### The Mizar article:

### Dimension of Real Unitary Space

**by****Noboru Endou,****Takashi Mitsuishi, and****Yasunari Shidama**

- Received October 9, 2002
Copyright (c) 2002 Association of Mizar Users

- MML identifier: RUSUB_4
- [ MML identifier index ]

environ vocabulary RLVECT_1, RLSUB_1, FINSET_1, RLVECT_2, FINSEQ_1, FUNCT_1, SEQ_1, RELAT_1, FINSEQ_4, BOOLE, FUNCT_2, ARYTM_1, TARSKI, CARD_1, RLVECT_3, BHSP_1, CONNSP_3, SUBSET_1, RLSUB_2, MATRLIN, VECTSP_9, RUSUB_4, ARYTM_3; notation TARSKI, XBOOLE_0, ENUMSET1, SUBSET_1, XREAL_0, REAL_1, NAT_1, FINSEQ_1, FUNCT_1, CARD_1, FUNCT_2, PRE_TOPC, STRUCT_0, RLVECT_1, FINSEQ_3, FINSEQ_4, FINSET_1, RLSUB_1, RLVECT_2, BHSP_1, RLVECT_3, RUSUB_1, RUSUB_2, RUSUB_3; constructors NAT_1, REAL_1, RLVECT_2, FINSEQ_4, DOMAIN_1, RLVECT_3, RUSUB_2, FINSEQ_3, RUSUB_3, PRE_TOPC, XREAL_0, MEMBERED; clusters SUBSET_1, FINSET_1, RELSET_1, STRUCT_0, FINSEQ_1, PRE_TOPC, RLVECT_1, RLSUB_1, NAT_1, MEMBERED; requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM; definitions XBOOLE_0, TARSKI; theorems FUNCT_1, FINSEQ_1, FINSEQ_2, FINSEQ_3, FINSEQ_4, FINSET_1, REAL_1, RLSUB_2, RLVECT_1, RLVECT_2, TARSKI, XBOOLE_0, XBOOLE_1, RUSUB_1, RUSUB_2, RLVECT_3, ENUMSET1, AXIOMS, RLVECT_5, NAT_1, SUBSET_1, RUSUB_3, RLSUB_1, CARD_1, CARD_2, MATRLIN, PRE_TOPC, VECTSP_9, XCMPLX_0, XCMPLX_1; schemes NAT_1, XBOOLE_0, FINSEQ_1, COMPLSP1; begin :: Finite-dimensional real unitary space theorem Th1: for V being RealUnitarySpace, A,B being finite Subset of V, v being VECTOR of V st v in Lin(A \/ B) & not v in Lin(B) holds ex w being VECTOR of V st w in A & w in Lin(A \/ B \ {w} \/ {v}) proof let V be RealUnitarySpace; let A, B be finite Subset of V; let v be VECTOR of V such that A1:v in Lin(A \/ B) and A2:not v in Lin(B); consider L being Linear_Combination of (A \/ B) such that A3:v = Sum(L) by A1,RUSUB_3:1; A4:Carrier(L) c= A \/ B by RLVECT_2:def 8; now assume A5: for w being VECTOR of V st w in A holds L.w = 0; now let x be set; assume A6: x in Carrier(L) & x in A; then consider u being VECTOR of V such that A7: x = u & L.u <> 0 by RLVECT_5:3; thus contradiction by A5,A6,A7; end; then Carrier(L) misses A by XBOOLE_0:3; then Carrier(L) c= B by A4,XBOOLE_1:73; then L is Linear_Combination of B by RLVECT_2:def 8; hence contradiction by A2,A3,RUSUB_3:1; end; then consider w being VECTOR of V such that A8:w in A and A9:L.w <> 0; take w; set a = L.w; consider F being FinSequence of the carrier of V such that A10:F is one-to-one and A11:rng F = Carrier(L) and A12:Sum(L) = Sum(L (#) F) by RLVECT_2:def 10; A13:w in rng F by A9,A11,RLVECT_5:3; then A14:F = (F -| w) ^ <* w *> ^ (F |-- w) by FINSEQ_4:66; reconsider Fw1 = (F -| w) as FinSequence of the carrier of V by A13,FINSEQ_4:53; reconsider Fw2 = (F |-- w) as FinSequence of the carrier of V by A13,FINSEQ_4:65; F = Fw1 ^ (<* w *> ^ Fw2) by A14,FINSEQ_1:45; then L (#) F = (L (#) Fw1) ^ (L (#) (<* w *> ^ Fw2)) by RLVECT_3:41 .= (L (#) Fw1) ^ ((L (#) <* w *>) ^ (L (#) Fw2)) by RLVECT_3:41 .= (L (#) Fw1) ^ (L (#) <* w *>) ^ (L (#) Fw2) by FINSEQ_1:45 .= (L (#) Fw1) ^ <* a*w *> ^ (L (#) Fw2) by RLVECT_2:42; then A15:Sum(L (#) F) = Sum((L (#) Fw1) ^ (<* a*w *> ^ (L (#) Fw2))) by FINSEQ_1:45 .= Sum(L (#) Fw1) + Sum(<* a*w *> ^ (L (#) Fw2)) by RLVECT_1:58 .= Sum(L (#) Fw1) + (Sum(<* a*w *>) + Sum(L (#) Fw2)) by RLVECT_1:58 .= Sum(L (#) Fw1) + (Sum(L (#) Fw2) + a*w) by RLVECT_1:61 .= (Sum(L (#) Fw1) + Sum(L (#) Fw2)) + a*w by RLVECT_1:def 6 .= Sum((L (#) Fw1) ^ (L (#) Fw2)) + a*w by RLVECT_1:58 .= Sum(L (#) (Fw1 ^ Fw2)) + a*w by RLVECT_3:41; set Fw = Fw1 ^ Fw2; consider K being Linear_Combination of V such that A16:Carrier(K) = rng Fw /\ Carrier(L) & L (#) Fw = K (#) Fw by RLVECT_5:8; F just_once_values w by A10,A13,FINSEQ_4:10; then Fw = F - {w} by FINSEQ_4:69; then A17:rng Fw = Carrier(L) \ {w} by A11,FINSEQ_3:72; then rng Fw c= Carrier(L) by XBOOLE_1:36; then A18:Carrier(K) = rng Fw by A16,XBOOLE_1:28; then A19:Carrier(K) c= A \/ B \ {w} by A4,A17,XBOOLE_1:33; then reconsider K as Linear_Combination of (A \/ B \ {w}) by RLVECT_2:def 8; Fw1 is one-to-one & Fw2 is one-to-one & rng Fw1 misses rng Fw2 by A10,A13,FINSEQ_4:67,68,72; then Fw is one-to-one by FINSEQ_3:98; then Sum(K (#) Fw) = Sum(K) by A18,RLVECT_2:def 10; then a"*v = a"*Sum(K) + a"*(a*w) by A3,A12,A15,A16,RLVECT_1:def 9 .= a"*Sum(K) + (a"*a)*w by RLVECT_1:def 9 .= a"*Sum(K) +1*w by A9,XCMPLX_0:def 7 .= a"*Sum(K) + w by RLVECT_1:def 9; then A20:w = a"*v - a"*Sum(K) by RLSUB_2:78 .= a"*(v - Sum(K)) by RLVECT_1:48 .= a"*(-Sum(K) + v) by RLVECT_1:def 11; v in {v} by TARSKI:def 1; then v in Lin({v}) by RUSUB_3:2; then consider Lv being Linear_Combination of {v} such that A21:v = Sum(Lv) by RUSUB_3:1; A22:w = a"*(Sum(-K) + Sum(Lv)) by A20,A21,RLVECT_3:3 .= a"*Sum(-K + Lv) by RLVECT_3:1 .= Sum(a"*(-K + Lv)) by RLVECT_3:2; A23:a" <> 0 by A9,XCMPLX_1:203; set LC = a"*(-K + Lv); A24:Carrier(Lv) c= {v} by RLVECT_2:def 8; Carrier (-K + Lv) c= Carrier(-K) \/ Carrier(Lv) by RLVECT_2:58; then A25:Carrier (-K + Lv) c= Carrier(K) \/ Carrier(Lv) by RLVECT_2:75; Carrier(K) \/ Carrier(Lv) c= A \/ B \ {w} \/ {v} by A19,A24,XBOOLE_1:13; then Carrier (-K + Lv) c= A \/ B \ {w} \/ {v} by A25,XBOOLE_1:1; then Carrier (LC) c= A \/ B \ {w} \/ {v} by A23,RLVECT_2:65; then LC is Linear_Combination of (A \/ B \ {w} \/ {v}) by RLVECT_2:def 8; hence thesis by A8,A22,RUSUB_3:1; end; Lm1: for X being set, x be set st x in X holds X \ {x} \/ {x} = X proof let X be set; let x be set; assume x in X; then A1: {x} is Subset of X by SUBSET_1:63; {x} \/ (X \ {x}) = {x} \/ X by XBOOLE_1:39 .= X by A1,XBOOLE_1:12; hence thesis; end; Lm2: for X being set, x being set st not x in X holds X \ {x} = X proof let X be set, x be set such that A1: not x in X; now assume X meets {x}; then consider y being set such that A2: y in X /\ {x} by XBOOLE_0:4; y in X & y in {x} by A2,XBOOLE_0:def 3; hence contradiction by A1,TARSKI:def 1; end; hence X \ {x} = X by XBOOLE_1:83; end; theorem Th2: for V being RealUnitarySpace, A,B being finite Subset of V st the UNITSTR of V = Lin(A) & B is linearly-independent holds Card B <= Card A & ex C being finite Subset of V st C c= A & Card C = Card A - Card B & the UNITSTR of V = Lin(B \/ C) proof let V be RealUnitarySpace; let A, B be finite Subset of V such that A1:the UNITSTR of V = Lin(A) and A2:B is linearly-independent; defpred P[Nat] means for n being Nat for A, B being finite Subset of V st card(A) = n & card(B) = $1 & the UNITSTR of V = Lin(A) & B is linearly-independent holds $1 <= n & ex C being finite Subset of V st C c= A & card(C) = n - $1 & the UNITSTR of V = Lin(B \/ C); A3:P[0] proof let n be Nat; let A, B be finite Subset of V such that A4: card(A) = n & card(B) = 0 & the UNITSTR of V = Lin(A) & B is linearly-independent; B = {} by A4,CARD_2:59; then A = B \/ A; hence thesis by A4,NAT_1:18; end; A5:for m being Nat st P[m] holds P[m + 1] proof let m be Nat such that A6: P[m]; let n be Nat; let A, B be finite Subset of V such that A7: card(A) = n and A8: card(B) = m + 1 and A9: the UNITSTR of V = Lin(A) and A10: B is linearly-independent; consider v being set such that A11: v in B by A8,CARD_1:47,XBOOLE_0:def 1; reconsider v as VECTOR of V by A11; {v} is Subset of B by A11,SUBSET_1:63; then A12: card(B \ {v}) = card(B) - card({v}) by CARD_2:63 .