Volume 3, 1991

University of Bialystok

Copyright (c) 1991 Association of Mizar Users

### The abstract of the Mizar article:

### Free Modules

**by****Michal Muzalewski**- Received October 18, 1991
- MML identifier: MOD_3

- [ Mizar article, MML identifier index ]

environ vocabulary FUNCSDOM, VECTSP_1, ARYTM_1, RLVECT_1, VECTSP_2, RLVECT_2, FINSEQ_1, FINSET_1, FUNCT_1, RELAT_1, SEQ_1, BOOLE, RLVECT_3, RLSUB_1, FUNCT_2, PRELAMB, ZFMISC_1, TARSKI, ORDERS_1, MOD_3, HAHNBAN, FINSEQ_4; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, NAT_1, FINSET_1, FINSEQ_1, RELAT_1, FUNCT_1, STRUCT_0, FUNCT_2, FRAENKEL, FINSEQ_4, RLVECT_1, ORDINAL1, ORDERS_1, RLVECT_2, VECTSP_1, FUNCSDOM, VECTSP_2, VECTSP_4, VECTSP_5, VECTSP_6, LMOD_5; constructors ORDERS_1, VECTSP_5, VECTSP_6, LMOD_5, RLVECT_2, FINSEQ_4, MEMBERED, XBOOLE_0; clusters VECTSP_2, VECTSP_4, RELSET_1, STRUCT_0, RLVECT_2, SUBSET_1, ARYTM_3, VECTSP_1, MEMBERED, ZFMISC_1, XBOOLE_0; requirements SUBSET, BOOLE; begin canceled; theorem :: MOD_3:2 for R being non degenerated add-associative right_zeroed right_complementable (non empty doubleLoopStr) holds 0.R <> -1_ R; reserve x,y for set, R for Ring, V for LeftMod of R, L for Linear_Combination of V, a for Scalar of R, v,u for Vector of V, F,G for FinSequence of the carrier of V, C for finite Subset of V; canceled 3; theorem :: MOD_3:6 Carrier(L) c= C implies ex F st F is one-to-one & rng F = C & Sum(L) = Sum(L (#) F); theorem :: MOD_3:7 Sum(a * L) = a * Sum(L); reserve X,Y,Z for set, A,B for Subset of V, T for finite Subset of V, l for Linear_Combination of A, f,g for Function of the carrier of V,the carrier of R; definition let R,V,A; func Lin(A) -> strict Subspace of V means :: MOD_3:def 1 the carrier of it = {Sum(l) : not contradiction}; end; canceled 3; theorem :: MOD_3:11 x in Lin(A) iff ex l st x = Sum(l); theorem :: MOD_3:12 x in A implies x in Lin(A); theorem :: MOD_3:13 Lin({}(the carrier of V)) = (0).V; theorem :: MOD_3:14 Lin(A) = (0).V implies A = {} or A = {0.V}; theorem :: MOD_3:15 for W being strict Subspace of V holds 0.R <> 1_ R & A = the carrier of W implies Lin(A) = W; theorem :: MOD_3:16 for V being strict LeftMod of R for A being Subset of V holds 0.R <> 1_ R & A = the carrier of V implies Lin(A) = V; theorem :: MOD_3:17 A c= B implies Lin(A) is Subspace of Lin(B); theorem :: MOD_3:18 Lin(A) = V & A c= B implies Lin(B) = V; theorem :: MOD_3:19 Lin(A \/ B) = Lin(A) + Lin(B); theorem :: MOD_3:20 Lin(A /\ B) is Subspace of Lin(A) /\ Lin(B); definition let R,V; let IT be Subset of V; attr IT is base means :: MOD_3:def 2 IT is linearly-independent & Lin(IT) = the VectSpStr of V; end; definition let R; let IT be LeftMod of R; attr IT is free means :: MOD_3:def 3 ex B being Subset of IT st B is base; end; theorem :: MOD_3:21 (0).V is free; definition let R; cluster strict free LeftMod of R; end; reserve R for Skew-Field; reserve a,b for Scalar of R; reserve V for LeftMod of R; reserve v,v1,v2,u for Vector of V; reserve f for Function of the carrier of V,the carrier of R; canceled; theorem :: MOD_3:23 {v} is linearly-independent iff v <> 0.V; theorem :: MOD_3:24 v1 <> v2 & {v1,v2} is linearly-independent iff v2 <> 0.V & for a holds v1 <> a * v2; theorem :: MOD_3:25 v1 <> v2 & {v1,v2} is linearly-independent iff for a,b st a * v1 + b * v2 = 0.V holds a = 0.R & b = 0.R; theorem :: MOD_3:26 for V being LeftMod of R for A being Subset of V holds A is linearly-independent implies ex B being Subset of V st A c= B & B is base; theorem :: MOD_3:27 for V being LeftMod of R for A being Subset of V holds Lin(A) = V implies ex B being Subset of V st B c= A & B is base; theorem :: MOD_3:28 for V being LeftMod of R holds V is free; definition let R; let V be LeftMod of R; canceled; mode Basis of V -> Subset of V means :: MOD_3:def 5 it is base; end; theorem :: MOD_3:29 for V being LeftMod of R for A being Subset of V holds A is linearly-independent implies ex I being Basis of V st A c= I; theorem :: MOD_3:30 for V being LeftMod of R for A being Subset of V holds Lin(A) = V implies ex I being Basis of V st I c= A;

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