool B; dom cL = Seg len cL by FINSEQ_1:def 3 ; hence P[e,u] by A4,A5,A8,A9; suppose A10: cL is empty; consider b being Element of B; set kL = <*0.R*b*0.R*>; now let i be set; assume i in dom kL; then i in Seg len kL by FINSEQ_1:def 3; then i in {1} by FINSEQ_1:4,57; then A11: i = 1 by TARSKI:def 1; take u = 0.R, v = 0.R, b; thus kL/.i = u*b*v by A11,FINSEQ_4:25; end; then reconsider kL as non empty LinearCombination of B by Def9; cL = <*>the carrier of R by A10; then A12: e = 0.R by A4,RLVECT_1:60 .= 0.R*b*0.R by BINOM:2 .= Sum kL by BINOM:3; defpred P2[Nat,Element of B] means ex u, v being Element of R st kL/.$1 = u*(($2 qua Element of B) qua Element of R)*v; A13: now let k be Nat; assume k in Seg len kL; then k in {1} by FINSEQ_1:4,57; then A14: k = 1 by TARSKI:def 1; take b; thus P2[k,b] proof take u=0.R, v=0.R; thus thesis by A14,FINSEQ_4:25; end; end; consider fsB being FinSequence of B such that A15: dom fsB = Seg len kL and A16: for k being Nat st k in Seg len kL holds P2[k,fsB/.k] from SeqExD(A13); take u = rng fsB; thus u in bool B; dom kL = Seg len kL by FINSEQ_1:def 3; hence P[e,u] by A12,A15,A16; end; consider f being Function of D, bool B such that A17: for e being set st e in D holds P[e,f.e] from FuncEx1(A3); reconsider rf = rng f as Subset-Family of B by SETFAM_1:def 7; reconsider C = union rf as Subset of B; consider r being set such that A18: r in rng f by XBOOLE_0:def 1; consider x being set such that A19: x in dom f & r = f.x by A18,FUNCT_1:def 5; A20: dom f = D by FUNCT_2:def 1; consider cL being non empty LinearCombination of B such that x = Sum cL and A21: ex fsB being FinSequence of B st dom fsB = dom cL & r = rng fsB & for i being Nat st i in dom cL ex u, v being Element of R st cL/.i = u*((fsB/.i qua Element of B) qua Element of R)*v by A17,A19; consider fsB being FinSequence of B such that A22: dom fsB = dom cL and A23: r = rng fsB and for i being Nat st i in dom cL ex u, v being Element of R st cL/.i = u*((fsB/.i qua Element of B) qua Element of R)*v by A21; r is non empty by A22,A23,RELAT_1:65; then consider x being set such that A24: x in r by XBOOLE_0:def 1; A25: rng f is finite by A20,FINSET_1:26; A26: now let r be set; assume r in rng f; then consider x being set such that A27: x in dom f & r = f.x by FUNCT_1:def 5; consider cL being non empty LinearCombination of B such that x = Sum cL and A28: ex fsB being FinSequence of B st dom fsB = dom cL & r = rng fsB & for i being Nat st i in dom cL ex u, v being Element of R st cL/.i = u*((fsB/.i qua Element of B) qua Element of R)*v by A17,A27; consider fsB being FinSequence of B such that dom fsB = dom cL and A29: r = rng fsB and for i being Nat st i in dom cL ex u, v being Element of R st cL/.i = u*((fsB/.i qua Element of B) qua Element of R)*v by A28; thus r is finite by A29; end; C c= the carrier of R by XBOOLE_1:1; then reconsider C as non empty finite Subset of R by A18,A24,A25,A26, FINSET_1:25,TARSKI:def 4; now let d be set; assume A30: d in D; then d in dom f by FUNCT_2:def 1; then f.d in rng f by FUNCT_1:def 5; then A31: f.d c= C by ZFMISC_1:92; consider cL being non empty LinearCombination of B such that A32: d = Sum cL and A33: ex fsB being FinSequence of B st dom fsB = dom cL & f.d = rng fsB & for i being Nat st i in dom cL ex u, v being Element of R st cL/.i = u*((fsB/.i qua Element of B) qua Element of R)*v by A17,A30; now let i be set; assume A34: i in dom cL; consider fsB being FinSequence of B such that A35: dom fsB = dom cL and A36: f.d = rng fsB and A37: for i being Nat st i in dom cL ex u, v being Element of R st cL/.i = u*((fsB/.i qua Element of B) qua Element of R)*v by A33; consider u, v being Element of R such that A38: cL/.i = u*((fsB/.i