### The Mizar article:

### Basis of Vector Space

**by****Wojciech A. Trybulec**

- Received July 27, 1990
Copyright (c) 1990 Association of Mizar Users

- MML identifier: VECTSP_7
- [ MML identifier index ]

environ vocabulary VECTSP_1, FINSET_1, RLVECT_2, FUNCT_1, RLVECT_3, RLVECT_1, BOOLE, FUNCT_2, ARYTM_1, RELAT_1, RLSUB_1, ZFMISC_1, TARSKI, ORDERS_1, HAHNBAN; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1, FUNCT_2, FRAENKEL, FINSET_1, STRUCT_0, ORDINAL1, ORDERS_1, RLVECT_1, VECTSP_1, RLVECT_2, VECTSP_4, VECTSP_5, VECTSP_6; constructors ORDERS_1, RLVECT_2, VECTSP_5, VECTSP_6, MEMBERED, XBOOLE_0; clusters VECTSP_1, VECTSP_4, RLVECT_2, STRUCT_0, RELSET_1, SUBSET_1, MEMBERED, ZFMISC_1, XBOOLE_0; requirements SUBSET, BOOLE; definitions XBOOLE_0, VECTSP_4, VECTSP_6, TARSKI; theorems FUNCT_1, FINSET_1, ORDERS_1, ORDERS_2, RLVECT_3, TARSKI, VECTSP_1, VECTSP_2, VECTSP_4, VECTSP_5, VECTSP_6, ZFMISC_1, RLVECT_1, FUNCT_2, RELAT_1, ORDINAL1, XBOOLE_0, XBOOLE_1; schemes FUNCT_1, RLVECT_2, FUNCT_2, XBOOLE_0; begin reserve x,y,X,Y,Z for set; reserve GF for Field; reserve a,b for Element of GF; reserve V for VectSp of GF; reserve v,v1,v2,u for Vector of V; reserve A,B,C for Subset of V; reserve T for finite Subset of V; reserve l for Linear_Combination of A; reserve f,g for Function of the carrier of V, the carrier of GF; definition let GF; let V; let IT be Subset of V; attr IT is linearly-independent means :Def1: for l being Linear_Combination of IT st Sum(l) = 0.V holds Carrier(l) = {}; antonym IT is linearly-dependent; end; canceled; theorem A c= B & B is linearly-independent implies A is linearly-independent proof assume that A1: A c= B and A2: B is linearly-independent; let l be Linear_Combination of A; assume A3: Sum(l) = 0.V; reconsider L = l as Linear_Combination of B by A1,VECTSP_6:25; Carrier(L) = {} by A2,A3,Def1; hence thesis; end; theorem Th3: A is linearly-independent implies not 0.V in A proof assume that A1: A is linearly-independent and A2: 0.V in A; deffunc F(set) = 0.GF; consider f such that A3: f.(0.V) = 1_ GF and A4: for v being Element of V st v <> 0.V holds f.v = F(v) from LambdaSep1; reconsider f as Element of Funcs(the carrier of V, the carrier of GF) by FUNCT_2:11; ex T st for v st not v in T holds f.v = 0.GF proof take T = {0.V}; let v; assume not v in T; then v <> 0.V by TARSKI:def 1; hence thesis by A4; end; then reconsider f as Linear_Combination of V by VECTSP_6:def 4; A5: Carrier(f) = {0.V} proof thus Carrier(f) c= {0.V} proof let x; assume x in Carrier(f); then consider v such that A6: v = x and A7: f.v <> 0.GF by VECTSP_6:19; v = 0.V by A4,A7; hence thesis by A6,TARSKI:def 1; end; let x; assume x in {0.V}; then x = 0.V & 0.GF <> 1_ GF by TARSKI:def 1,VECTSP_1:def 21; then x in {v : f.v <> 0.GF} by A3; hence thesis by VECTSP_6:def 5; end; then Carrier(f) c= A by A2,ZFMISC_1:37; then reconsider f as Linear_Combination of A by VECTSP_6:def 7; Sum(f) = f.(0.V) * 0.V by A5,VECTSP_6:46 .= 0.V by VECTSP_1:60; hence contradiction by A1,A5,Def1; end; theorem Th4: {}(the carrier of V) is linearly-independent proof let l be Linear_Combination of {}(the carrier of V); Carrier(l) c= {} by VECTSP_6:def 7; hence thesis by XBOOLE_1:3; end; theorem {v} is linearly-independent iff v <> 0.V proof thus {v} is linearly-independent implies v <> 0.V proof assume {v} is linearly-independent; then not 0.V in {v} by Th3; hence thesis by TARSKI:def 1; end; assume A1: v <> 0.V; let l be Linear_Combination of {v}; assume A2: Sum(l) = 0.V; A3: Carrier(l) c= {v} by VECTSP_6:def 7; now per cases by A3,ZFMISC_1:39; suppose Carrier(l) = {}; hence thesis; suppose A4: Carrier(l) = {v}; A5: 0.V = l.v * v by A2,VECTSP_6:43; now assume v in Carrier(l); then v in {u : l.u <> 0.GF} by VECTSP_6:def 5; then ex u st v = u & l.u <> 0.GF; hence contradiction by A1,A5,VECTSP_1:60; end; hence thesis by A4,TARSKI:def 1; end; hence thesis; end; theorem Th6: {v1,v2} is linearly-independent implies v1 <> 0.V & v2 <> 0.V proof assume A1: {v1,v2} is linearly-independent; v1 in {v1,v2} & v2 in {v1,v2} by TARSKI:def 2; hence thesis by A1,Th3; end; theorem {v,0.V} is linearly-dependent & {0.V,v} is linearly-dependent by Th6; theorem Th8: v1 <> v2 & {v1,v2} is