= m + 1 - 1 by A8,CARD_1:79 .= m by XCMPLX_1:26; set Bv = B \ {v}; A13: Bv c= B by XBOOLE_1:36; then A14: Bv is linearly-independent by A10,RLVECT_3:6; A15: not v in Lin(Bv) by A10,A11,RUSUB_3:25; now assume m = n; then consider C being finite Subset of V such that A16: C c= A & card(C) = m - m & the UNITSTR of V = Lin(Bv \/ C) by A6,A7,A9,A12,A14; card(C) = 0 by A16,XCMPLX_1:14; then C = {} by CARD_2:59; then Bv is Basis of V by A14,A16,RUSUB_3:def 2; hence contradiction by A15,RUSUB_3:21; end; then m <> n & m <= n by A6,A7,A9,A12,A14; then A17: m < n by REAL_1:def 5; consider C being finite Subset of V such that A18: C c= A and A19: card(C) = n - m and A20: the UNITSTR of V = Lin(Bv \/ C) by A6,A7,A9,A12,A14; v in Lin(Bv \/ C) by A20,RLVECT_1:def 1; then consider w being VECTOR of V such that A21: w in C and A22: w in Lin(C \/ Bv \ {w} \/ {v}) by A15,Th1; set Cw = C \ {w}; Cw c= C by XBOOLE_1:36; then A23: Cw c= A by A18,XBOOLE_1:1; {w} is Subset of C by A21,SUBSET_1:63; then A24: card(Cw) = card(C) - card({w}) by CARD_2:63 .= n - m - 1 by A19,CARD_1:79 .= n - (m + 1) by XCMPLX_1:36; A25: C = Cw \/ {w} by A21,Lm1; A26: C \/ Bv \ {w} \/ {v} = (Cw \/ (Bv \ {w})) \/ {v} by XBOOLE_1:42 .= Cw \/ ((Bv \ {w}) \/ {v}) by XBOOLE_1:4; Bv \ {w} c= Bv by XBOOLE_1:36; then (Bv \ {w}) \/ {v} c= Bv \/ {v} by XBOOLE_1:9; then Cw \/ ((Bv \ {w}) \/ {v}) c= Cw \/ (Bv \/ {v}) by XBOOLE_1:9; then Cw \/ ((Bv \ {w}) \/ {v}) c= B \/ Cw by A11,Lm1; then Lin(C \/ Bv \ {w} \/ {v}) is Subspace of Lin(B \/ Cw) by A26,RUSUB_3:7; then A27: w in Lin(B \/ Cw) by A22,RUSUB_1:1; now let x be set; assume x in Bv \/ C; then x in Bv or x in C by XBOOLE_0:def 2; then x in B or x in Cw or x in {w} by A13,A25,XBOOLE_0:def 2; then x in B \/ Cw or x in {w} by XBOOLE_0:def 2; then x in Lin(B \/ Cw) or x = w by RUSUB_3:2,TARSKI:def 1; hence x in the carrier of Lin(B \/ Cw) by A27,RLVECT_1:def 1; end; then Bv \/ C c= the carrier of Lin(B \/ Cw) by TARSKI:def 3; then Lin(Bv \/ C) is Subspace of Lin(B \/ Cw) by RUSUB_3:27; then the carrier of Lin(Bv \/ C) c= the carrier of Lin(B \/ Cw) & the carrier of Lin(B \/ Cw) c= the carrier of V by RUSUB_1:def 1; then the carrier of Lin(B \/ Cw) = the carrier of V by A20,XBOOLE_0:def 10; then the UNITSTR of V = Lin(B \/ Cw) by A20,RUSUB_1:24; hence thesis by A17,A23,A24,NAT_1:38; end; for m being Nat holds P[m] from Ind(A3, A5); hence thesis by A1,A2; end; definition let V be RealUnitarySpace; attr V is finite-dimensional means :Def1: ex A being finite Subset of V st A is Basis of V; end; definition cluster strict finite-dimensional RealUnitarySpace; existence proof consider V being RealUnitarySpace; take (0).V; thus (0).V is strict; take A = {}( the carrier of (0).V ); A1:A is linearly-independent by RLVECT_3:8; Lin(A) = (0).(0).V by RUSUB_3:3; then Lin(A) = the UNITSTR of (0).V by RUSUB_1:30; hence A is Basis of (0).V by A1,RUSUB_3:def 2; end; end; definition let V be RealUnitarySpace; redefine attr V is finite-dimensional means :Def2: ex I being finite Subset of V st I is Basis of V; compatibility by Def1; end; theorem Th3: for V being RealUnitarySpace st V is finite-dimensional holds for I being Basis of V holds I is finite proof let V be RealUnitarySpace; assume V is finite-dimensional; then consider A being finite Subset of V such that A1:A is Basis of V by Def2; let B be Basis of V; consider p being FinSequence such that A2:rng p = A by FINSEQ_1:73; reconsider p as FinSequence of the carrier of V by A2,FINSEQ_1:def 4; set Car = {Carrier(L) where L is Linear_Combination of B: ex i being Nat st i in dom p & Sum(L) = p.i}; set C = union Car; A3:C c= B proof let x be set; assume x in C; then consider R being set such that A4: x in R and A5: R in Car by TARSKI:def 4; consider L being Linear_Combination of B such that A6: R = Carrier(L) and ex i being Nat st i in dom p & Sum(L) = p.i by A5; R c= B by A6,RLVECT_2:def 8; hence x in B by A4; end; then C c= the carrier of V by XBOOLE_1:1; then reconsider C as Subset of V; A7:B is linearly-independent by RUSUB_3:def 2; then A8:C is linearly-independent by A3,RLVECT_3:6; for v being VECTOR of V holds v in (Omega).V iff v in Lin(C) proof let v be VECTOR of V; hereby assume v in (Omega).V; then v in the UNITSTR of V by RUSUB_1:def 3; then v in Lin(A) by A1,RUSUB_3:def 2; then consider LA being Linear_Combination of A such that A9: v = Sum(LA) by RUSUB_3:1; Carrier(LA) c= the carrier of Lin(C) proof let w be set; assume A10: w in Carrier(LA); then A11: w in Carrier(LA) & Carrier(LA) c= A by RLVECT_2:def 8; reconsider w'= w as VECTOR of V by A10; w' in Lin(B) by RUSUB_3:21; then consider LB being Linear_Combination of B such that A12: w = Sum(LB) by RUSUB_3:1; ex i being set st i in dom p & w = p.i by A2,A11,FUNCT_1:def 5; then A13: Carrier(LB) in Car by A12; Carrier(LB) c= C proof let x be set; assume x in Carrier(LB); hence x in C by A13,TARSKI:def 4; end; then LB is Linear_Combination of C by RLVECT_2:def 8; then w in Lin(C) by A12,RUSUB_3:1; hence w in the carrier of Lin(C) by RLVECT_1:def 1; end; then consider LC being Linear_Combination of C such that A14: Sum(LA) = Sum(LC) by RUSUB_3:17; thus v in Lin(C) by A9,A14,RUSUB_3:1; end; assume v in Lin(C); v in the carrier of the UNITSTR of V; then v in the carrier of (Omega).V by RUSUB_1:def 3; hence thesis by RLVECT_1:def 1; end; then (Omega).V = Lin(C) by RUSUB_1:25; then the UNITSTR of V = Lin(C) by RUSUB_1:def 3; then A15:C is Basis of V by A8,RUSUB_3:def 2; B c= C proof assume not B c= C; then consider v being set such that A16: v in B and A17: not v in C by TARSKI:def 3; set D = B \ C; A18: D misses C by XBOOLE_1:79; reconsider B as Subset of V; reconsider D as non empty Subset of V by A16,A17,XBOOLE_0:def 4; C \/ (B \ C) = C \/ B by XBOOLE_1:39 .= B by A3,XBOOLE_1:12; then B = C \/ D; hence contradiction by A7,A15,A18,RUSUB_3:26; end; then A19:B = C by A3,XBOOLE_0:def 10; defpred P[set, set] means ex L being Linear_Combination of B st $2 = Carrier(L) & Sum(L) = p.$1; A20:for i being Nat, y1,y2 being set st i in Seg len p & P[i, y1] & P[i, y2] holds y1 = y2 proof let i be Nat; let y1,y2 be set; assume that i in Seg len p and A21: P[i, y1] and A22: P[i, y2]; consider L1 being Linear_Combination of B such that A23: y1 = Carrier(L1) and A24: Sum(L1) = p.i by A21; consider L2 being Linear_Combination of B such that A25: y2 = Carrier(L2) and A26: Sum(L2) = p.i by A22; Carrier(L1) c= B & Carrier(L2) c= B by RLVECT_2:def 8; hence y1 = y2 by A7,A23,A24,A25,A26,RLVECT_5:1; end; A27:for i being Nat st i in Seg len p ex x being set st P[i, x] proof let i be Nat; assume i in Seg len p; then i in dom p by FINSEQ_1:def 3; then p.i in the carrier of V by FINSEQ_2:13; then p.i in Lin(B) by RUSUB_3:21; then consider L being Linear_Combination of B such that A28: p.i = Sum(L) by RUSUB_3:1; P[i, Carrier(L)] by A28; hence thesis; end; ex q being FinSequence st dom q = Seg len p & for i being Nat st i in Seg len p holds P[i, q.i] from SeqEx(A20, A27); then consider q being FinSequence such that A29:dom q = Seg len p & for i being Nat st i in Seg len p holds P[i, q.i]; A30:dom p = dom q by A29,FINSEQ_1:def 3; now let x be set; assume x in