qua Element of B) qua Element of R)*v by A34,A37; fsB/.i = fsB.i by A34,A35,FINSEQ_4:def 4; then fsB/.i in f.d by A34,A35,A36,FUNCT_1:def 5; hence ex u,v being Element of R, a being Element of C st cL/.i = u*a*v by A31,A38; end; then reconsider cL'= cL as LinearCombination of C by Def9; d = Sum cL' by A32; hence d in C-Ideal by Th60; end; then D c= C-Ideal by TARSKI:def 3; then D-Ideal c= (C-Ideal)-Ideal by Th57 ; then A39: B-Ideal c= C-Ideal by A1,Th44; take C; C-Ideal c= B-Ideal by Th57; hence thesis by A39,XBOOLE_0:def 10; end; theorem :: Lemma_4_5_ii_iii: for R being (non empty doubleLoopStr) st for B being non empty Subset of R ex C being non empty finite Subset of R st C c= B & C-Ideal = B-Ideal for a being sequence of R ex m being Nat st a.(m+1) in (rng (a|Segm(m+1)))-Ideal proof let R be (non empty doubleLoopStr); assume A1: for B being non empty Subset of R ex C being non empty finite Subset of R st C c= B & C-Ideal = B-Ideal; let a being sequence of R; reconsider B = rng a as non empty Subset of R; consider C being non empty finite Subset of R such that A2: C c= B & C-Ideal = B-Ideal by A1; defpred P[set,set] means $1 = a.$2; A3: dom a = NAT by FUNCT_2:def 1; A4: for e being set st e in C ex u being set st u in NAT & P[e,u] proof let e be set; assume e in C; then consider u being set such that A5: u in dom a & e = a.u by A2,FUNCT_1:def 5; take u; thus u in NAT by A5; thus P[e,u] by A5; end; consider f being Function of C, NAT such that A6: for e being set st e in C holds P[e,f.e] from FuncEx1(A4); set Rf = rng f; A7: dom f = C by FUNCT_2:def 1; then reconsider Rf as non empty finite Subset of NAT by FINSET_1:26; reconsider m = max Rf as Nat by ORDINAL2:def 21; take m; set D = rng (a | Segm(m+1)); A8: C c= D proof let X be set; assume A9: X in C; then A10: f.X in Rf by A7,FUNCT_1:def 5; reconsider fx=f.X as Nat; fx <= m by A10,PRE_CIRC:def 1; then A11: fx < m+1 by NAT_1:38; m + 1 > 0 by NAT_1:19; then fx in Segm(m+1) by A11,GR_CY_1:10; then a.fx in rng (a | Segm(m+1)) by A3,FUNCT_1:73; hence X in D by A6,A9; end; then reconsider D as non empty Subset of R by XBOOLE_1:3 ; A12: B-Ideal c= D-Ideal by A2,A8,Th57; D c= B by RELAT_1:99; then D-Ideal c= B-Ideal by Th57; then A13: D-Ideal = B-Ideal by A12,XBOOLE_0:def 10; A14: B c= B-Ideal by Def15; a.(m+1) in B by FUNCT_2:6; hence a.(m+1) in (rng (a|Segm(m+1)))-Ideal by A13,A14; end; definition let X, Y be non empty set, f be Function of X, Y, A be non empty Subset of X; cluster f|A -> non empty; coherence proof dom f = X by FUNCT_2:def 1; then (dom f) /\ A is non empty by XBOOLE_1:28; then (dom f) meets A by XBOOLE_0:def 7; hence f|A is non empty by RELAT_1:95; end; end; theorem :: Lemma_4_5_iii_iv: for R being (non empty doubleLoopStr) st for a being sequence of R ex m being Nat st a.(m+1) in (rng (a|Segm(m+1)))-Ideal holds not ex F being Function of NAT, bool (the carrier of R) st (for i being Nat holds F.i is Ideal of R) & (for j,k being Nat st j < k holds F.j c< F.k) proof let R being (non empty doubleLoopStr); assume A1: for a being sequence of R ex m being Nat st a.(m+1) in (rng (a|Segm(m+1)))-Ideal; given F being Function of NAT, bool (the carrier of R) such that A2: for i being Nat holds F.i is Ideal of R and A3: for j,k being Nat st j < k holds F.j c< F.k; defpred P[set,set] means ex n being Nat st n = $1 & (n = 0 implies $2 in F.0) & (n > 0 implies ex k being Nat st n = k+1 & $2 in F.n \ F.k); A4: for e being set st e in NAT ex u being set st u in the carrier of R & P[e,u] proof let e be set such that A5: e in NAT; reconsider n=e as Nat by A5; per cases by NAT_1:19; suppose A6: n = 0; F.0 is Ideal of R by A2; then consider u being set such that A7: u in F.0 by XBOOLE_0:def 1; take