linearly-independent iff v2 <> 0.V & for a holds v1 <> a * v2 proof thus v1 <> v2 & {v1,v2} is linearly-independent implies v2 <> 0.V & for a holds v1 <> a * v2 proof assume that A1: v1 <> v2 and A2: {v1,v2} is linearly-independent; thus v2 <> 0.V by A2,Th6; let a; assume A3: v1 = a * v2; deffunc F(set) = 0.GF; consider f such that A4: f.v1 = - 1_ GF & f.v2 = a and A5: for v being Element of V st v <> v1 & v <> v2 holds f.v = F(v) from LambdaSep2(A1); reconsider f as Element of Funcs(the carrier of V, the carrier of GF) by FUNCT_2:11; now let v; assume not v in ({v1,v2}); then v <> v1 & v <> v2 by TARSKI:def 2; hence f.v = 0.GF by A5; end; then reconsider f as Linear_Combination of V by VECTSP_6:def 4; Carrier(f) c= {v1,v2} proof let x; assume x in Carrier(f); then x in {u : f.u <> 0.GF} by VECTSP_6:def 5; then A6: ex u st x = u & f.u <> 0.GF; assume not x in {v1,v2}; then x <> v1 & x <> v2 by TARSKI:def 2; hence thesis by A5,A6; end; then reconsider f as Linear_Combination of {v1,v2} by VECTSP_6:def 7; A7: now assume not v1 in Carrier(f); then not v1 in {u : f.u <> 0.GF} by VECTSP_6:def 5; then 0.GF = - 1_ GF by A4; hence contradiction by VECTSP_6:82; end; set w = a * v2; Sum(f) = (- 1_ GF) * w + w by A1,A3,A4,VECTSP_6:44 .= (- w) + w by VECTSP_1:59 .= - (w - w) by VECTSP_1:64 .= - 0.V by VECTSP_1:66 .= 0.V by RLVECT_1:25; hence thesis by A2,A7,Def1; end; assume A8: v2 <> 0.V; assume A9: for a holds v1 <> a * v2; then A10: v1 <> 1_ GF * v2 & 1_ GF * v2 = v2 by VECTSP_1:def 26; hence v1 <> v2; let l be Linear_Combination of {v1,v2}; assume that A11: Sum(l) = 0.V and A12: Carrier(l) <> {}; consider x being Element of Carrier(l); x in Carrier(l) by A12; then x in {u : l.u <> 0.GF} by VECTSP_6:def 5; then A13: ex u st x = u & l.u <> 0.GF; Carrier(l) c= {v1,v2} by VECTSP_6:def 7; then A14: x in {v1,v2} by A12,TARSKI:def 3; A15: 0.V = l.v1 * v1 + l.v2 * v2 by A10,A11,VECTSP_6:44; now per cases by A13,A14,TARSKI:def 2; suppose A16: l.v1 <> 0.GF; 0.V = (l.v1)" * (l.v1 * v1 + l.v2 * v2) by A15,VECTSP_1:60 .= (l.v1)" * (l.v1 * v1) + (l.v1)" * (l.v2 * v2) by VECTSP_1:def 26 .= (l.v1)" * l.v1 * v1 + (l.v1)" * (l.v2 * v2) by VECTSP_1:def 26 .= (l.v1)" * l.v1 * v1 + (l.v1)" * l.v2 * v2 by VECTSP_1:def 26 .= 1_ GF * v1 + (l.v1)" * l.v2 * v2 by A16,VECTSP_1:def 22 .= v1 + (l.v1)" * l.v2 * v2 by VECTSP_1:def 26; then v1 = - ((l.v1)" * l.v2 * v2) by VECTSP_1:63 .= (- 1_ GF) * ((l.v1)" * l.v2 * v2) by VECTSP_1:59 .= ((- 1_ GF) * ((l.v1)" * l.v2)) * v2 by VECTSP_1:def 26; hence thesis by A9; suppose A17: l.v2 <> 0.GF & l.v1 = 0.GF; 0.V = (l.v2)" * (l.v1 * v1 + l.v2 * v2) by A15,VECTSP_1:60 .= (l.v2)" * (l.v1 * v1) + (l.v2)" * (l.v2 * v2) by VECTSP_1:def 26 .= (l.v2)" * l.v1 * v1 + (l.v2)" * (l.v2 * v2) by VECTSP_1:def 26 .= (l.v2)" * l.v1 * v1 + (l.v2)" * l.v2 * v2 by VECTSP_1:def 26 .= (l.v2)" * l.v1 * v1 + 1_ GF * v2 by A17,VECTSP_1:def 22 .= (l.v2)" * l.v1 * v1 + v2 by VECTSP_1:def 26 .= 0.GF * v1 + v2 by A17,VECTSP_1:44 .= 0.V + v2 by VECTSP_1:60 .= v2 by RLVECT_1:10; hence thesis by A8; end; hence thesis; end; theorem v1 <> v2 & {v1,v2} is linearly-independent iff for a,b st a * v1 + b * v2 = 0.V holds a = 0.GF & b = 0.GF proof thus v1 <> v2 & {v1,v2} is linearly-independent implies for a,b st a * v1 + b * v2 = 0.V holds a = 0.GF & b = 0.GF proof assume that A1: v1 <> v2 and A2: {v1,v2} is linearly-independent; let a,b; assume that A3: a * v1 + b * v2 = 0.V and A4: a <> 0.GF or b <> 0.GF; now per cases by A4; suppose A5: a <> 0.GF; 0.V = a" * (a * v1 + b * v2) by A3,VECTSP_1:60 .= a" * (a * v1) + a" * (b * v2) by VECTSP_1:def 26 .= (a" * a) * v1 + a" * (b * v2) by VECTSP_1:def 26 .= (a" * a) * v1 + (a" * b) * v2 by VECTSP_1:def 26 .= 1_ GF * v1 + (a" * b) * v2 by A5,VECTSP_1:def 22 .= v1 + (a" * b) * v2 by VECTSP_1:def 26; then v1 = - ((a" * b) * v2) by VECTSP_1:63 .= (- 1_ GF) * ((a" * b) * v2) by VECTSP_1:59 .= (- 1_ GF) * (a" * b) * v2 by VECTSP_1:def 26; hence thesis by A1,A2,Th8; suppose A6: b <> 0.GF; 0.V = b" * (a * v1 + b * v2) by A3,VECTSP_1:60 .= b" * (a * v1) + b" * (b * v2) by VECTSP_1:def 26 .= (b" * a) * v1 + b" * (b * v2) by VECTSP_1:def 26 .= (b" * a) * v1 + (b" * b) * v2 by VECTSP_1:def 26 .= (b" * a) * v1 + 1_ GF* v2 by A6,VECTSP_1:def 22 .