rng q; then consider i being set such that A31: i in dom q and A32: x = q.i by FUNCT_1:def 5; reconsider i as Nat by A31; consider L being Linear_Combination of B such that A33: x = Carrier(L) and A34: Sum(L) = p.i by A29,A31,A32; thus x in Car by A30,A31,A33,A34; end; then A35:rng q c= Car by TARSKI:def 3; now let x be set; assume x in Car; then consider L being Linear_Combination of B such that A36: x = Carrier(L) and A37: ex i being Nat st i in dom p & Sum(L) = p.i; consider i being Nat such that A38: i in dom p & Sum(L) = p.i by A37; P[i, x] & P[i, q.i] by A29,A30,A36,A38; then x = q.i by A20,A29,A30,A38; hence x in rng q by A30,A38,FUNCT_1:def 5; end; then Car c= rng q by TARSKI:def 3; then A39:Car is finite by A35,XBOOLE_0:def 10; for R being set st R in Car holds R is finite proof let R be set; assume R in Car; then consider L being Linear_Combination of B such that A40: R = Carrier(L) and ex i being Nat st i in dom p & Sum(L) = p.i; thus R is finite by A40; end; hence B is finite by A19,A39,FINSET_1:25; end; theorem for V being RealUnitarySpace, A being Subset of V st V is finite-dimensional & A is linearly-independent holds A is finite proof let V be RealUnitarySpace; let A be Subset of V; assume A1:V is finite-dimensional & A is linearly-independent; then consider B being Basis of V such that A2:A c= B by RUSUB_3:15; B is finite by A1,Th3; hence A is finite by A2,FINSET_1:13; end; theorem Th5: for V being RealUnitarySpace, A,B being Basis of V st V is finite-dimensional holds Card A = Card B proof let V be RealUnitarySpace; let A, B be Basis of V; assume V is finite-dimensional; then reconsider A'= A, B'= B as finite Subset of V by Th3; A1:the UNITSTR of V = Lin(A) by RUSUB_3:def 2; B' is linearly-independent by RUSUB_3:def 2; then A2:Card B' <= Card A' by A1,Th2; A3:the UNITSTR of V = Lin(B) by RUSUB_3:def 2; A' is linearly-independent by RUSUB_3:def 2; then Card A' <= Card B' by A3,Th2; hence Card A = Card B by A2,AXIOMS:21; end; theorem Th6: for V being RealUnitarySpace holds (0).V is finite-dimensional proof let V be RealUnitarySpace; reconsider V'= (0).V as strict RealUnitarySpace; the carrier of V' = {0.V} by RUSUB_1:def 2 .= {0.V'} by RUSUB_1:4 .= the carrier of (0).V' by RUSUB_1:def 2; then A1:V' = (0).V' by RUSUB_1:26; reconsider I = {}(the carrier of V') as finite Subset of V' ; A2:I is linearly-independent by RLVECT_3:8; Lin(I) = (0).V' by RUSUB_3:3; then I is Basis of V' by A1,A2,RUSUB_3:def 2; hence thesis by Def2; end; theorem Th7: for V being RealUnitarySpace, W being Subspace of V st V is finite-dimensional holds W is finite-dimensional proof let V be RealUnitarySpace; let W be Subspace of V; assume A1:V is finite-dimensional; consider A being Basis of W; consider I being Basis of V such that A2:A c= I by RUSUB_3:24; I is finite by A1,Th3; then A is finite by A2,FINSET_1:13; hence thesis by Def1; end; definition let V be RealUnitarySpace; cluster finite-dimensional strict Subspace of V; existence proof take (0).V; thus thesis by Th6; end; end; definition let V be finite-dimensional RealUnitarySpace; cluster -> finite-dimensional Subspace of V; correctness by Th7; end; definition let V be finite-dimensional RealUnitarySpace; cluster strict Subspace of V; existence proof (0).V is strict finite-dimensional Subspace of V; hence thesis; end; end; begin :: Dimension of real unitary space definition let V be RealUnitarySpace; assume A1: V is finite-dimensional; func dim V -> Nat means :Def3: for I being Basis of V holds it = Card I; existence proof consider A being finite Subset of V such that A2:A is Basis of V by A1,Def2; consider n being Nat such that A3:n = Card A; for I being Basis of V holds Card I = n by A1,A2,A3,Th5; hence thesis; end; uniqueness proof let n, m be Nat; assume that A4: for I being Basis of V holds Card I = n and A5: for I being Basis of V holds Card I = m; consider A being finite Subset of V such that A6: A is Basis of V by A1,Def2; Card A = n & Card A = m by A4,A5,A6; hence n = m; end; end; theorem Th8: for V being finite-dimensional RealUnitarySpace, W being Subspace of V holds dim W <= dim V proof let V be finite-dimensional RealUnitarySpace; let W be Subspace of V; reconsider V'= V as RealUnitarySpace; consider I being Basis of V'; A1:Lin(I) = the UNITSTR of V' by RUSUB_3:def 2; reconsider I as finite Subset of V by Th3; A2:dim V = Card I by Def3; consider A being Basis of W; reconsider A as Subset of W; A is linearly-independent by RUSUB_3:def 2; then consider B being Subset of V such that A3:B is linearly-independent & B = A by RUSUB_3:22; reconsider A'= A as finite Subset of V by A3,Th3; Card A' <= Card I by A1,A3,Th2; hence dim W <= dim V by A2,Def3; end; theorem Th9: for V being finite-dimensional RealUnitarySpace, A being Subset of V st A is linearly-independent holds Card A = dim Lin(A) proof let V be finite-dimensional RealUnitarySpace; let A be Subset of V such that A1:A is linearly-independent; set W = Lin(A); now let x be set; assume x in A; then x in W by RUSUB_3:2; hence x in the carrier of W by RLVECT_1:def 1; end; then A c= the carrier of W by TARSKI:def 3; then consider B being Subset of W such that A2:B is linearly-independent & B = A by A1,RUSUB_3:23; W = Lin(B) by A2,RUSUB_3:28; then reconsider B as Basis of W by A2,RUSUB_3:def 2; Card B = dim W by Def3; hence Card A = dim Lin(A) by A2; end; theorem Th10: for V being finite-dimensional RealUnitarySpace holds dim V = dim (Omega).V proof let V be finite-dimensional RealUnitarySpace; consider I being finite Subset of V such that A1:I is Basis of V by Def2; A2:Card I = dim V by A1,Def3; A3:I is linearly-independent by A1,RUSUB_3:def 2; (Omega).V = the UNITSTR of V by RUSUB_1:def 3 .= Lin(I) by A1,RUSUB_3:def 2; hence thesis by A2,A3,Th9; end; theorem for V being finite-dimensional RealUnitarySpace, W being Subspace of V holds dim V = dim W iff (Omega).V = (Omega).W proof let V be finite-dimensional RealUnitarySpace; let W be Subspace of V; hereby assume A1: dim V = dim W; consider A being Basis of W; consider B being Basis of V such that A2: A c= B by RUSUB_3:24; A3: Card A = dim V by A1,Def3 .= Card B by Def3; A c= the carrier of W & the carrier of W c= the carrier of V by RUSUB_1:def 1; then A c= the carrier of V & A is finite by Th3,XBOOLE_1:1; then reconsider A'= A as finite Subset of V; reconsider B'= B as finite Subset of V by Th3; A4: now assume A <> B; then A c< B by A2,XBOOLE_0:def 8; then Card A' < Card B' by CARD_2:67; hence contradiction by A3; end; reconsider A as Subset of W; reconsider B as Subset of V; (Omega).V = the UNITSTR of V by RUSUB_1:def 3 .= Lin(B) by RUSUB_3:def 2 .= Lin(A) by A4,RUSUB_3:28 .= the UNITSTR of W by RUSUB_3:def 2 .= (Omega).W by RUSUB_1:def 3; hence (Omega).V = (Omega).W; end; assume (Omega).V = (Omega).W; then A5:the UNITSTR of V = (Omega).W by RUSUB_1:def 3 .= the UNITSTR of W by RUSUB_1:def 3; consider A being finite Subset of V such that A6:A is Basis of V by Def2; consider B being finite Subset of W such that A7:B is Basis of W by Def2; A8:A is linearly-independent by A6,RUSUB_3:def 2; A9:B is linearly-independent by A7,RUSUB_3:def 2; A10:Lin(A) = the UNITSTR of W by A5,A6,RUSUB_3:def 2 .