:: VECTSP_9 semantic presentation begin registration let S be non empty 1-sorted ; cluster non empty for Element of K27( the carrier of S); existence not for b1 being Subset of S holds b1 is empty proof set A = the non empty Subset of S; the non empty Subset of S is Subset of S ; hence not for b1 being Subset of S holds b1 is empty ; ::_thesis: verum end; end; Lm1: for X, x being set st x in X holds (X \ {x}) \/ {x} = X proof let X, x be set ; ::_thesis: ( x in X implies (X \ {x}) \/ {x} = X ) assume x in X ; ::_thesis: (X \ {x}) \/ {x} = X then A1: {x} is Subset of X by SUBSET_1:41; {x} \/ (X \ {x}) = {x} \/ X by XBOOLE_1:39 .= X by A1, XBOOLE_1:12 ; hence (X \ {x}) \/ {x} = X ; ::_thesis: verum end; theorem Th1: :: VECTSP_9:1 for GF being Field for V being VectSp of GF for L being Linear_Combination of V for F, G being FinSequence of the carrier of V for P being Permutation of (dom F) st G = F * P holds Sum (L (#) F) = Sum (L (#) G) proof let GF be Field; ::_thesis: for V being VectSp of GF for L being Linear_Combination of V for F, G being FinSequence of the carrier of V for P being Permutation of (dom F) st G = F * P holds Sum (L (#) F) = Sum (L (#) G) let V be VectSp of GF; ::_thesis: for L being Linear_Combination of V for F, G being FinSequence of the carrier of V for P being Permutation of (dom F) st G = F * P holds Sum (L (#) F) = Sum (L (#) G) let L be Linear_Combination of V; ::_thesis: for F, G being FinSequence of the carrier of V for P being Permutation of (dom F) st G = F * P holds Sum (L (#) F) = Sum (L (#) G) let F, G be FinSequence of the carrier of V; ::_thesis: for P being Permutation of (dom F) st G = F * P holds Sum (L (#) F) = Sum (L (#) G) set p = L (#) F; set q = L (#) G; let P be Permutation of (dom F); ::_thesis: ( G = F * P implies Sum (L (#) F) = Sum (L (#) G) ) assume A1: G = F * P ; ::_thesis: Sum (L (#) F) = Sum (L (#) G) A2: len G = len F by A1, FINSEQ_2:44; len F = len (L (#) F) by VECTSP_6:def_5; then A3: dom F = dom (L (#) F) by FINSEQ_3:29; then reconsider r = (L (#) F) * P as FinSequence of the carrier of V by FINSEQ_2:47; len r = len (L (#) F) by A3, FINSEQ_2:44; then A4: dom r = dom (L (#) F) by FINSEQ_3:29; A5: len (L (#) F) = len F by VECTSP_6:def_5; then A6: dom F = dom (L (#) F) by FINSEQ_3:29; len (L (#) G) = len G by VECTSP_6:def_5; then A7: dom (L (#) F) = dom (L (#) G) by A5, A2, FINSEQ_3:29; A8: dom F = dom G by A2, FINSEQ_3:29; now__::_thesis:_for_k_being_Nat_st_k_in_dom_(L_(#)_G)_holds_ (L_(#)_G)_._k_=_r_._k let k be Nat; ::_thesis: ( k in dom (L (#) G) implies (L (#) G) . k = r . k ) assume A9: k in dom (L (#) G) ; ::_thesis: (L (#) G) . k = r . k set l = P . k; ( dom P = dom F & rng P = dom F ) by FUNCT_2:52, FUNCT_2:def_3; then A10: P . k in dom F by A7, A6, A9, FUNCT_1:def_3; then reconsider l = P . k as Element of NAT ; G /. k = G . k by A7, A8, A6, A9, PARTFUN1:def_6 .= F . (P . k) by A1, A7, A8, A6, A9, FUNCT_1:12 .= F /. l by A10, PARTFUN1:def_6 ; then (L (#) G) . k = (L . (F /. l)) * (F /. l) by A9, VECTSP_6:def_5 .= (L (#) F) . (P . k) by A6, A10, VECTSP_6:def_5 .= r . k by A7, A4, A9, FUNCT_1:12 ; hence (L (#) G) . k = r . k ; ::_thesis: verum end; hence Sum (L (#) F) = Sum (L (#) G) by A3, A7, A4, FINSEQ_1:13, RLVECT_2:7; ::_thesis: verum end; theorem Th2: :: VECTSP_9:2 for GF being Field for V being VectSp of GF for L being Linear_Combination of V for F being FinSequence of the carrier of V st Carrier L misses rng F holds Sum (L (#) F) = 0. V proof let GF be Field; ::_thesis: for V being VectSp of GF for L being Linear_Combination of V for F being FinSequence of the carrier of V st Carrier L misses rng F holds Sum (L (#) F) = 0. V let V be VectSp of GF; ::_thesis: for L being Linear_Combination of V for F being