= (b" * a) * v1 + v2 by VECTSP_1:def 26; then v2 = - ((b" * a) * v1) by VECTSP_1:63 .= (- 1_ GF) * ((b" * a) * v1) by VECTSP_1:59 .= (- 1_ GF) * (b" * a) * v1 by VECTSP_1:def 26; hence thesis by A1,A2,Th8; end; hence thesis; end; assume A7: for a,b st a * v1 + b * v2 = 0.V holds a = 0.GF & b = 0.GF; A8: now assume A9: v2 = 0.V; 0.V = 0.V + 0.V by RLVECT_1:10 .= 0.GF * v1 + 0.V by VECTSP_1:60 .= 0.GF * v1 + 1_ GF * v2 by A9,VECTSP_1:60; then 0.GF = 1_ GF by A7; hence contradiction by VECTSP_1:def 21; end; now let a; assume v1 = a * v2; then v1 = 0.V + a * v2 by RLVECT_1:10; then 0.V = v1 - a * v2 by VECTSP_2:22 .= v1 + ((- a) * v2) by VECTSP_1:68 .= 1_ GF * v1 + (- a) * v2 by VECTSP_1:def 26; then 1_ GF = 0.GF by A7; hence contradiction by VECTSP_1:def 21; end; hence thesis by A8,Th8; end; definition let GF; let V; let A; func Lin(A) -> strict Subspace of V means :Def2: the carrier of it = {Sum(l) : not contradiction}; existence proof set A1 = {Sum(l) : not contradiction}; A1 c= the carrier of V proof let x; assume x in A1; then ex l st x = Sum(l); hence thesis; end; then reconsider A1 as Subset of V; reconsider l = ZeroLC(V) as Linear_Combination of A by VECTSP_6:26; Sum(l) = 0.V by VECTSP_6:41; then A1: 0.V in A1; A1 is lineary-closed proof thus for v,u st v in A1 & u in A1 holds v + u in A1 proof let v,u; assume that A2: v in A1 and A3: u in A1; consider l1 being Linear_Combination of A such that A4: v = Sum(l1) by A2; consider l2 being Linear_Combination of A such that A5: u = Sum(l2) by A3; reconsider f = l1 + l2 as Linear_Combination of A by VECTSP_6:52; v + u = Sum(f) by A4,A5,VECTSP_6:77; hence thesis; end; let a,v; assume v in A1; then consider l such that A6: v = Sum(l); reconsider f = a * l as Linear_Combination of A by VECTSP_6:61; a * v = Sum(f) by A6,VECTSP_6:78; hence thesis; end; hence thesis by A1,VECTSP_4:42; end; uniqueness by VECTSP_4:37; end; canceled 2; theorem Th12: x in Lin(A) iff ex l st x = Sum(l) proof thus x in Lin(A) implies ex l st x = Sum(l) proof assume x in Lin(A); then x in the carrier of Lin(A) by RLVECT_1:def 1; then x in {Sum(l) : not contradiction} by Def2; hence thesis; end; given k being Linear_Combination of A such that A1: x = Sum(k); x in {Sum(l): not contradiction} by A1; then x in the carrier of Lin(A) by Def2; hence thesis by RLVECT_1:def 1; end; theorem Th13: x in A implies x in Lin(A) proof assume A1: x in A; then reconsider v = x as Vector of V; deffunc F(set) = 0.GF; consider f being Function of the carrier of V, the carrier of GF such that A2: f.v = 1_ GF and A3: for u st u <> v holds f.u = F(u) from LambdaSep1; reconsider f as Element of Funcs(the carrier of V, the carrier of GF) by FUNCT_2:11; ex T st for u st not u in T holds f.u = 0.GF proof take T = {v}; let u; assume not u in T; then u <> v by TARSKI:def 1; hence thesis by A3; end; then reconsider f as Linear_Combination of V by VECTSP_6:def 4; A4: Carrier(f) c= {v} proof let x; assume x in Carrier(f); then x in {u : f.u <> 0.GF} by VECTSP_6:def 5; then consider u such that A5: x = u and A6: f.u <> 0.GF; u = v by A3,A6; hence thesis by A5,TARSKI:def 1; end; then reconsider f as Linear_Combination of {v} by VECTSP_6:def 7; A7: Sum(f) = 1_ GF * v by A2,VECTSP_6:43 .= v by VECTSP_1:def 26; {v} c= A by A1,ZFMISC_1:37; then Carrier(f) c= A by A4,XBOOLE_1:1; then reconsider f as Linear_Combination of A by VECTSP_6:def 7; Sum(f) = v by A7; hence thesis by Th12; end; reserve l0 for Linear_Combination of {}(the carrier of V); theorem Lin({}(the carrier of V)) = (0).V proof set A = Lin({}(the carrier of V)); now let v; thus v in A implies v in (0).V proof assume v in A; then v in the carrier of A & the carrier of A = {Sum(l0) : not contradiction} by Def2,RLVECT_1: def 1; then ex l0 st v = Sum(l0); then v = 0.V by VECTSP_6:42; hence thesis by VECTSP_4:46; end; assume v in (0).V; then v = 0.V by VECTSP_4:46; hence v in A by VECTSP_4:25; end; hence thesis by VECTSP_4:38; end; theorem Lin(A) = (0).V implies A = {} or A = {0.V} proof assume that A1: Lin(A) = (0).V and A2: A <> {}; thus A c= {0.V} proof let x; assume x in A; then x in Lin(A) by Th13; then x = 0.V by A1,VECTSP_4:46; hence thesis by TARSKI:def 1; end; let x; assume x in {0.V}; then A3: x = 0.V by TARSKI:def 1; consider y being Element of A; A4: y in A by A2; y in Lin(A) by A2,Th13; hence thesis by A1,A3,A4,VECTSP_4:46; end; theorem Th16: for W being strict Subspace of V st A = the carrier of W holds Lin(A) = W proof let W be strict Subspace of V; assume A1: A = the carrier of W; now let v; A2: 0.GF <> 1_ GF by VECTSP_1:def 21; thus v in Lin(A) implies v in W proof assume v in Lin(A); then A3: ex l st v = Sum(l) by Th12; A is lineary-closed & A <> {} by A1,VECTSP_4:41; then v in the carrier of W by A1,A2,A3,VECTSP_6:40; hence thesis by RLVECT_1:def 1; end; v in W iff v in the carrier of W by RLVECT_1:def 1; hence v in W implies v in Lin(A) by A1,Th13; end; hence thesis by VECTSP_4:38; end; theorem for V being strict VectSp of GF, A being Subset of V st A = the carrier of V holds Lin(A) = V proof let V be strict VectSp of GF, A be Subset of V such that A1: A = the carrier of V; (Omega).V = V by VECTSP_4:def 4; hence Lin(A) = V by A1,Th16; end; theorem Th18: A c= B implies Lin(A) is Subspace of Lin(B) proof assume A1: A c= B; now let v; assume v in Lin(A); then consider l such that A2: v = Sum(l) by Th12; reconsider l as Linear_Combination of B by A1,VECTSP_6:25; Sum(l) = v by A2; hence v in Lin(B) by Th12; end; hence thesis by VECTSP_4:36; end; theorem for V being strict VectSp of GF, A,B being Subset of V st Lin(A) = V & A c= B holds Lin(B) = V proof let V be strict VectSp of GF, A,B be Subset of V; assume Lin(A) = V & A c= B; then V is Subspace of Lin(B) by Th18; hence thesis by VECTSP_4:33; end; theorem Lin(A \/ B) = Lin(A) + Lin(B) proof A c= A \/ B & B c= A \/ B by XBOOLE_1:7; then Lin(A) is Subspace of Lin(A \/ B) & Lin(B) is Subspace of Lin(A \/ B) by Th18; then A1: Lin(A) + Lin(B) is Subspace of Lin(A \/ B) by VECTSP_5:40; now let v; assume v in Lin(A \/ B); then consider l being Linear_Combination of A \/ B such that A2: v = Sum(l) by Th12; set C = Carrier(l) /\ A; set D = Carrier(l) \ A; deffunc F(set) = 0.GF; deffunc G(set) = l.$1; defpred P[set] means $1 in C; A3: now let x; assume x in the carrier of V; then reconsider v = x as Vector of V; f.v in the carrier of GF; hence P[x] implies G(x) in the carrier of GF; assume not P[x]; thus F(x) in the carrier of GF; end; consider f being Function of the carrier of V, the carrier of GF such that A4: for x st x in the carrier of V holds (P[x] implies f.x = G(x)) & (not P[x] implies f.x = F(x)) from Lambda1C(A3); reconsider f as Element of Funcs(the carrier of V, the carrier of GF) by FUNCT_2:11; deffunc G(set) = l.$1; defpred C[set] means $1 in D; A5: now let x; assume x in the carrier of V; then reconsider v = x as Vector of V; g.v in the carrier of GF; hence C[x] implies G(x) in the carrier of GF; assume not C[x]; thus F(x) in the carrier of GF; end; consider g being Function of the carrier of V, the carrier of GF such that A6: for x st x in the carrier of V holds (C[x] implies g.x = G(x)) & (not C[x] implies g.x = F(x)) from Lambda1C(A5); reconsider g as Element of Funcs(the carrier of V, the carrier of GF) by FUNCT_2:11; C c= Carrier(l) & Carrier(l) is finite by XBOOLE_1:17; then reconsider C as finite Subset of V by FINSET_1:13; for u holds not u in C implies f.u = 0.GF by A4; then reconsider f as Linear_Combination of V by VECTSP_6:def 4; A7: Carrier(f) c= C proof let x; assume x in Carrier(f); then x in {u : f.u <> 0.GF} by VECTSP_6:def 5; then A8: ex u st x = u & f.u <> 0.GF; assume not x in C; hence thesis by A4,A8; end; C c= A by XBOOLE_1:17; then Carrier(f) c= A by A7,XBOOLE_1:1; then reconsider f as Linear_Combination of A by VECTSP_6:def 7; D c= Carrier(l) & Carrier(l) is finite by XBOOLE_1:36; then reconsider D as finite Subset of V by FINSET_1:13; for u holds not u in D implies g.u = 0.GF by A6; then reconsider g as Linear_Combination of V by VECTSP_6:def 4; A9: Carrier(g) c= D proof let x; assume x in Carrier(g); then x in {u : g.u <> 0.GF} by VECTSP_6:def 5; then A10: ex u st x = u & g.u <> 0.GF; assume not x in D; hence thesis by A6,A10; end; D c= B proof let x; assume x in D; then x in Carrier(l) & not x in A & Carrier(l) c= A \/ B by VECTSP_6:def 7,XBOOLE_0: def 4 ; hence thesis by XBOOLE_0:def 2; end; then Carrier(g) c= B by A9,XBOOLE_1:1; then reconsider g as Linear_Combination of B by VECTSP_6:def 7; l = f + g proof let v; now per cases; suppose A11: v in C; A12: now assume v in D; then not v in A by XBOOLE_0:def 4; hence contradiction by A11,XBOOLE_0:def 3; end; thus (f + g).v = f.v + g.v by VECTSP_6:def 11 .