= Lin(B) by A7,RUSUB_3:def 2; reconsider A as Subset of V; reconsider B as Subset of W; dim V = Card A by A6,Def3 .= dim Lin(B) by A8,A10,Th9 .= Card B by A9,Th9 .= dim W by A7,Def3; hence dim V = dim W; end; theorem Th12: for V being finite-dimensional RealUnitarySpace holds dim V = 0 iff (Omega).V = (0).V proof let V be finite-dimensional RealUnitarySpace; hereby assume A1: dim V = 0; consider I being finite Subset of V such that A2: I is Basis of V by Def2; Card I = 0 by A1,A2,Def3; then A3: I = {}(the carrier of V) by CARD_2:59; (Omega).V = the UNITSTR of V by RUSUB_1:def 3 .= Lin(I) by A2,RUSUB_3:def 2 .= (0).V by A3,RUSUB_3:3; hence (Omega).V = (0).V; end; assume (Omega).V = (0).V; then A4:the UNITSTR of V = (0).V by RUSUB_1:def 3; consider I being finite Subset of V such that A5:I is Basis of V by Def2; Lin(I) = (0).V by A4,A5,RUSUB_3:def 2; then A6:I = {} or I = {0.V} by RUSUB_3:4; now assume I = {0.V}; then I is linearly-dependent by RLVECT_3:9; hence contradiction by A5,RUSUB_3:def 2; end; hence dim V = 0 by A5,A6,Def3,CARD_1:47; end; theorem for V being finite-dimensional RealUnitarySpace holds dim V = 1 iff ex v being VECTOR of V st v <> 0.V & (Omega).V = Lin{v} proof let V be finite-dimensional RealUnitarySpace; hereby consider I being finite Subset of V such that A1: I is Basis of V by Def2; assume dim V = 1; then Card I = 1 by A1,Def3; then consider v being set such that A2: I = {v} by CARD_2:60; v in I by A2,TARSKI:def 1; then reconsider v as VECTOR of V; {v} is linearly-independent & Lin{v} = the UNITSTR of V by A1,A2,RUSUB_3:def 2; then v <> 0.V & (Omega).V = Lin{v} by RLVECT_3:9,RUSUB_1:def 3; hence ex v being VECTOR of V st v <> 0.V & (Omega).V = Lin{v}; end; given v being VECTOR of V such that A3:v <> 0.V and A4:(Omega).V = Lin{v}; {v} is linearly-independent & Lin{v} = the UNITSTR of V by A3,A4,RLVECT_3:9,RUSUB_1:def 3; then {v} is Basis of V & Card {v} = 1 by CARD_1:79,RUSUB_3:def 2; hence dim V = 1 by Def3; end; theorem for V being finite-dimensional RealUnitarySpace holds dim V = 2 iff ex u,v being VECTOR of V st u <> v & {u, v} is linearly-independent & (Omega).V = Lin{u, v} proof let V be finite-dimensional RealUnitarySpace; hereby consider I being finite Subset of V such that A1: I is Basis of V by Def2; assume dim V = 2; then A2: Card I = 2 by A1,Def3; then consider u being set such that A3: u in I by CARD_1:47,XBOOLE_0:def 1; reconsider u as VECTOR of V by A3; now assume I c= {u}; then card I <= card {u} by CARD_1:80; then 2 <= 1 by A2,CARD_1:79; hence contradiction; end; then consider v being set such that A4: v in I & not v in {u} by TARSKI:def 3; reconsider v as VECTOR of V by A4; A5: v <> u by A4,TARSKI:def 1; for x being set st x in {u,v} holds x in I by A3,A4,TARSKI:def 2; then A6: {u, v} c= I by TARSKI:def 3; now assume not I c= {u, v}; then consider w being set such that A7: w in I & not w in {u, v} by TARSKI:def 3; A8: w <> u & w <> v by A7,TARSKI:def 2; for x being set st x in {u,v,w} holds x in I by A3,A4,A7,ENUMSET1:13; then {u, v, w} c= I by TARSKI:def 3; then card {u, v, w} <= card I by CARD_1:80; then 3 <= 2 by A2,A5,A8,CARD_2:77; hence contradiction; end; then A9: I = {u, v} by A6,XBOOLE_0:def 10; then A10: Lin{u, v} = the UNITSTR of V by A1,RUSUB_3:def 2 .= (Omega).V by RUSUB_1:def 3; {u, v} is linearly-independent by A1,A9,RUSUB_3:def 2; hence ex u,v being VECTOR of V st u <> v & {u,v} is linearly-independent & (Omega).V = Lin{u, v} by A5,A10; end; given u,v being VECTOR of V such that A11:u <> v and A12:{u, v} is linearly-independent and A13:(Omega).V = Lin{u, v}; Lin{u, v} = the UNITSTR of V by A13,RUSUB_1:def 3; then {u, v} is Basis of V & Card {u, v} = 2 by A11,A12,CARD_2:76,RUSUB_3:def 2; hence dim V = 2 by Def3; end; theorem Th15: for V being finite-dimensional RealUnitarySpace, W1,W2 being Subspace of V holds dim(W1 + W2) + dim(W1 /\ W2) = dim W1 + dim W2 proof let V be finite-dimensional RealUnitarySpace; let W1,W2 be Subspace of V; reconsider V as RealUnitarySpace; reconsider W1, W2 as Subspace of V; A1:W1 /\ W2 is finite-dimensional by Th7; then consider I being finite Subset of W1 /\ W2 such that A2:I is Basis of W1 /\ W2 by Def2; A3:Card I = dim(W1 /\ W2) by A1,A2,Def3; W1 /\ W2 is Subspace of W1 by RUSUB_2:16; then consider I1 being Basis of W1 such that A4:I c= I1 by A2,RUSUB_3:24; reconsider I1 as finite Subset of W1 by Th3; W1 /\ W2 is Subspace of W2 by RUSUB_2:16; then consider I2 being Basis of W2 such that A5:I c= I2 by A2,RUSUB_3:24; reconsider I2 as finite Subset of W2 by Th3; A6:Card I2 = dim W2 by Def3; A7:W1 + W2 is finite-dimensional by Th7; set A = I1 \/ I2; now let v be set; assume v in A; then A8: v in I1 or v in I2 by XBOOLE_0:def 2; then A9: v in the carrier of W1 or v in the carrier of W2; the carrier of W1 c= the carrier of V & the carrier of W2 c= the carrier of V by RUSUB_1:def 1; then reconsider v'= v as VECTOR of V by A9; v' in W1 or v' in W2 by A8,RLVECT_1:def 1; then v' in W1 + W2 by RUSUB_2:2; hence v in the carrier of W1 + W2 by RLVECT_1:def 1; end; then A c= the carrier of W1 + W2 & A is finite by TARSKI:def 3; then reconsider A as finite Subset of W1 + W2; A10:I c= I1 /\ I2 by A4,A5,XBOOLE_1:19; now assume not I1 /\ I2 c= I; then consider x being set such that A11: x in I1 /\ I2 and A12: not x in I by TARSKI:def 3; x in I1 & x in I2 by A11,XBOOLE_0:def 3; then x in Lin(I1) & x in Lin(I2) by RUSUB_3:2; then x in the UNITSTR of W1 & x in the UNITSTR of W2 by RUSUB_3:def 2; then A13: x in the carrier of W1 & x in the carrier of W2 by RLVECT_1:def 1; then x in (the carrier of W1) /\ (the carrier of W2) by XBOOLE_0:def 3; then x in the carrier of W1 /\ W2 by RUSUB_2:def 2; then x in the UNITSTR of W1 /\ W2 by RLVECT_1:def 1; then A14: x in Lin(I) by A2,RUSUB_3:def 2; A15: the carrier of W1 c= the carrier of V by RUSUB_1:def 1; then reconsider x'= x as VECTOR of V by A13; I c= the carrier of W1 /\ W2 & the carrier of W1 /\ W2 c= the carrier of V by RUSUB_1:def 1; then I c= the carrier of V by XBOOLE_1:1; then reconsider I'= I as Subset of V; now let y be set; assume y in I \/ {x}; then A16: y in I or y in {x} by XBOOLE_0:def 2; I c= the carrier of W1 /\ W2 & the carrier of W1 /\ W2 c= the carrier of V by RUSUB_1:def 1; then I c= the carrier of V by XBOOLE_1:1; then y in the carrier of V or y = x by A16,TARSKI:def 1; hence y in the carrier of V by A13,A15; end; then I \/ {x} c= the carrier of V by TARSKI:def 3; then reconsider Ix = I \/ {x} as Subset of V; I1 is linearly-independent & I1 is Subset of W1 by RUSUB_3:def 2; then consider I1' being Subset of V such that A17: I1' is linearly-independent & I1'= I1 by RUSUB_3:22; now let y be set; assume y in I \/ {x}; then y in I or y in {x} by XBOOLE_0:def 2; then y in I1 or y = x by A4,TARSKI:def 1; hence y in I1' by A11,A17,XBOOLE_0:def 3; end; then Ix c= I1' by TARSKI:def 3; then A18: Ix is linearly-independent by A17,RLVECT_3:6; x in {x} by TARSKI:def 1; then A19: x' in Ix by XBOOLE_0:def 2; Ix \ {x} = I \ {x} by XBOOLE_1:40 .