Volume 2, 1990

University of Bialystok

Copyright (c) 1990 Association of Mizar Users

### The abstract of the Mizar article:

### Subspaces and Cosets of Subspaces in Vector Space

**by****Wojciech A. Trybulec**- Received July 27, 1990
- MML identifier: VECTSP_4

- [ Mizar article, MML identifier index ]

environ vocabulary RLVECT_1, VECTSP_1, ARYTM_1, BINOP_1, LATTICES, RLSUB_1, BOOLE, RELAT_1, FUNCT_1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1, FUNCT_2, STRUCT_0, DOMAIN_1, BINOP_1, RLVECT_1, VECTSP_1; constructors GROUP_2, DOMAIN_1, BINOP_1, PARTFUN1, MEMBERED, XBOOLE_0; clusters FUNCT_1, VECTSP_1, STRUCT_0, RELSET_1, SUBSET_1, MEMBERED, ZFMISC_1, XBOOLE_0; requirements SUBSET, BOOLE; begin reserve x,y,y1,y2 for set; definition let GF be non empty HGrStr; let V be non empty VectSpStr over GF; let V1 be Subset of V; attr V1 is lineary-closed means :: VECTSP_4:def 1 (for v,u being Element of V st v in V1 & u in V1 holds v + u in V1) & (for a being Element of GF, v being Element of V st v in V1 holds a * v in V1); end; canceled 3; theorem :: VECTSP_4:4 for GF be add-associative right_zeroed right_complementable Abelian associative left_unital distributive (non empty doubleLoopStr), V be Abelian add-associative right_zeroed right_complementable VectSp-like (non empty VectSpStr over GF), V1 be Subset of V st V1 <> {} & V1 is lineary-closed holds 0.V in V1; theorem :: VECTSP_4:5 for GF be add-associative right_zeroed right_complementable Abelian associative left_unital distributive (non empty doubleLoopStr), V be Abelian add-associative right_zeroed right_complementable VectSp-like (non empty VectSpStr over GF), V1 be Subset of V st V1 is lineary-closed for v being Element of V st v in V1 holds - v in V1; theorem :: VECTSP_4:6 for GF be add-associative right_zeroed right_complementable Abelian associative left_unital distributive (non empty doubleLoopStr), V be Abelian add-associative right_zeroed right_complementable VectSp-like (non empty VectSpStr over GF), V1 be Subset of V st V1 is lineary-closed for v,u being Element of V st v in V1 & u in V1 holds v - u in V1; theorem :: VECTSP_4:7 for GF be add-associative right_zeroed right_complementable Abelian associative left_unital distributive (non empty doubleLoopStr), V be Abelian add-associative right_zeroed right_complementable VectSp-like (non empty VectSpStr over GF) holds {0.V} is lineary-closed; theorem :: VECTSP_4:8 for GF be add-associative right_zeroed right_complementable Abelian associative left_unital distributive (non empty doubleLoopStr), V be Abelian add-associative right_zeroed right_complementable VectSp-like (non empty VectSpStr over GF), V1 be Subset of V st the carrier of V = V1 holds V1 is lineary-closed; theorem :: VECTSP_4:9 for GF be add-associative right_zeroed right_complementable Abelian associative left_unital distributive (non empty doubleLoopStr), V be Abelian add-associative right_zeroed right_complementable VectSp-like (non empty VectSpStr over GF), V1,V2,V3 be Subset of V st V1 is lineary-closed & V2 is lineary-closed & V3 = {v + u where v is Element of V, u is Element of V : v in V1 & u in V2} holds V3 is lineary-closed; theorem :: VECTSP_4:10 for GF be add-associative right_zeroed right_complementable Abelian associative left_unital distributive (non empty doubleLoopStr), V be Abelian add-associative right_zeroed right_complementable VectSp-like (non empty VectSpStr over GF), V1,V2 be Subset of V st V1 is lineary-closed & V2 is lineary-closed holds V1 /\ V2 is lineary-closed; definition let GF be add-associative right_zeroed right_complementable Abelian associative left_unital distributive (non empty doubleLoopStr), V be Abelian add-associative right_zeroed right_complementable VectSp-like (non empty VectSpStr over GF); mode