= l.v + g.v by A4,A11 .= l.v + 0.GF by A6,A12 .= l.v by RLVECT_1:10; suppose A13: not v in C; now per cases; suppose A14: v in Carrier(l); A15: now assume not v in D; then not v in Carrier(l) or v in A by XBOOLE_0:def 4; hence contradiction by A13,A14,XBOOLE_0:def 3; end; thus (f + g). v = f.v + g.v by VECTSP_6:def 11 .= g.v + 0.GF by A4,A13 .= g.v by RLVECT_1:10 .= l.v by A6,A15; suppose A16: not v in Carrier(l); then A17: not v in C & not v in D by XBOOLE_0:def 3,def 4; thus (f + g).v = f.v + g.v by VECTSP_6:def 11 .= 0.GF + g.v by A4,A17 .= 0.GF + 0.GF by A6,A17 .= 0.GF by RLVECT_1:10 .= l.v by A16,VECTSP_6:20; end; hence (f + g).v = l.v; end; hence thesis; end; then A18: v = Sum(f) + Sum(g) by A2,VECTSP_6:77; Sum(f) in Lin(A) & Sum(g) in Lin(B) by Th12; hence v in Lin(A) + Lin(B) by A18,VECTSP_5:5; end; then Lin(A \/ B) is Subspace of Lin(A) + Lin(B) by VECTSP_4:36; hence thesis by A1,VECTSP_4:33; end; theorem Lin(A /\ B) is Subspace of Lin(A) /\ Lin(B) proof A /\ B c= A & A /\ B c= B by XBOOLE_1:17; then Lin(A /\ B) is Subspace of Lin(A) & Lin(A /\ B) is Subspace of Lin(B) by Th18; hence thesis by VECTSP_5:24; end; theorem Th22: for V being VectSp of GF, A being Subset of V st A is linearly-independent holds ex B being Subset of V st A c= B & B is linearly-independent & Lin(B) = the VectSpStr of V proof let V be VectSp of GF, A be Subset of V; assume A1: A is linearly-independent; defpred P[set] means ex B being Subset of V st B = $1 & A c= B & B is linearly-independent; consider Q being set such that A2: for Z holds Z in Q iff Z in bool(the carrier of V) & P[Z] from Separation; A3: Q <> {} by A1,A2; now let Z; assume that A4: Z <> {} and A5: Z c= Q and A6: Z is c=-linear; set W = union Z; W c= the carrier of V proof let x; assume x in W; then consider X such that A7: x in X and A8: X in Z by TARSKI:def 4; X in bool(the carrier of V) by A2,A5,A8; hence thesis by A7; end; then reconsider W as Subset of V; consider x being Element of Z; x in Q by A4,A5,TARSKI:def 3; then A9: ex B being Subset of V st B = x & A c= B & B is linearly-independent by A2; x c= W by A4,ZFMISC_1:92; then A10: A c= W by A9,XBOOLE_1:1; W is linearly-independent proof let l be Linear_Combination of W; assume that A11: Sum(l) = 0.V and A12: Carrier(l) <> {}; deffunc F(set) = {C where C is Subset of V: $1 in C & C in Z}; consider f being Function such that A13: dom f = Carrier(l) and A14: for x st x in Carrier(l) holds f.x = F(x) from Lambda; reconsider M = rng f as non empty set by A12,A13,RELAT_1:65; consider F being Choice_Function of M; A15: now assume {} in M; then consider x such that A16: x in dom f and A17: f.x = {} by FUNCT_1:def 5; Carrier(l) c= W by VECTSP_6:def 7; then consider X such that A18: x in X and A19: X in Z by A13, A16,TARSKI:def 4; reconsider X as Subset of V by A2,A5,A19; X in {C where C is Subset of V : x in C & C in Z} by A18,A19; hence contradiction by A13,A14,A16,A17; end; set S = rng F; A20: dom F = M & M <> {} by A15,RLVECT_3:35; then A21: S <> {} by RELAT_1:65; A22: now let X; assume X in S; then consider x such that A23: x in dom F and A24: F.x = X by FUNCT_1:def 5; consider y such that A25: y in dom f and A26: f.y = x by A20,A23,FUNCT_1:def 5; A27: X in x by A15,A20,A23,A24,ORDERS_1:def 1; x = {C where C is Subset of V : y in C & C in Z} by A13,A14,A25,A26; then ex C being Subset of V st C = X & y in C & C in Z by A27; hence X in Z; end; A28: now let X,Y; assume X in S & Y in S; then X in Z & Y in Z by A22; then X,Y are_c=-comparable by A6,ORDINAL1:def 9; hence X c= Y or Y c= X by XBOOLE_0:def 9; end; dom F is finite by A13,A20,FINSET_1:26; then S is finite by FINSET_1:26; then union S in S by A21,A28,RLVECT_3:34; then union S in Z by A22; then consider B being Subset of V such that A29: B = union S and A c= B and A30: B is linearly-independent by A2,A5; Carrier(l) c= union S proof let x; assume A31: x in Carrier(l); then A32: f.x in M by A13,FUNCT_1:def 