= I by A12,Lm2; then not x' in Lin(I') by A18,A19,RUSUB_3:25; hence contradiction by A14,RUSUB_3:28; end; then I = I1 /\ I2 by A10,XBOOLE_0:def 10; then A20:Card A = Card I1 + Card I2 - Card I by CARD_2:64; A c= the carrier of W1 + W2 & the carrier of W1 + W2 c= the carrier of V by RUSUB_1:def 1; then A c= the carrier of V by XBOOLE_1:1; then reconsider A'= A as Subset of V; A21: Lin(A') = Lin(A) by RUSUB_3:28; now let x be set; assume x in the carrier of W1 + W2; then x in W1 + W2 by RLVECT_1:def 1; then consider w1, w2 being VECTOR of V such that A22: w1 in W1 and A23: w2 in W2 and A24: x = w1 + w2 by RUSUB_2:1; reconsider w1 as VECTOR of W1 by A22,RLVECT_1:def 1; reconsider w2 as VECTOR of W2 by A23,RLVECT_1:def 1; w1 in Lin(I1) by RUSUB_3:21; then consider K1 being Linear_Combination of I1 such that A25: w1 = Sum(K1) by RUSUB_3:1; consider L1 being Linear_Combination of V such that A26: Carrier(L1) = Carrier(K1) & Sum(L1) = Sum(K1) by RUSUB_3:19; w2 in Lin(I2) by RUSUB_3:21; then consider K2 being Linear_Combination of I2 such that A27: w2 = Sum(K2) by RUSUB_3:1; consider L2 being Linear_Combination of V such that A28: Carrier(L2) = Carrier(K2) & Sum(L2) = Sum(K2) by RUSUB_3:19; set L = L1 + L2; Carrier(L1) c= I1 & Carrier(L2) c= I2 by A26,A28,RLVECT_2:def 8; then Carrier(L) c= Carrier(L1) \/ Carrier(L2) & Carrier(L1) \/ Carrier(L2) c= I1 \/ I2 by RLVECT_2:58,XBOOLE_1:13; then Carrier(L) c= I1 \/ I2 by XBOOLE_1:1; then reconsider L as Linear_Combination of A' by RLVECT_2:def 8; x = Sum(L) by A24,A25,A26,A27,A28,RLVECT_3:1; then x in Lin(A') by RUSUB_3:1; hence x in the carrier of Lin(A') by RLVECT_1:def 1; end; then the carrier of W1 + W2 c= the carrier of Lin(A') by TARSKI:def 3; then W1 + W2 is Subspace of Lin(A') by RUSUB_1:22; then A29: Lin(A) = W1 + W2 by A21,RUSUB_1:20; for L being Linear_Combination of A st Sum(L) = 0.(W1 + W2) holds Carrier(L) = {} proof let L be Linear_Combination of A; assume A30: Sum(L) = 0.(W1 + W2); A31: W1 is Subspace of W1 + W2 & W2 is Subspace of W1 + W2 by RUSUB_2:7; reconsider W1'= W1 as Subspace of W1 + W2 by RUSUB_2:7; reconsider W2'= W2 as Subspace of W1 + W2 by RUSUB_2:7; A32: Carrier(L) c= I1 \/ I2 by RLVECT_2:def 8; consider F being FinSequence of the carrier of W1 + W2 such that A33: F is one-to-one and A34: rng F = Carrier(L) and A35: Sum(L) = Sum(L (#) F) by RLVECT_2:def 10; set B = Carrier(L) /\ I1; reconsider B as Subset of rng F by A34,XBOOLE_1:17; consider P being Permutation of dom F such that A36: (F - B`) ^ (F - B) = F*P by A33,MATRLIN:8; reconsider F1 = F - B`, F2 = F - B as FinSequence of the carrier of W1 + W2 by FINSEQ_3:93; A37: F1 is one-to-one & F2 is one-to-one by A33,FINSEQ_3:94; consider L1 being Linear_Combination of W1 + W2 such that A38: Carrier(L1) = rng F1 /\ Carrier(L) and A39: L1 (#) F1 = L (#) F1 by RLVECT_5:8; Carrier(L1) c= rng F1 by A38,XBOOLE_1:17; then A40: Sum(L (#) F1) = Sum(L1) by A37,A39,RLVECT_5:7; rng F c= rng F; then reconsider X = rng F as Subset of rng F; X \ B` = X /\ B`` by SUBSET_1:32 .= B by XBOOLE_1:28; then rng F1 = B by FINSEQ_3:72; then A41: Carrier(L1) = I1 /\ (Carrier(L) /\ Carrier(L)) by A38,XBOOLE_1:16 .= Carrier(L) /\ I1; then A42: Carrier(L1) c= I1 & I1 c= the carrier of W1 by XBOOLE_1:17; then Carrier(L1) c= the carrier of W1' by XBOOLE_1:1; then consider K1 being Linear_Combination of W1' such that Carrier(K1) = Carrier(L1) and A43: Sum(K1) = Sum(L1) by RUSUB_3:20; consider L2 being Linear_Combination of W1 + W2 such that A44: Carrier(L2) = rng F2 /\ Carrier(L) and A45: L2 (#) F2 = L (#) F2 by RLVECT_5:8; Carrier(L2) c= rng F2 by A44,XBOOLE_1:17; then A46: Sum(L (#) F2) = Sum(L2) by A37,A45,RLVECT_5:7; A47: Carrier(L) \ I1 c= Carrier(L) by XBOOLE_1:36; rng F2 = Carrier(L) \ (Carrier(L) /\ I1) by A34,FINSEQ_3:72 .= Carrier(L) \ I1 by XBOOLE_1:47; then A48: Carrier(L2) = Carrier(L) \ I1 by A44,A47,XBOOLE_1:28; then Carrier(L2) c= I2 & I2 c= the carrier of W2 by A32,XBOOLE_1:43; then Carrier(L2) c= the carrier of W2' by XBOOLE_1:1; then consider K2 being Linear_Combination of W2' such that Carrier(K2) = Carrier(L2) and A49: Sum(K2) = Sum(L2) by RUSUB_3:20; A50: 0.(W1 + W2) = Sum(L (#) (F1^F2)) by A30,A35,A36,RLVECT_5:5 .= Sum((L (#) F1) ^ (L (#) F2)) by RLVECT_3:41 .= Sum(L1) + Sum(L2) by A40,A46,RLVECT_1:58; then Sum(L1) = - Sum(L2) by RLVECT_1:def 10 .= - Sum(K2) by A49,RUSUB_1:9; then Sum(K1) in W2 & Sum(K1) in W1 by A43,RLVECT_1:def 1; then Sum(K1) in W1 /\ W2 by RUSUB_2:3; then Sum(K1) in Lin(I) by A2,RUSUB_3:def 2; then consider KI being Linear_Combination of I such that A51: Sum(K1) = Sum(KI) by RUSUB_3:1; W1 /\ W2 is Subspace of W1 + W2 by RUSUB_2:22; then consider LI being Linear_Combination of W1 + W2 such that A52: Carrier(LI) = Carrier(KI) and A53: Sum(LI) = Sum(KI) by RUSUB_3:19; A54: Carrier(LI + L2) c= Carrier(LI) \/ Carrier(L2) by RLVECT_2:58; A55: I \/ I2 = I2 by A5,XBOOLE_1:12; Carrier(LI) c= I & Carrier(L2) c= I2 by A32,A48,A52,RLVECT_2:def 8,XBOOLE_1:43; then Carrier(LI) \/ Carrier(L2) c= I2 by A55,XBOOLE_1:13; then A56: Carrier(LI + L2) c= I2 & I2 c= the carrier of W2 by A54,XBOOLE_1:1; then Carrier(LI + L2) c= the carrier of W2 by XBOOLE_1:1; then consider K being Linear_Combination of W2 such that A57: Carrier(K) = Carrier(LI + L2) and A58: Sum(K) = Sum(LI + L2) by A31,RUSUB_3:20; reconsider K as Linear_Combination of I2 by A56,A57,RLVECT_2:def 8; I1 is Subset of W1 & I1 is linearly-independent by RUSUB_3:def 2; then consider I1' being Subset of W1 + W2 such that A59: I1' is linearly-independent & I1'= I1 by A31,RUSUB_3:22; Carrier(LI) c= I by A52,RLVECT_2:def 8; then Carrier(LI) c= I1' by A4,A59,XBOOLE_1:1; then A60: LI = L1 by A42,A43,A51,A53,A59,RLVECT_5:1; A61: I2 is linearly-independent by RUSUB_3:def 2; 0.W2 = Sum(LI) + Sum(L2) by A43,A50,A51,A53,RUSUB_1:5 .= Sum(K) by A58,RLVECT_3:1; then A62: {} = Carrier(L1 + L2) by A57,A60,A61,RLVECT_3:def 1; A63: Carrier(L) = Carrier(L1) \/ Carrier(L2) by A41,A48,XBOOLE_1:51; A64: I1 misses (Carrier(L) \ I1) by XBOOLE_1:79; Carrier(L1) /\ Carrier(L2) = Carrier(L) /\ (I1 /\ (Carrier(L) \ I1)) by A41,A48,XBOOLE_1:16 .= Carrier(L) /\ {} by A64,XBOOLE_0:def 7 .= {}; then A65: Carrier(L1) misses Carrier(L2) by XBOOLE_0:def 7; now assume not Carrier(L) c= Carrier(L1 + L2); then consider x being set such that A66: x in Carrier(L) and A67: not x in Carrier(L1 + L2) by TARSKI:def 3; reconsider x as VECTOR of W1 + W2 by A66; A68: 0 = (L1 + L2).x by A67,RLVECT_2:28 .= L1.x + L2.x by RLVECT_2:def 12; per cases by A63,A66,XBOOLE_0:def 2; suppose A69: x in Carrier(L1); then consider v being VECTOR of W1 + W2 such that A70: x = v & L1.v <> 0 by RLVECT_5:3; not x in Carrier(L2) by A65,A69,XBOOLE_0:3; then L2.x = 0 by RLVECT_2:28; hence contradiction by A68,A70; suppose A71: x in Carrier(L2); then consider v being VECTOR of W1 + W2 such that A72: x = v & L2.v <> 0 by RLVECT_5:3; not x in Carrier(L1) by A65,A71,XBOOLE_0:3; then L1.x = 0 by RLVECT_2:28; hence contradiction by A68,A72; end; hence Carrier(L) = {} by A62,XBOOLE_1:3; end; then A is linearly-independent by RLVECT_3:def 1; then A is Basis of W1 + W2 by A29,RUSUB_3:def 2; then Card A = dim(W1 + W2) by A7,Def3; then dim(W1 + W2) + dim(W1 /\ W2) = Card I1 + Card I2 + - Card I + Card I by A3,A20,XCMPLX_0:def 8 .= Card I1 + Card I2 + (- Card I + Card I) by XCMPLX_1:1 .= Card I1 + Card I2 + 0 by XCMPLX_0:def 6 .