Subspace of V -> Abelian add-associative right_zeroed right_complementable VectSp-like (non empty VectSpStr over GF) means :: VECTSP_4:def 2 the carrier of it c= the carrier of V & the Zero of it = the Zero of V & the add of it = (the add of V) | [:the carrier of it,the carrier of it:] & the lmult of it = (the lmult of V) | [:the carrier of GF, the carrier of it:]; end; reserve GF for add-associative right_zeroed right_complementable Abelian associative left_unital distributive (non empty doubleLoopStr), V,X,Y for Abelian add-associative right_zeroed right_complementable VectSp-like (non empty VectSpStr over GF); reserve a for Element of GF; reserve u,u1,u2,v,v1,v2 for Element of V; reserve W,W1,W2 for Subspace of V; reserve V1 for Subset of V; reserve w,w1,w2 for Element of W; canceled 5; theorem :: VECTSP_4:16 x in W1 & W1 is Subspace of W2 implies x in W2; theorem :: VECTSP_4:17 x in W implies x in V; theorem :: VECTSP_4:18 w is Element of V; theorem :: VECTSP_4:19 0.W = 0.V; theorem :: VECTSP_4:20 0.W1 = 0.W2; theorem :: VECTSP_4:21 w1 = v & w2 = u implies w1 + w2 = v + u; theorem :: VECTSP_4:22 w = v implies a * w = a * v; theorem :: VECTSP_4:23 w = v implies - v = - w; theorem :: VECTSP_4:24 w1 = v & w2 = u implies w1 - w2 = v - u; theorem :: VECTSP_4:25 0.V in W; theorem :: VECTSP_4:26 0.W1 in W2; theorem :: VECTSP_4:27 0.W in V; theorem :: VECTSP_4:28 u in W & v in W implies u + v in W; theorem :: VECTSP_4:29 v in W implies a * v in W; theorem :: VECTSP_4:30 v in W implies - v in W; theorem :: VECTSP_4:31 u in W & v in W implies u - v in W; theorem :: VECTSP_4:32 V is Subspace of V; theorem :: VECTSP_4:33 for X,V being strict Abelian add-associative right_zeroed right_complementable VectSp-like (non empty VectSpStr over GF) holds V is Subspace of X & X is Subspace of V implies V = X; theorem :: VECTSP_4:34 V is Subspace of X & X is Subspace of Y implies V is Subspace of Y; theorem :: VECTSP_4:35 the carrier of W1 c= the carrier of W2 implies W1 is Subspace of W2; theorem :: VECTSP_4:36 (for v st v in W1 holds v in W2) implies W1 is Subspace of W2; definition let GF be add-associative right_zeroed right_complementable Abelian associative left_unital distributive (non empty doubleLoopStr), V be Abelian add-associative right_zeroed right_complementable VectSp-like (non empty VectSpStr over GF); cluster strict Subspace of V; end; theorem :: VECTSP_4:37 for W1,W2 being strict Subspace of V st the carrier of W1 = the carrier of W2 holds W1 = W2; theorem :: VECTSP_4:38 for W1,W2 being strict Subspace of V st (for v holds v in W1 iff v in W2) holds W1 = W2; theorem :: VECTSP_4:39 for V being strict Abelian add-associative right_zeroed right_complementable VectSp-like (non empty VectSpStr over GF), W being strict Subspace of V holds the carrier of W = the carrier of V implies W = V; theorem :: VECTSP_4:40 for V being strict Abelian add-associative right_zeroed right_complementable VectSp-like (non empty VectSpStr over GF), W being strict Subspace of V holds (for v being Element of V holds v in W) implies W = V; theorem :: VECTSP_4:41 the carrier of W = V1 implies V1 is lineary-closed; theorem :: VECTSP_4:42 V1 <> {} & V1 is lineary-closed implies ex W being strict Subspace of V st V1 = the carrier of W; definition let GF; let V; func (0).V -> strict Subspace of V means :: VECTSP_4:def 3 the carrier of it = {0.V}; end; definition let GF; let V; func (Omega).V -> strict Subspace of V equals :: VECTSP_4:def 4 the VectSpStr of V; end; canceled 3; theorem :: VECTSP_4:46 x in (0).V iff x = 0.V; theorem :: VECTSP_4:47 (0).W = (0).V; theorem :: VECTSP_4:48 (0).W1 = (0).W2; theorem :: VECTSP_4:49 (0).W is Subspace of V; theorem :: VECTSP_4:50 (0).V is Subspace of W; theorem :: VECTSP_4:51 (0).W1 is Subspace of W2; canceled; theorem :: VECTSP_4:53 for V being strict Abelian add-associative right_zeroed right_complementable VectSp-like (non empty VectSpStr over GF) holds V is Subspace of (Omega).V; definition let GF; let V; let v,W; func v + W -> Subset of V equals :: VECTSP_4:def 5 {v + u : u in W}; end; definition let GF; let V; let W; mode Coset of W -> Subset of V means :: VECTSP_4:def 6 ex v st it = v + W; end; reserve B,C for Coset of W; canceled 3; theorem :: VECTSP_4:57 x in v + W iff ex u st u in W & x = v + u; theorem :: VECTSP_4:58 0.V in v + W iff v in W; theorem :: VECTSP_4:59 v in v + W; theorem :: VECTSP_4:60 0.V + W = the carrier of W; theorem :: VECTSP_4:61 v + (0).V = {v}; theorem :: VECTSP_4:62 v + (Omega).V = the carrier of V; theorem :: VECTSP_4:63 0.V in v + W iff v + W = the carrier of W; theorem :: VECTSP_4:64 v in W iff v + W = the carrier of W; theorem :: VECTSP_4:65 v in W implies (a * v) + W = the carrier of W; theorem :: VECTSP_4:66 for GF being Field, V being VectSp of GF, a being Element of GF, v being Element of V, W being Subspace of V st a <> 0.GF & (a * v) + W = the carrier of W holds v in W; theorem :: VECTSP_4:67 for GF being Field, V being VectSp of GF, v being Element of V, W being Subspace of V holds v in W iff - v + W = the carrier of W; theorem :: VECTSP_4:68 u in W iff v + W = (v + u) + W; theorem :: VECTSP_4:69 u in W iff v + W = (v - u) + W; theorem :: VECTSP_4:70 v in u + W iff u + W = v + W; theorem :: VECTSP_4:71 u in v1 + W & u in v2 + W implies v1 + W = v2 + W; theorem :: VECTSP_4:72 for GF being Field, V being VectSp of GF, a being Element of GF, v being Element of V, W being Subspace of V st a <> 1_ GF & a * v in v + W holds v in W; theorem :: VECTSP_4:73 v in W implies a * v in v + W; theorem :: VECTSP_4:74 v in W implies - v in v + W; theorem :: VECTSP_4:75 u + v in v + W iff u in W; theorem :: VECTSP_4:76 v - u in v + W iff u in W; canceled; theorem :: VECTSP_4:78 u in v + W iff (ex v1 st v1 in W & u = v - v1); theorem :: VECTSP_4:79 (ex v st v1 in v + W & v2 in v + W) iff v1 - v2 in W; theorem :: VECTSP_4:80 v + W = u + W implies (ex v1 st v1 in W & v + v1 = u); theorem :: VECTSP_4:81 v + W = u + W implies (ex v1 st v1 in W & v - v1 = u); theorem :: VECTSP_4:82 for W1,W2 being strict Subspace of V holds v + W1 = v + W2 iff W1 = W2; theorem :: VECTSP_4:83 for W1,W2 being strict Subspace of V st v + W1 = u + W2 holds W1 = W2; theorem :: VECTSP_4:84 ex C st v in C; theorem :: VECTSP_4:85 C is lineary-closed iff C = the carrier of W; theorem :: VECTSP_4:86 for W1,W2 being strict Subspace of V, C1 being Coset of W1, C2 being Coset of W2 st C1 = C2 holds W1 = W2; theorem :: VECTSP_4:87 {v} is Coset of (0).V; theorem :: VECTSP_4:88 V1 is Coset of (0).V implies (ex v st V1 = {v}); theorem :: VECTSP_4:89 the carrier of W is Coset of W; theorem :: VECTSP_4:90 the carrier of V is Coset of (Omega).V; theorem :: VECTSP_4:91 V1 is Coset of (Omega).V implies V1 = the carrier of V; theorem :: VECTSP_4:92 0.V in C iff C = the carrier of W; theorem :: VECTSP_4:93 u in C iff C = u + W; theorem :: VECTSP_4:94 u in C & v in C implies (ex v1 st v1 in W & u + v1 = v); theorem :: VECTSP_4:95 u in C & v in C implies (ex v1 st v1 in W & u - v1 = v); theorem :: VECTSP_4:96 (ex C st v1 in C & v2 in C) iff v1 - v2 in W; theorem :: VECTSP_4:97 u in B & u in C implies B = C; :: :: Auxiliary theorems. :: canceled 5; theorem :: VECTSP_4:103 for GF be add-associative right_zeroed right_complementable Abelian commutative associative left_unital distributive (non empty doubleLoopStr), V be Abelian add-associative right_zeroed right_complementable VectSp-like (non empty VectSpStr over GF), a,b being Element of GF, v being Element of V holds (a - b) * v = a * v - b * v;

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