Volume 14, 2002

University of Bialystok

Copyright (c) 2002 Association of Mizar Users

### The abstract of the Mizar article:

### Dimension of Real Unitary Space

**by****Noboru Endou,****Takashi Mitsuishi, and****Yasunari Shidama**- Received October 9, 2002
- MML identifier: RUSUB_4

- [ Mizar article, 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; begin :: Finite-dimensional real unitary space theorem :: RUSUB_4:1 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}); theorem :: RUSUB_4:2 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); definition let V be RealUnitarySpace; attr V is finite-dimensional means :: RUSUB_4:def 1 ex A being finite Subset of V st A is Basis of V; end; definition cluster strict finite-dimensional RealUnitarySpace; end; definition let V be RealUnitarySpace; redefine attr V is finite-dimensional means :: RUSUB_4:def 2 ex I being finite Subset of V st I is Basis of V; end; theorem :: RUSUB_4:3 for V being RealUnitarySpace st V is finite-dimensional holds for I being Basis of V holds I is finite; theorem :: RUSUB_4:4 for V being RealUnitarySpace, A being Subset of V st V is finite-dimensional & A is linearly-independent holds A is finite; theorem :: RUSUB_4:5 for V being RealUnitarySpace, A,B being Basis of V st V is finite-dimensional holds Card A = Card B; theorem :: RUSUB_4:6 for V being RealUnitarySpace holds (0).V is finite-dimensional; theorem :: RUSUB_4:7 for V being RealUnitarySpace, W being Subspace of V st V is finite-dimensional holds W is finite-dimensional; definition let V be RealUnitarySpace; cluster finite-dimensional strict Subspace of V; end; definition let V be finite-dimensional RealUnitarySpace; cluster -> finite-dimensional Subspace of V; end; definition let V be finite-dimensional RealUnitarySpace; cluster strict Subspace of V; end; begin :: Dimension of real unitary space definition let V be RealUnitarySpace; assume V is finite-dimensional; func dim V -> Nat means :: RUSUB_4:def 3 for I being Basis of V holds it = Card I; end; theorem :: RUSUB_4:8 for V being finite-dimensional RealUnitarySpace, W being Subspace of V holds dim W <= dim V; theorem :: RUSUB_4:9 for V being finite-dimensional RealUnitarySpace, A being Subset of V st A is linearly-independent holds Card A = dim Lin(A); theorem :: RUSUB_4:10 for V being finite-dimensional RealUnitarySpace holds dim V = dim (Omega).V; theorem :: RUSUB_4:11 for V being finite-dimensional RealUnitarySpace, W being Subspace of V holds dim V = dim W iff (Omega).V = (Omega).W; theorem :: RUSUB_4:12 for V being finite-dimensional RealUnitarySpace holds dim V = 0 iff (Omega).V = (0).V; theorem :: RUSUB_4:13 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}; theorem :: RUSUB_4:14 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}; theorem :: RUSUB_4:15 for V being finite-dimensional RealUnitarySpace, W1,W2 being Subspace of V holds dim(W1 + W2) + dim(W1 /\ W2) = dim W1 + dim W2; theorem :: RUSUB_4:16 for V being finite-dimensional RealUnitarySpace, W1,W2 being Subspace of V holds dim(W1 /\ W2) >= dim W1 + dim W2 - dim V; theorem :: RUSUB_4:17 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; begin theorem :: RUSUB_4:18 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; definition let V be finite-dimensional RealUnitarySpace, n be Nat; func n Subspaces_of V -> set means :: RUSUB_4:def 4 for x being set holds x in it iff ex W being strict Subspace of V st W = x & dim W = n; end; theorem :: RUSUB_4:19 for V being finite-dimensional RealUnitarySpace, n being Nat st n <= dim V holds n Subspaces_of V is non empty; theorem :: RUSUB_4:20 for V being finite-dimensional RealUnitarySpace, n being Nat st dim V < n holds n Subspaces_of V = {}; theorem :: RUSUB_4:21 for V being finite-dimensional RealUnitarySpace, W being Subspace of V, n being Nat holds n Subspaces_of W c= n Subspaces_of V; begin :: Affine set definition let V be non empty RLSStruct, S be Subset of V; attr S is Affine means :: RUSUB_4:def 5 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 :: RUSUB_4:22 for V being non empty RLSStruct holds [#]V is Affine & {}V is Affine; theorem :: RUSUB_4:23 for V being RealLinearSpace-like (non empty RLSStruct), v being VECTOR of V holds {v} is Affine; definition let V be non empty RLSStruct; cluster non empty Affine Subset of V; cluster empty Affine Subset of V; end; definition let V be RealLinearSpace, W be Subspace of V; func Up(W) -> non empty Subset of V equals :: RUSUB_4:def 6 the carrier of W; end; definition let V be RealUnitarySpace, W be Subspace of V; func Up(W) -> non empty Subset of V equals :: RUSUB_4:def 7 the carrier of W; end; theorem :: RUSUB_4:24 for V being RealLinearSpace, W being Subspace of V holds Up(W) is Affine & 0.V in the carrier of W; theorem :: RUSUB_4:25 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 ; definition let V be non empty RLSStruct, S be non empty Subset of V; attr S is Subspace-like means :: RUSUB_4:def 8 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 :: RUSUB_4:26 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); theorem :: RUSUB_4:27 for V being RealLinearSpace, W being Subspace of V holds Up(W) is Subspace-like; theorem :: RUSUB_4:28 for V being RealLinearSpace, W being strict Subspace of V holds W = Lin(Up(W)); theorem :: RUSUB_4:29 for V being RealUnitarySpace, A being non empty Affine Subset of V st 0.V in A holds A = the carrier of Lin(A); theorem :: RUSUB_4:30 for V being RealUnitarySpace, W being Subspace of V holds Up(W) is Subspace-like; theorem :: RUSUB_4:31 for V being RealUnitarySpace, W being strict Subspace of V holds W = Lin(Up(W)); definition let V be non empty LoopStr, M be Subset of V, v be Element of V; func v + M -> Subset of V equals :: RUSUB_4:def 9 {v + u where u is Element of V: u in M}; end; theorem :: RUSUB_4:32 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; theorem :: RUSUB_4:33 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; theorem :: RUSUB_4:34 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; definition let V be non empty LoopStr, M,N be Subset of V; func M + N -> Subset of V equals :: RUSUB_4:def 10 {u + v where u,v is Element of V: u in M & v in N}; end; theorem :: RUSUB_4:35 for V be Abelian (non empty LoopStr), M,N be Subset of V holds N + M = M + N; definition let V be Abelian (non empty LoopStr), M,N be Subset of V; redefine func M + N; commutativity; end; theorem :: RUSUB_4:36 for V being non empty LoopStr, M being Subset of V, v being Element of V holds {v} + M = v + M; theorem :: RUSUB_4:37 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; theorem :: RUSUB_4:38 for V being non empty RLSStruct, M,N being Affine Subset of V holds M /\ N is Affine; theorem :: RUSUB_4:39 for V being Abelian add-associative RealLinearSpace-like (non empty RLSStruct), M,N being Affine Subset of V holds M + N is Affine;

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