= dim W1 + dim W2 by A6,Def3; hence thesis; end; theorem for V being finite-dimensional RealUnitarySpace, W1,W2 being Subspace of V holds dim(W1 /\ W2) >= dim W1 + dim W2 - dim V proof let V be finite-dimensional RealUnitarySpace; let W1,W2 be Subspace of V; A1:dim W1 + dim W2 - dim V = dim(W1 + W2) + dim(W1 /\ W2) - dim V by Th15 .= dim(W1 + W2) + (dim(W1 /\ W2) - dim V) by XCMPLX_1:29 ; A2:dim(W1 + W2) <= dim V by Th8; dim V + (dim(W1 /\ W2) - dim V) = dim V + (dim(W1 /\ W2) + -dim V) by XCMPLX_0:def 8 .= dim(W1 /\ W2) + (dim V + -dim V) by XCMPLX_1:1 .= dim(W1 /\ W2) + 0 by XCMPLX_0:def 6 .= dim(W1 /\ W2); hence thesis by A1,A2,AXIOMS:24; end; theorem for V being finite-dimensional RealUnitarySpace, W1,W2 being Subspace of V st V is_the_direct_sum_of W1, W2 holds dim V = dim W1 + dim W2 proof let V be finite-dimensional RealUnitarySpace; let W1,W2 be Subspace of V; assume V is_the_direct_sum_of W1, W2; then A1:the UNITSTR of V = W1 + W2 & W1 /\ W2 = (0).V by RUSUB_2:def 4; then (Omega).(W1 /\ W2) = (0).V by RUSUB_1:def 3 .= (0).(W1 /\ W2) by RUSUB_1:30; then dim(W1 /\ W2) = 0 by Th12; then dim W1 + dim W2 = dim(W1 + W2) + 0 by Th15 .= dim (Omega).V by A1,RUSUB_1:def 3 .= dim V by Th10; hence dim V = dim W1 + dim W2; end; begin :: Fixed-dimensional subspace family Lm3: for V being finite-dimensional RealUnitarySpace, n being Nat st n <= dim V holds ex W being strict Subspace of V st dim W = n proof let V be finite-dimensional RealUnitarySpace; let n be Nat; assume A1:n <= dim V; consider I being finite Subset of V such that A2:I is Basis of V by Def2; n <= Card I by A1,A2,Def3; then consider A being finite Subset of I such that A3:Card A = n by VECTSP_9:1; A c= the carrier of V by XBOOLE_1:1; then reconsider A as Subset of V; reconsider W = Lin(A) as strict finite-dimensional Subspace of V; I is linearly-independent by A2,RUSUB_3:def 2; then A is linearly-independent by RLVECT_3:6; then dim W = n by A3,Th9; hence thesis; end; theorem for V being finite-dimensional RealUnitarySpace, W being Subspace of V, n being Nat holds n <= dim V iff ex W being strict Subspace of V st dim W = n by Lm3,Th8; definition let V be finite-dimensional RealUnitarySpace, n be Nat; func n Subspaces_of V -> set means :Def4: for x being set holds x in it iff ex W being strict Subspace of V st W = x & dim W = n; existence proof set S = {Lin(A) where A is Subset of V: A is linearly-independent & Card A = n}; A1: for x being set holds x in S iff ex W being strict Subspace of V st W = x & dim W = n proof let x be set; hereby assume x in S; then consider A being Subset of V such that A2: x = Lin(A) and A3: A is linearly-independent and A4: Card A = n; reconsider W = x as strict Subspace of V by A2; dim W = n by A2,A3,A4,Th9; hence ex W being strict Subspace of V st W = x & dim W = n; end; given W being strict Subspace of V such that A5: W = x and A6: dim W = n; consider A being finite Subset of W such that A7: A is Basis of W by Def2; reconsider A as Subset of W; A is linearly-independent by A7,RUSUB_3:def 2; then consider B being Subset of V such that A8: B is linearly-independent and A9: B = A by RUSUB_3:22; A10: x = Lin(A) by A5,A7,RUSUB_3:def 2 .= Lin(B) by A9,RUSUB_3:28; then Card B = n by A5,A6,A8,Th9; hence x in S by A8,A10; end; take S; thus thesis by A1; end; uniqueness proof defpred P[set] means ex W being strict Subspace of V st W = $1 & dim W = n; for X1,X2 being set st (for x being set holds x in X1 iff P[x]) & (for x being set holds x in X2 iff P[x]) holds X1 = X2 from SetEq; hence thesis; end; end; theorem for V being finite-dimensional RealUnitarySpace, n being Nat st n <= dim V holds n Subspaces_of V is non empty proof let V be finite-dimensional RealUnitarySpace; let n be Nat; assume n <= dim V; then consider W being strict Subspace of V such that A1:dim W = n by Lm3; thus n Subspaces_of V is non empty by A1,Def4; end; theorem for V being finite-dimensional RealUnitarySpace, n being Nat st dim V < n holds n Subspaces_of V = {} proof let V be finite-dimensional RealUnitarySpace; let n be Nat; assume that A1:dim V < n and A2:n Subspaces_of V <> {}; consider x being set such that A3:x in n Subspaces_of V by A2,XBOOLE_0:def 1; consider W being strict Subspace of V such that W = x and A4:dim W = n by A3,Def4; thus contradiction by A1,A4,Th8; end; theorem for V being finite-dimensional RealUnitarySpace, W being Subspace of V, n being Nat holds n Subspaces_of W c= n Subspaces_of V proof let V be finite-dimensional RealUnitarySpace; let W be Subspace of V; let n be Nat; let x be set; assume x in n Subspaces_of W; then consider W1 being strict Subspace of W such that A1:W1 = x and A2:dim W1 = n by Def4; reconsider W1 as strict Subspace of V by RUSUB_1:21; W1 in n Subspaces_of V by A2,Def4; hence x in n Subspaces_of V by A1; end; begin :: Affine set definition let V be non empty RLSStruct, S be Subset of V; attr S is Affine means :Def5: for x,y being VECTOR of V, a being Real st x in S & y in S holds (1 - a) * x + a * y in S; end; theorem Th22: for V being non empty RLSStruct holds [#]V is Affine & {}V is Affine proof let V be non empty RLSStruct; for x,y being VECTOR of V, a being Real st x in [#]V & y in [#]V holds (1-a) * x + a * y in [#]V by PRE_TOPC:13; hence [#]V is Affine by Def5; for x,y being VECTOR of V, a being Real st x in {}V & y in {}V holds (1-a) * x + a * y in {}V; hence thesis by Def5; end; theorem for V being RealLinearSpace-like (non empty RLSStruct), v being VECTOR of V holds {v} is Affine proof let V be RealLinearSpace-like (non empty RLSStruct); let v be VECTOR of V; for x,y being VECTOR of V, a being Real st x in {v} & y in {v} holds (1-a)*x + a*y in {v} proof let x,y being VECTOR of V; let a be Real; assume x in {v} & y in {v}; then x = v & y = v by TARSKI:def 1; then (1-a)*x + a*y = ((1-a)+a)*v by RLVECT_1:def 9 .= (1-(a-a))*v by XCMPLX_1:37 .= (1-0)*v by XCMPLX_1:14 .= v by RLVECT_1:def 9; hence thesis by TARSKI:def 1; end; hence thesis by Def5; end; definition let V be non empty RLSStruct; cluster non empty Affine Subset of V; existence proof take [#]V; thus thesis by Th22; end; cluster empty Affine Subset of V; existence proof take {}V; thus thesis by Th22; end; end; definition let V be RealLinearSpace, W be Subspace of V; func Up(W) -> non empty Subset of V equals :Def6: the carrier of W; coherence by RLSUB_1:def 2; end; definition let V be RealUnitarySpace, W be Subspace of V; func Up(W) -> non empty Subset of V equals :Def7: the carrier of W; coherence by RUSUB_1:def 1; end; theorem for V being RealLinearSpace, W being Subspace of V holds Up(W) is Affine & 0.V in the carrier of W proof let V be RealLinearSpace; let W be Subspace of V; for x,y being VECTOR of V, a being Real st x in Up(W) & y in Up(W) holds (1 - a) * x + a * y in Up(W) proof let x,y be VECTOR of V; let a be Real; assume x in Up(W) & y in Up(W); then x in the carrier of W & y in the carrier of W by Def6; then x in W & y in W by RLVECT_1:def 1; then (1 - a) * x in W & a * y in W by RLSUB_1:29; then A1: (1 - a) * x + a * y in W by RLSUB_1:28; reconsider z = (1 - a) * x + a * y as VECTOR of V; z in the carrier of W by A1,RLVECT_1:def 1; hence thesis by Def6; end; hence Up(W) is Affine by Def5; 0.V in W by RLSUB_1:25; hence thesis by RLVECT_1:def 1; end; theorem Th25: for V being RealLinearSpace, A being Affine Subset of V st 0.V