5; set X = f.x; A33: F.X in X by A15,A32,ORDERS_1:def 1; f.x = {C where C is Subset of V : x in C & C in Z} by A14,A31; then A34: ex C being Subset of V st F.X = C & x in C & C in Z by A33; F.X in S by A20,A32,FUNCT_1:def 5; hence x in union S by A34,TARSKI:def 4; end; then l is Linear_Combination of B by A29,VECTSP_6:def 7; hence thesis by A11,A12,A30,Def1; end; hence union Z in Q by A2,A10; end; then consider X such that A35: X in Q and A36: for Z st Z in Q & Z <> X holds not X c= Z by A3,ORDERS_2:79; consider B being Subset of V such that A37: B = X and A38: A c= B and A39: B is linearly-independent by A2,A35; take B; thus A c= B & B is linearly-independent by A38,A39; A40: the VectSpStr of V = (Omega).V by VECTSP_4:def 4; assume Lin(B) <> the VectSpStr of V; then consider v being Vector of V such that A41: not (v in Lin(B) iff v in (Omega).V) by A40,VECTSP_4:38; A42: B \/ {v} is linearly-independent proof let l be Linear_Combination of B \/ {v}; assume A43: Sum(l) = 0.V; now per cases; suppose A44: v in Carrier(l); deffunc F(Vector of V) = l.$1; consider f being Function of the carrier of V, the carrier of GF such that A45: f.v = 0.GF and A46: for u being Vector of V st u <> v holds f.u = F(u) from LambdaSep1; reconsider f as Element of Funcs(the carrier of V, the carrier of GF) by FUNCT_2:11; now let u be Vector of V; assume not u in Carrier(l) \ {v}; then not u in Carrier(l) or u in {v} by XBOOLE_0:def 4; then (l.u = 0.GF & u <> v) or u = v by TARSKI:def 1,VECTSP_6:20; hence f.u = 0.GF by A45,A46; end; then reconsider f as Linear_Combination of V by VECTSP_6:def 4; Carrier(f) c= B proof let x; assume x in Carrier(f); then x in {u where u is Vector of V : f.u <> 0.GF} by VECTSP_6:def 5; then consider u being Vector of V such that A47: u = x and A48: f.u <> 0.GF; f.u = l.u by A45,A46,A48; then u in {v1 where v1 is Vector of V : l.v1 <> 0.GF} by A48; then u in Carrier(l) & Carrier(l) c= B \/ {v} by VECTSP_6:def 5,def 7 ; then u in B or u in {v} by XBOOLE_0:def 2; hence thesis by A45,A47,A48,TARSKI:def 1; end; then reconsider f as Linear_Combination of B by VECTSP_6:def 7; deffunc G(set) = 0.GF; consider g being Function of the carrier of V, the carrier of GF such that A49: g.v = - l.v and A50: for u being Vector of V st u <> v holds g.u = G(u) from LambdaSep1; reconsider g as Element of Funcs(the carrier of V, the carrier of GF) by FUNCT_2:11; now let u be Vector of V; assume not u in {v}; then u <> v by TARSKI:def 1; hence g.u = 0.GF by A50; end; then reconsider g as Linear_Combination of V by VECTSP_6:def 4; Carrier(g) c= {v} proof let x; assume x in Carrier(g); then x in {u where u is Vector of V : g.u <> 0.GF} by VECTSP_6:def 5; then ex u being Vector of V st x = u & g.u <> 0.GF; then x = v by A50; hence thesis by TARSKI:def 1; end; then reconsider g as Linear_Combination of {v} by VECTSP_6:def 7; A51: f - g = l proof let u be Vector of V; now per cases; suppose A52: v = u; thus (f - g).u = f.u - g.u by VECTSP_6:73 .= 0.GF + (- (- l.v)) by A45,A49,A52,RLVECT_1 :def 11 .= l.v + 0.GF by RLVECT_1:30 .= l.u by A52,RLVECT_1:10; suppose A53: v <> u; thus (f - g).u = f.u - g.u by VECTSP_6:73 .= l.u - g.u by A46,A53 .= l.u - 0.GF by A50,A53 .= l.u by RLVECT_1:26; end; hence thesis; end; A54: Sum(g) = (- l.v) * v by A49,VECTSP_6:43; 0.V = Sum(f) - Sum(g) by A43,A51,VECTSP_6:80; then Sum(f) = 0.V + Sum(g) by VECTSP_2:22 .= (- l.v) * v by A54,RLVECT_1:10; then A55: (- l.v) * v in Lin(B) by Th12; l.v <> 0.GF by A44,VECTSP_6:20; then A56: - l.v <> 0.GF by VECTSP_2:34; (- l.v)" * ((- l.v) * v) = (- l.v)" * (- l.v) * v by VECTSP_1: def 26 .= 1_ GF * v by A56,VECTSP_1:def 22 .