u; thus u in the carrier of R by A7; take n; thus n=e; thus thesis by A6,A7; suppose n > 0; then consider k being Nat such that A8: n=k+1 by NAT_1:22; n > k by A8,NAT_1:38; then F.k c< F.n by A3; then not (F.n c= F.k) by XBOOLE_1:60; then F.n \ F.k is non empty by XBOOLE_1:37; then consider u being set such that A9: u in F.n \ F.k by XBOOLE_0:def 1; take u; thus u in the carrier of R by A9; take n; thus n=e; thus thesis by A8,A9; end; consider f being Function of NAT, the carrier of R such that A10: for e being set st e in NAT holds P[e, f.e] from FuncEx1(A4); reconsider f as sequence of R by NORMSP_1:def 3; defpred P[Nat] means rng (f|Segm($1+1)) c= F.$1; A11: P[0] proof let y be set; assume y in rng (f|Segm(0+1)); then consider x being set such that A12: x in dom (f|Segm(1)) and A13: y = (f|Segm(1)).x by FUNCT_1:def 5; A14: x in Segm(1) & x in dom f by A12,RELAT_1:86; ex n being Nat st n = x & (n = 0 implies f.x in F.0) & (n > 0 implies ex k being Nat st n = k+1 & f.x in F.n \ F.k) by A10,A12; hence thesis by A13,A14,FUNCT_1:72,GR_CY_1:13,TARSKI:def 1; end; A15: for k being Nat st P[k] holds P[k+1] proof let k be Nat such that A16: rng (f|Segm(k+1)) c= F.k; let y be set such that A17: y in rng (f|Segm((k+1)+1)); consider x being set such that A18: x in dom (f|Segm((k+1)+1)) and A19: y = (f|Segm((k+1)+1)).x by A17,FUNCT_1:def 5; A20: x in Segm((k+1)+1) & x in dom f by A18,RELAT_1:86; reconsider nx = x as Nat by A18; 0 < (k+1)+1 by NAT_1:19; then nx < (k+1)+1 by A20,GR_CY_1:10; then A21: nx <= k+1 by NAT_1:38; per cases by A21,AXIOMS:21; suppose A22: nx < k+1; 0 < k+1 by NAT_1:19; then A23: nx in Segm(k+1) by A22,GR_CY_1:10; y = f.nx by A18,A19,FUNCT_1:70; then y in rng(f|Segm(k+1)) by A20,A23,FUNCT_1:73; then A24: y in F.k by A16; k < k+1 by NAT_1:38; then F.k c< F.(k+1) by A3; then F.k c= F.(k+1) by XBOOLE_0:def 8; hence y in F.(k+1) by A24; suppose A25: nx = k+1; A26: y = f.nx by A18,A19,FUNCT_1:70; consider n being Nat such that A27: n = nx and (n = 0 implies f.nx in F.0) and A28: (n > 0 implies ex k being Nat st n = k+1 & f.nx in F.n \ F.k) by A10; consider m being Nat such that A29: n = m+1 & f.nx in F.n \ F.m by A25,A27,A28,NAT_1:19; thus y in F.(k+1) by A25,A26,A27,A29,XBOOLE_0:def 4; end; A30: for m being Nat holds P[m] from Ind(A11,A15); consider m being Nat such that A31: f.(m+1) in (rng (f|Segm(m+1)))-Ideal by A1; F.m is Ideal of R by A2; then A32: F.m = (F.m)-Ideal by Th44; (rng (f|Segm(m+1))) c= F.m by A30; then (rng (f|Segm(m+1)))-Ideal c= F.m by A32,Th57; then A33: f.(m+1) in F.m by A31; consider n being Nat such that A34: n = m+1 and n = 0 implies f.(m+1) in F.0 and A35: n > 0 implies ex k being Nat st n = k+1 & f.(m+1) in F.n \ F.k by A10; consider k being Nat such that A36: n = k+1 and A37: f.(m+1) in F.n \ F.k by A34,A35,NAT_1:19; not f.(m+1) in F.k by A37,XBOOLE_0:def 4; hence contradiction by A33,A34,A36,XCMPLX_1:2; end; theorem :: Lemma_4_5_iv_i: for R being (non empty doubleLoopStr) st not ex F being Function of NAT, bool (the carrier of R) st (for i being Nat holds F.i is Ideal of R) & (for j,k being Nat st j < k holds F.j c< F.k) holds R is Noetherian proof let R being (non empty doubleLoopStr) such that A1: not ex F being Function of NAT, bool (the carrier of R) st (for i being Nat holds F.i is Ideal of R) & (for j,k being Nat st j < k holds F.j c< F.k) and A2: not R is Noetherian; consider I being Ideal of R such that A3: I is not finitely_generated by A2,Def27; consider e being set such that A4: e in I by XBOOLE_0:def 1; reconsider e as Element of R by A4; set D = { S where S is Element of bool the carrier of R : S is non empty finite Subset of I }; {e} c= I by A4,ZFMISC_1:37; then A5: {e} in D; D c= bool the