in A holds for x being VECTOR of V, a being Real st x in A holds a * x in A proof let V be RealLinearSpace; let A be Affine Subset of V; assume A1:0.V in A; for x being VECTOR of V, a being Real st x in A holds a * x in A proof let x be VECTOR of V; let a be Real; assume x in A; then (1-a) * 0.V + a * x in A by A1,Def5; then 0.V + a * x in A by RLVECT_1:23; hence thesis by RLVECT_1:10; end; hence thesis; end; definition let V be non empty RLSStruct, S be non empty Subset of V; attr S is Subspace-like means :Def8: the Zero of V in S & for x,y being Element of V, a being Real st x in S & y in S holds x + y in S & a * x in S; end; theorem Th26: for V being RealLinearSpace, A being non empty Affine Subset of V st 0.V in A holds A is Subspace-like & A = the carrier of Lin(A) proof let V be RealLinearSpace; let A be non empty Affine Subset of V; assume A1:0.V in A; then A2:the Zero of V in A by RLVECT_1:def 2; A3:for x,y being Element of V, a being Real st x in A & y in A holds x + y in A & a * x in A proof let x,y be Element of V; let a be Real; assume A4: x in A & y in A; reconsider x,y as VECTOR of V; A5: (1 - 1/2) * x + (1/2) * y in A by A4,Def5; 2 * ( (1-1/2) * x + (1/2) * y ) = 2*( (1-1/2) * x) + 2*((1/2) * y) by RLVECT_1:def 9 .= ( 2*(1-1/2) ) * x + 2*((1/2) * y) by RLVECT_1:def 9 .= ( 2 - 1 ) * x + ( 2*(1/2) ) * y by RLVECT_1:def 9 .= x + 1 * y by RLVECT_1:def 9 .= x + y by RLVECT_1:def 9; hence thesis by A1,A4,A5,Th25; end; hence A is Subspace-like by A2,Def8; for x being set st x in A holds x in the carrier of Lin(A) proof let x be set; assume x in A; then x in Lin(A) by RLVECT_3:18; hence thesis by RLVECT_1:def 1; end; then A6:A c= the carrier of Lin(A) by TARSKI:def 3; for x being set st x in the carrier of Lin(A) holds x in A proof let x be set; assume x in the carrier of Lin(A); then x in Lin(A) by RLVECT_1:def 1; then consider l being Linear_Combination of A such that A7: x = Sum(l) by RLVECT_3:17; A8: for v,u being VECTOR of V st v in A & u in A holds v + u in A by A3; for a being Real, v being VECTOR of V st v in A holds a * v in A by A3; then A is lineary-closed by A8,RLSUB_1:def 1; hence x in A by A7,RLVECT_2:47; end; then the carrier of Lin(A) c= A by TARSKI:def 3; hence thesis by A6,XBOOLE_0:def 10; end; theorem for V being RealLinearSpace, W being Subspace of V holds Up(W) is Subspace-like proof let V be RealLinearSpace; let W be Subspace of V; 0.V in W by RLSUB_1:25; then 0.V in the carrier of W by RLVECT_1:def 1; then 0.V in Up(W) by Def6; hence the Zero of V in Up(W) by RLVECT_1:def 2; thus for x,y being Element of V, a being Real st x in Up(W) & y in Up(W) holds x + y in Up(W) & a * x in Up(W) proof let x,y be Element of V; let a be Real; assume x in Up(W) & y in Up(W); then A1: x in the carrier of W & y in the carrier of W by Def6; reconsider x,y as Element of V; x in W & y in W by A1,RLVECT_1:def 1; then x + y in W & a * x in W by RLSUB_1:28,29; then x + y in the carrier of W & a * x in the carrier of W by RLVECT_1:def 1; hence thesis by Def6; end; end; theorem for V being RealLinearSpace, W being strict Subspace of V holds W = Lin(Up(W)) proof let V be RealLinearSpace; let W be strict Subspace of V; Up(W) = the carrier of W by Def6; hence thesis by RLVECT_3:21; end; theorem for V being RealUnitarySpace, A being non empty Affine Subset of V st 0.V in A holds A = the carrier of Lin(A) proof let V be RealUnitarySpace; let A be non empty Affine Subset of V; assume 0.V in A; then A1:A is Subspace-like by Th26; for x being set st x in A holds x in the carrier of Lin(A) proof let x be set; assume x in A; then x in Lin(A) by RUSUB_3:2; hence thesis by RLVECT_1:def 1; end; then A2:A c= the carrier of Lin(A) by TARSKI:def 3; for x being set st x in the carrier of Lin(A) holds x in A proof let x be set; assume x in the carrier of Lin(A); then x in Lin(A) by RLVECT_1:def 1; then consider l being Linear_Combination of A such that A3: x = Sum(l) by RUSUB_3:1; A4: for v,u being VECTOR of V st v in A & u in A holds v + u in A by A1,Def8; for a being Real, v being VECTOR of V st v in A holds a * v in A by A1,Def8; then A is lineary-closed by A4,RLSUB_1:def 1; hence x in A by A3,RLVECT_2:47; end; then the carrier of Lin(A) c= A by TARSKI:def 3; hence thesis by A2,XBOOLE_0:def 10; end; theorem for V being RealUnitarySpace, W being Subspace of V holds Up(W) is Subspace-like proof let V be RealUnitarySpace; let W be Subspace of V; 0.V in W by RUSUB_1:11; then 0.V in the carrier of W by RLVECT_1:def 1; then 0.V in Up(W) by Def7; hence the Zero of V in Up(W) by RLVECT_1:def 2; thus for x,y being Element of V, a being Real st x in Up(W) & y in Up(W) holds x + y in Up(W) & a * x in Up(W) proof let x,y be Element of V; let a be Real; assume x in Up(W) & y in Up(W); then A1: x in the carrier of W & y in the carrier of W by Def7; reconsider x,y as Element of V; x in W & y in W by A1,RLVECT_1:def 1; then x + y in W & a * x in W by RUSUB_1:14,15; then x + y in the carrier of W & a * x in the carrier of W by RLVECT_1:def 1; hence thesis by Def7; end; end; theorem for V being RealUnitarySpace, W being strict Subspace of V holds W = Lin(Up(W)) proof let V be RealUnitarySpace; let W be strict Subspace of V; Up(W) = the carrier of W by Def7; hence thesis by RUSUB_3:5; end; definition let V be non empty LoopStr, M be Subset of V, v be Element of V; func v + M -> Subset of V equals :Def9: {v + u where u is Element of V: u in M}; coherence proof defpred P[set] means ex u being Element of V st $1 = v + u & u in M; consider X being set such that A1:for x being set holds x in X iff x in the carrier of V & P[x] from Separation; X c= the carrier of V proof let x be set; assume x in X; hence x in the carrier of V by A1; end; then reconsider X as Subset of V; reconsider X as Subset of V; set Y = {v + u where u is Element of V: u in M}; X = Y proof thus X c= Y proof let x be set; assume x in X; then ex u being Element of V st x = v + u & u in M by A1 ; hence thesis; end; thus Y c= X proof let x be set; assume x in Y; then ex u being Element of V st x = v + u & u in M; hence thesis by A1; end; end; hence thesis; end; end; theorem for V being RealLinearSpace, W being strict Subspace of V, M being Subset of V, v being VECTOR of V st Up(W) = M holds v + W = v + M proof let V be RealLinearSpace; let W be strict Subspace of V; let M be Subset of V; let v be VECTOR of V; assume Up(W) = M; then A1:the carrier of W = M by Def6; for x being set st x in v + W holds x in v + M proof let x be set; assume x in v + W; then x in {v + u where u is VECTOR of V : u in W} by RLSUB_1:def 5; then consider u being VECTOR of V such that A2: x = v + u & u in W; u in M by A1,A2,RLVECT_1:def 1; then x in {v + u' where u' is Element of V : u' in M} by A2; hence thesis by Def9; end; then A3:v + W c= v + M by TARSKI:def 3; for x being set st x in v + M holds x in v + W proof let x be set; assume x in v + M; then x in {v + u where u is Element of V : u in M} by Def9 ; then consider u being Element of V such that A4: x = v + u & u in M; u in W by A1,A4,RLVECT_1:def 1; then x in {v + u' where u' is VECTOR of V : u' in W} by A4; hence thesis by RLSUB_1:def 5; end; then v + M c= v + W by TARSKI:def 3; hence thesis by A3,XBOOLE_0:def 10; end; theorem Th33: for V being Abelian add-associative RealLinearSpace-like (non empty RLSStruct), M being Affine Subset of V, v being VECTOR of V holds v + M is Affine proof let V be Abelian add-associative RealLinearSpace-like (non empty RLSStruct); let M be Affine Subset of V; let v be VECTOR of V; for x,y being VECTOR of V, a being Real st x in v + M & y in v + M holds (1-a)*x + a*y in v + M proof let x,y be VECTOR of V; let a be Real; assume x in v + M & y in v + M; then A1: x in {v + u where u is Element of V : u in M} & y in {v + u where u is Element of V : u in M} by Def9; then consider x' being Element of V such that A2: x = v + x' & x' in M; consider y' being Element of V such that A3: y = v + y' & y' in M by A1; A4: (1 - a) * x' + a * y' in M by A2,A3,Def5; (1 - a) * x + a * y = (1-a)*v + (1-a)*x' + a * (v + y') by A2,A3, RLVECT_1:def 9 .