= v by VECTSP_1:def 26; hence thesis by A40,A41,A55,RLVECT_1:3,VECTSP_4:29; suppose A57: not v in Carrier(l); Carrier(l) c= B proof let x; assume A58: x in Carrier(l); Carrier(l) c= B \/ {v} by VECTSP_6:def 7; then x in B or x in {v} by A58,XBOOLE_0:def 2; hence thesis by A57,A58,TARSKI:def 1; end; then l is Linear_Combination of B by VECTSP_6:def 7; hence thesis by A39,A43,Def1; end; hence thesis; end; v in {v} by TARSKI:def 1; then A59: v in B \/ {v} & not v in B by A40,A41,Th13,RLVECT_1:3,XBOOLE_0 :def 2; A60: B c= B \/ {v} by XBOOLE_1:7; then A c= B \/ {v} by A38,XBOOLE_1:1; then B \/ {v} in Q by A2,A42; hence contradiction by A36,A37,A59,A60; end; theorem Th23: Lin(A) = V implies ex B st B c= A & B is linearly-independent & Lin(B) = V proof assume A1: Lin(A) = V; defpred P[set] means ex B st B = $1 & B c= A & B is linearly-independent; consider Q being set such that A2: for Z holds Z in Q iff Z in bool(the carrier of V) & P[Z] from Separation; {}(the carrier of V) in bool(the carrier of V) & {}(the carrier of V) c= A & {}(the carrier of V) is linearly-independent by Th4,XBOOLE_1:2; then A3: Q <> {} by A2; now let Z; assume that Z <> {} and A4: Z c= Q and A5: Z is c=-linear; set W = union Z; W c= the carrier of V proof let x; assume x in W; then consider X such that A6: x in X and A7: X in Z by TARSKI:def 4; X in bool(the carrier of V) by A2,A4,A7; hence thesis by A6; end; then reconsider W as Subset of V; A8: W c= A proof let x; assume x in W; then consider X such that A9: x in X and A10: X in Z by TARSKI:def 4; ex B st B = X & B c= A & B is linearly-independent by A2,A4,A10 ; hence thesis by A9; end; W is linearly-independent proof let l be Linear_Combination of W; assume that A11: Sum(l) = 0.V and A12: Carrier(l) <> {}; deffunc F(set) = {C : $1 in C & C in Z}; consider f being Function such that A13: dom f = Carrier(l) and A14: for x st x in Carrier(l) holds f.x = F(x) from Lambda; reconsider M = rng f as non empty set by A12,A13,RELAT_1:65; consider F being Choice_Function of M; A15: now assume {} in M; then consider x such that A16: x in dom f and A17: f.x = {} by FUNCT_1:def 5 ; Carrier(l) c= W by VECTSP_6:def 7; then consider X such that A18: x in X and A19: X in Z by A13, A16,TARSKI:def 4; reconsider X as Subset of V by A2,A4,A19; X in {C : x in C & C in Z} by A18,A19; hence contradiction by A13,A14,A16,A17; end; set S = rng F; A20: dom F = M & M <> {} by A15,RLVECT_3:35; then A21: S <> {} by RELAT_1:65; A22: now let X; assume X in S; then consider x such that A23: x in dom F and A24: F.x = X by FUNCT_1:def 5; consider y such that A25: y in dom f and A26: f.y = x by A20,A23,FUNCT_1:def 5; A27: X in x by A15,A20,A23,A24,ORDERS_1:def 1; x = {C : y in C & C in Z} by A13,A14,A25,A26; then ex C st C = X & y in C & C in Z by A27; hence X in Z; end; A28: now let X,Y; assume X in S & Y in S; then X in Z & Y in Z by A22; then X,Y are_c=-comparable by A5,ORDINAL1:def 9; hence X c= Y or Y c= X by XBOOLE_0:def 9; end; dom F is finite by A13,A20,FINSET_1:26; then S is finite by FINSET_1:26; then union S in S by A21,A28,RLVECT_3:34; then union S in Z by A22; then consider B such that A29: B = union S and B c= A and A30: B is linearly-independent by A2,A4; Carrier(l) c= union S proof let x; assume A31: x in Carrier(l); then A32: f.x in M by A13,FUNCT_1:def 5; set X = f.x; A33: F.X in X by A15,A32,ORDERS_1:def 1; f.x = {C : x in C & C in Z} by A14,A31; then A34: ex C st F.X = C & x in C & C in Z by A33; F.X in S by A20,A32,FUNCT_1:def 5; hence x in union S by A34,TARSKI:def 4; end; then l is Linear_Combination of B by A29,VECTSP_6:def 7; hence thesis by A11,A12,A30,Def1; end; hence union Z in Q by A2,A8; end; then consider X such that A35: X in Q and A36: for Z st Z in Q & Z <> X holds not X c= Z by A3,ORDERS_2:79; consider B such that A37: B = X and A38: B c= A and A39: B is linearly-independent by A2,A35; take B; thus B c= A & B is linearly-independent by A38,A39; assume A40: Lin(B) <> V; now assume A41: for v st v in A holds v in Lin(B); now let v; assume v in Lin(A); then consider l such that A42: v = Sum(l) by Th12; A43: Carrier(l) c= the carrier of Lin(B) proof let x; assume A44: x in Carrier(l); then reconsider a = x as Vector of V; Carrier(l) c= A by VECTSP_6:def 7; then a in Lin(B) by A41,A44; hence thesis by RLVECT_1:def 1; end; reconsider F = the carrier of Lin(B) as Subset of V by VECTSP_4:def 2 ; reconsider l as Linear_Combination of F by A43,VECTSP_6:def 7; Sum(l) = v by A42; then v in Lin(F) by Th12; hence v in Lin(B) by Th16; end; then Lin(A) is Subspace of Lin(B) by VECTSP_4:36; hence contradiction by A1,A40,VECTSP_4:33; end; then consider v such that A45: v in A and A46: not v in Lin(B); A47: B \/ {v} is linearly-independent proof let l be Linear_Combination of B \/ {v}; assume A48: Sum(l) = 0.V; now per cases; suppose A49: v in Carrier(l); deffunc F(Vector of V) = l.