carrier of R proof let x be set; assume x in D; then ex s being Element of bool the carrier of R st x = s & s is non empty finite Subset of I; hence x in bool the carrier of R; end; then reconsider D as non empty Subset of bool the carrier of R by A5; reconsider e'={e} as Element of D by A5; defpred P[Nat,Element of D,Element of D] means ex r being Element of R st r in I \ $2-Ideal & $3 = $2 \/ {r}; A6: for n be Nat for x be Element of D ex y be Element of D st P[n,x,y] proof let n be Nat, x be Element of D; x in D; then consider x' being Subset of R such that A7: x' = x and A8: x' is non empty finite Subset of I; A9: I <> x'-Ideal by A3,A8,Def26; x'-Ideal c= I-Ideal by A8,Th57; then x'-Ideal c= I by Th44; then not I c= x'-Ideal by A9,XBOOLE_0:def 10; then consider r being set such that A10: r in I & not r in x'-Ideal by TARSKI:def 3; reconsider x1'=x' as non empty finite Subset of I by A8; set y=x1' \/ {r}; A11: y c= I proof let x be set; assume x in y; then x in x1' or x in {r} by XBOOLE_0:def 2; hence thesis by A10,TARSKI:def 1; end; then y is Element of bool the carrier of R by XBOOLE_1:1; then y in D by A11; then reconsider y as Element of D; take y; reconsider r as Element of R by A10; take r; thus thesis by A7,A10,XBOOLE_0:def 4; end; consider f be Function of NAT, D such that A12: f.0 = e' & for n be Element of NAT holds P[n,f.n,f.(n+1)] from RecExD(A6); defpred Q[Nat,Element of bool the carrier of R] means ex c being Subset of R st c = f.$1 & $2 = c-Ideal; A13: for x being Nat ex y being Element of bool the carrier of R st Q[x,y] proof let x be Nat; f.x in D; then consider c being Subset of R such that A14: c = f.x & c is non empty finite Subset of I; reconsider y = c-Ideal as Subset of R; take y; take c; thus thesis by A14; end; consider F being Function of NAT, bool the carrier of R such that A15: for x being Element of NAT holds Q[x,F.x] from FuncExD(A13); A16: for i being Nat holds F.i is Ideal of R proof let i be Nat; ex c being Subset of R st c = f.i & F.i = c-Ideal by A15 ; hence thesis; end; for j,k being Nat st j < k holds F.j c< F.k proof let j,k be Nat; defpred P[Nat] means F.j c< F.(j+1+$1); A17: for i being Nat holds F.i c< F.(i+1) proof let i be Nat; consider c being Subset of R such that A18: c = f.i and A19: F.i = c-Ideal by A15; c in D by A18; then consider c' being Subset of R such that A20: c' = c and A21: c' is non empty finite Subset of I; consider c1 being Subset of R such that A22: c1 = f.(i+1) and A23: F.(i+1) = c1-Ideal by A15; c1 in D by A22; then consider c1' being Subset of R such that A24: c1' = c1 and A25: c1' is non empty finite Subset of I; consider r being Element of R such that A26: r in I \ c-Ideal and A27: c1 = c \/ {r} by A12,A18,A22; c c= c1 by A27,XBOOLE_1:7; hence F.i c= F.(i+1) by A19,A20,A21,A23,A24,A25,Th57; A28: c1 c= c1-Ideal by A24,A25,Def15; r in {r} by TARSKI:def 1; then r in c1 by A27,XBOOLE_0:def 2; hence F.i <> F.(i+1) by A19,A23,A26,A28,XBOOLE_0:def 4; end; then A29: P[0]; A30: for i being Nat st P[i] holds P[i+1] proof let i be Nat such that A31: F.j c= F.(j+1+i) and A32: F.j <> F.(j+1+i); F.(j+1+i) c< F.((j+1+i)+1) by A17; then F.(j+1+i) c= F.((j+1+i)+1) by XBOOLE_0:def 8; then A33: F.(j+1+i) c= F.(j+1+(i+1)) by XCMPLX_1:1; hence F.j c= F.(j+1+(i+1)) by A31,XBOOLE_1:1; not F.(j+1+i) c= F.j by A31,A32,XBOOLE_0:def 10; then consider x being set such that A34: x in F.(j+1+i) and A35: not x in F.j by TARSKI:def 3; thus thesis by A33,A34,A35; end; A36: for i being Nat holds P[i] from Ind(A29, A30); assume j < k; then j+1 <= k by NAT_1:38; then consider i being Nat such that A37: k = j + 1 + i by NAT_1:28; thus thesis by A36,A37; end; hence contradiction by A1,A16; end;

Back to top