= (1-a)*v + (1-a)*x' + (a*v + a*y') by RLVECT_1:def 9 .= (1-a)*v + (1-a)*x' + a*v + a*y' by RLVECT_1:def 6 .= (1-a)*x' + ((1-a)*v + a*v) + a*y' by RLVECT_1:def 6 .= (1-a)*x' + (1-a+a)*v + a*y' by RLVECT_1:def 9 .= (1-a)*x' + (1-(a-a))*v + a*y' by XCMPLX_1:37 .= (1-a)*x' + 1*v + a*y' by XCMPLX_1:17 .= (1-a)*x' + v + a*y' by RLVECT_1:def 9 .= v + ((1-a)*x' + a*y') by RLVECT_1:def 6; then (1 - a) * x + a * y in {v + u where u is Element of V : u in M} by A4; hence thesis by Def9; end; hence thesis by Def5; end; theorem for V being RealUnitarySpace, W being strict Subspace of V, M being Subset of V, v being VECTOR of V st Up(W) = M holds v + W = v + M proof let V be RealUnitarySpace; let W be strict Subspace of V; let M be Subset of V; let v be VECTOR of V; assume Up(W) = M; then A1:the carrier of W = M by Def7; for x being set st x in v + W holds x in v + M proof let x be set; assume x in v + W; then x in {v + u where u is VECTOR of V : u in W} by RUSUB_1:def 4; then consider u being VECTOR of V such that A2: x = v + u & u in W; u in M by A1,A2,RLVECT_1:def 1; then x in {v + u' where u' is Element of V : u' in M} by A2; hence thesis by Def9; end; then A3:v + W c= v + M by TARSKI:def 3; for x being set st x in v + M holds x in v + W proof let x be set; assume x in v + M; then x in {v + u where u is Element of V : u in M} by Def9 ; then consider u being Element of V such that A4: x = v + u & u in M; u in W by A1,A4,RLVECT_1:def 1; then x in {v + u' where u' is VECTOR of V : u' in W} by A4; hence thesis by RUSUB_1:def 4; end; then v + M c= v + W by TARSKI:def 3; hence thesis by A3,XBOOLE_0:def 10; end; definition let V be non empty LoopStr, M,N be Subset of V; func M + N -> Subset of V equals :Def10: {u + v where u,v is Element of V: u in M & v in N}; coherence proof deffunc F(Element of V,Element of V) = $1+$2; defpred P[set,set] means $1 in M & $2 in N; {F(u,v) where u,v is Element of V : P[u,v]} is Subset of V from SubsetFD2; hence thesis; end; end; theorem Th35: for V be Abelian (non empty LoopStr), M,N be Subset of V holds N + M = M + N proof let V be Abelian (non empty LoopStr); let M,N be Subset of V; for x being set st x in N + M holds x in M + N proof let x be set; assume x in N + M; then x in {u + v where u,v is Element of V: u in N & v in M} by Def10; then consider u1,v1 being Element of V such that A1: x = u1 + v1 & u1 in N & v1 in M; x in {u + v where u,v is Element of V: u in M & v in N} by A1; hence thesis by Def10; end; then A2:N + M c= M + N by TARSKI:def 3; for x being set st x in M + N holds x in N + M proof let x be set; assume x in M + N; then x in {u + v where u,v is Element of V: u in M & v in N} by Def10; then consider u1,v1 being Element of V such that A3: x = u1 + v1 & u1 in M & v1 in N; x in {u + v where u,v is Element of V: u in N & v in M} by A3; hence thesis by Def10; end; then M + N c= N + M by TARSKI:def 3; hence thesis by A2,XBOOLE_0:def 10; end; definition let V be Abelian (non empty LoopStr), M,N be Subset of V; redefine func M + N; commutativity by Th35; end; theorem Th36: for V being non empty LoopStr, M being Subset of V, v being Element of V holds {v} + M = v + M proof let V be non empty LoopStr; let M be Subset of V; let v be Element of V; for x being set st x in {v} + M holds x in v + M proof let x be set; assume x in {v} + M; then x in {v1 + u1 where v1,u1 is Element of V : v1 in {v} & u1 in M} by Def10; then consider v1,u1 being Element of V such that A1: x = v1 + u1 & v1 in {v} & u1 in M; v1 = v by A1,TARSKI:def 1; then x in {v + u' where u' is Element of V: u' in M} by A1 ; hence thesis by Def9; end; then A2:{v} + M c= v + M by TARSKI:def 3; for x being set st x in v + M holds x in {v} + M proof let x be set; assume x in v + M; then x in {v + u where u is Element of V: u in M} by Def9; then consider u being Element of V such that A3: x = v + u & u in M; v in {v} by TARSKI:def 1; then x in {v1 + u1 where v1,u1 is Element of V : v1 in {v} & u1 in M} by A3; hence thesis by Def10; end; then v + M c= {v} + M by TARSKI:def 3; hence thesis by A2,XBOOLE_0:def 10; end; theorem for V being Abelian add-associative RealLinearSpace-like (non empty RLSStruct), M being Affine Subset of V, v being VECTOR of V holds {v} + M is Affine proof let V be Abelian add-associative RealLinearSpace-like (non empty RLSStruct); let M be Affine Subset of V; let v be VECTOR of V; {v} + M = v + M by Th36; hence thesis by Th33; end; theorem for V being non empty RLSStruct, M,N being Affine Subset of V holds M /\ N is Affine proof let V be non empty RLSStruct; let M,N be Affine Subset of V; for x,y being VECTOR of V, a being Real st x in M /\ N & y in M /\ N holds (1 - a) * x + a * y in M /\ N proof let x,y be VECTOR of V; let a be Real; assume x in M /\ N & y in M /\ N; then x in M & x in N & y in M & y in N by XBOOLE_0:def 3; then (1 - a) * x + a * y in M & (1 - a) * x + a * y in N by Def5; hence thesis by XBOOLE_0:def 3; end; hence thesis by Def5; end; theorem for V being Abelian add-associative RealLinearSpace-like (non empty RLSStruct), M,N being Affine Subset of V holds M + N is Affine proof let V be Abelian add-associative RealLinearSpace-like (non empty RLSStruct); let M,N be Affine Subset of V; for x,y being VECTOR of V, a being Real st x in M + N & y in M + N holds (1 - a) * x + a * y in M + N proof let x,y be VECTOR of V; let a be Real; assume x in M + N & y in M + N; then A1: x in {u + v where u,v is Element of V: u in M & v in N } & y in {u + v where u,v is Element of V: u in M & v in N} by Def10; then consider u1,v1 being Element of V such that A2: x = u1 + v1 & u1 in M & v1 in N; consider u2,v2 being Element of V such that A3: y = u2 + v2 & u2 in M & v2 in N by A1; A4: (1 - a) * u1 + a * u2 in M by A2,A3,Def5; A5: (1 - a) * v1 + a * v2 in N by A2,A3,Def5; (1 - a) * x + a * y = (1 - a) * u1 + (1 - a) * v1 + a * (u2 + v2) by A2,A3,RLVECT_1:def 9 .= (1 - a) * u1 + (1 - a) * v1 + ( a * u2 + a * v2 ) by RLVECT_1:def 9 .= (1 - a) * u1 + (1 - a) * v1 + a * u2 + a * v2 by RLVECT_1:def 6 .= (1 - a) * v1 + ((1 - a) * u1 + a * u2) + a * v2 by RLVECT_1:def 6 .= ((1 - a) * u1 + a * u2) + ((1 - a) * v1 + a * v2) by RLVECT_1:def 6; then (1 - a) * x + a * y in {u + v where u,v is Element of V: u in M & v in N} by A4,A5; hence thesis by Def10; end; hence thesis by Def5; end;

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