$1; consider f such that A50: f.v = 0.GF and A51: for u st u <> v holds f.u = F(u) from LambdaSep1; reconsider f as Element of Funcs(the carrier of V, the carrier of GF) by FUNCT_2:11; now let u; assume not u in Carrier(l) \ {v}; then not u in Carrier(l) or u in {v} by XBOOLE_0:def 4; then (l.u = 0.GF & u <> v) or u = v by TARSKI:def 1,VECTSP_6: 20; hence f.u = 0.GF by A50,A51; end; then reconsider f as Linear_Combination of V by VECTSP_6:def 4; Carrier(f) c= B proof let x; assume x in Carrier(f); then x in {u : f.u <> 0.GF} by VECTSP_6:def 5; then consider u such that A52: u = x and A53: f.u <> 0.GF; f.u = l.u by A50,A51,A53; then u in {v1 : l.v1 <> 0.GF} by A53; then u in Carrier(l) & Carrier(l) c= B \/ {v} by VECTSP_6:def 5,def 7 ; then u in B or u in {v} by XBOOLE_0:def 2; hence thesis by A50,A52,A53,TARSKI:def 1; end; then reconsider f as Linear_Combination of B by VECTSP_6:def 7; deffunc G(Vector of V) = 0.GF; consider g such that A54: g.v = - l.v and A55: for u st u <> v holds g.u = G(u) from LambdaSep1; reconsider g as Element of Funcs(the carrier of V, the carrier of GF) by FUNCT_2:11; now let u; assume not u in {v}; then u <> v by TARSKI:def 1; hence g.u = 0.GF by A55; end; then reconsider g as Linear_Combination of V by VECTSP_6:def 4; Carrier(g) c= {v} proof let x; assume x in Carrier(g); then x in {u : g.u <> 0.GF} by VECTSP_6:def 5; then ex u st x = u & g.u <> 0.GF; then x = v by A55; hence thesis by TARSKI:def 1; end; then reconsider g as Linear_Combination of {v} by VECTSP_6:def 7; A56: f - g = l proof let u; now per cases; suppose A57: v = u; thus (f - g).u = f.u - g.u by VECTSP_6:73 .= 0.GF + (- (- l.v)) by A50,A54,A57,RLVECT_1 :def 11 .= l.v + 0.GF by RLVECT_1:30 .= l.u by A57,RLVECT_1:10; suppose A58: v <> u; thus (f - g).u = f.u - g.u by VECTSP_6:73 .= l.u - g.u by A51,A58 .= l.u - 0.GF by A55,A58 .= l.u by RLVECT_1:26; end; hence thesis; end; A59: Sum(g) = (- l.v) * v by A54,VECTSP_6:43; 0.V = Sum(f) - Sum(g) by A48,A56,VECTSP_6:80; then Sum(f) = 0.V + Sum(g) by VECTSP_2:22 .= (- l.v) * v by A59,RLVECT_1:10; then A60: (- l.v) * v in Lin(B) by Th12; l.v <> 0.GF by A49,VECTSP_6:20; then A61: - l.v <> 0.GF by VECTSP_2:34; (- l.v)" * ((- l.v) * v) = (- l.v)" * (- l.v) * v by VECTSP_1: def 26 .= 1_ GF * v by A61,VECTSP_1:def 22 .= v by VECTSP_1:def 26; hence thesis by A46,A60,VECTSP_4:29; suppose A62: not v in Carrier(l); Carrier(l) c= B proof let x; assume A63: x in Carrier(l); Carrier(l) c= B \/ {v} by VECTSP_6:def 7; then x in B or x in {v} by A63,XBOOLE_0:def 2; hence thesis by A62,A63,TARSKI:def 1; end; then l is Linear_Combination of B by VECTSP_6:def 7; hence thesis by A39,A48,Def1; end; hence thesis; end; v in {v} by TARSKI:def 1; then A64: v in B \/ {v} & not v in B by A46,Th13,XBOOLE_0:def 2; A65: B c= B \/ {v} by XBOOLE_1:7; {v} c= A by A45,ZFMISC_1:37; then B \/ {v} c= A by A38,XBOOLE_1:8; then B \/ {v} in Q by A2,A47; hence contradiction by A36,A37,A64,A65; end; definition let GF; let V be VectSp of GF; mode Basis of V -> Subset of V means :Def3: it is linearly-independent & Lin(it) = the VectSpStr of V; existence proof {}(the carrier of V) is linearly-independent by Th4; then ex B being Subset of V st {}(the carrier of V) c= B & B is linearly-independent & Lin(B) = the VectSpStr of V by Th22; hence thesis; end; end; canceled 3; theorem for V being VectSp of GF, A being Subset of V st A is linearly-independent holds ex I being Basis of V st A c= I proof let V be VectSp of GF, A be Subset of V; assume A is linearly-independent; then consider B being Subset of V such that A1: A c= B and A2: B is linearly-independent & Lin(B) = the VectSpStr of V by Th22; reconsider B as Basis of V by A2,Def3; take B; thus thesis by A1; end; theorem for V being VectSp of GF, A being Subset of V st Lin(A) = V holds ex I being Basis of V st I c= A proof let V be VectSp of GF, A be Subset of V; assume Lin(A) = V; then consider B being Subset of V such that A1: B c= A and A2: B is linearly-independent & Lin(B) = V by Th23; reconsider B as Basis of V by A2,Def3; take B; thus thesis by A1; end;

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