FinSequence of the carrier of V st Carrier L misses rng F holds Sum (L (#) F) = 0. V let L be Linear_Combination of V; ::_thesis: for F being FinSequence of the carrier of V st Carrier L misses rng F holds Sum (L (#) F) = 0. V defpred S1[ FinSequence] means for G being FinSequence of the carrier of V st G = $1 & Carrier L misses rng G holds Sum (L (#) G) = 0. V; A1: for p being FinSequence for x being set st S1[p] holds S1[p ^ <*x*>] proof let p be FinSequence; ::_thesis: for x being set st S1[p] holds S1[p ^ <*x*>] let x be set ; ::_thesis: ( S1[p] implies S1[p ^ <*x*>] ) assume A2: S1[p] ; ::_thesis: S1[p ^ <*x*>] let G be FinSequence of the carrier of V; ::_thesis: ( G = p ^ <*x*> & Carrier L misses rng G implies Sum (L (#) G) = 0. V ) assume A3: G = p ^ <*x*> ; ::_thesis: ( not Carrier L misses rng G or Sum (L (#) G) = 0. V ) then reconsider p = p, x9 = <*x*> as FinSequence of the carrier of V by FINSEQ_1:36; x in {x} by TARSKI:def_1; then A4: x in rng x9 by FINSEQ_1:38; rng x9 c= the carrier of V by FINSEQ_1:def_4; then reconsider x = x as Vector of V by A4; assume Carrier L misses rng G ; ::_thesis: Sum (L (#) G) = 0. V then A5: {} = (Carrier L) /\ (rng G) by XBOOLE_0:def_7 .= (Carrier L) /\ ((rng p) \/ (rng <*x*>)) by A3, FINSEQ_1:31 .= (Carrier L) /\ ((rng p) \/ {x}) by FINSEQ_1:38 .= ((Carrier L) /\ (rng p)) \/ ((Carrier L) /\ {x}) by XBOOLE_1:23 ; then (Carrier L) /\ (rng p) = {} ; then Carrier L misses rng p by XBOOLE_0:def_7; then A6: Sum (L (#) p) = 0. V by A2; now__::_thesis:_not_x_in_Carrier_L A7: x in {x} by TARSKI:def_1; assume x in Carrier L ; ::_thesis: contradiction then x in (Carrier L) /\ {x} by A7, XBOOLE_0:def_4; hence contradiction by A5; ::_thesis: verum end; then A8: L . x = 0. GF by VECTSP_6:2; Sum (L (#) G) = Sum ((L (#) p) ^ (L (#) x9)) by A3, VECTSP_6:13 .= (Sum (L (#) p)) + (Sum (L (#) x9)) by RLVECT_1:41 .= (0. V) + (Sum <*((L . x) * x)*>) by A6, VECTSP_6:10 .= Sum <*((L . x) * x)*> by RLVECT_1:4 .= (0. GF) * x by A8, RLVECT_1:44 .= 0. V by VECTSP_1:15 ; hence Sum (L (#) G) = 0. V ; ::_thesis: verum end; A9: S1[ {} ] proof let G be FinSequence of the carrier of V; ::_thesis: ( G = {} & Carrier L misses rng G implies Sum (L (#) G) = 0. V ) assume G = {} ; ::_thesis: ( not Carrier L misses rng G or Sum (L (#) G) = 0. V ) then A10: L (#) G = <*> the carrier of V by VECTSP_6:9; assume Carrier L misses rng G ; ::_thesis: Sum (L (#) G) = 0. V thus Sum (L (#) G) = 0. V by A10, RLVECT_1:43; ::_thesis: verum end; A11: for p being FinSequence holds S1[p] from FINSEQ_1:sch_3(A9, A1); let F be FinSequence of the carrier of V; ::_thesis: ( Carrier L misses rng F implies Sum (L (#) F) = 0. V ) assume Carrier L misses rng F ; ::_thesis: Sum (L (#) F) = 0. V hence Sum (L (#) F) = 0. V by A11; ::_thesis: verum end; theorem Th3: :: VECTSP_9:3 for GF being Field for V being VectSp of GF for F being FinSequence of the carrier of V st F is one-to-one holds for L being Linear_Combination of V st Carrier L c= rng F holds Sum (L (#) F) = Sum L proof let GF be Field; ::_thesis: for V being VectSp of GF for F being FinSequence of the carrier of V st F is one-to-one holds for L being Linear_Combination of V st Carrier L c= rng F holds Sum (L (#) F) = Sum L let V be VectSp of GF; ::_thesis: for F being FinSequence of the carrier of V st F is one-to-one holds for L being Linear_Combination of V st Carrier L c= rng F holds Sum (L (#) F) = Sum L let F be FinSequence of the carrier of V; ::_thesis: ( F is one-to-one implies for L being Linear_Combination of V st Carrier L c= rng F holds Sum (L (#) F) = Sum L ) assume A1: F is one-to-one ; ::_thesis: for L being Linear_Combination of V st Carrier L c= rng F holds Sum (L (#) F) = Sum L rng F c= rng F ; then reconsider X = rng F as Subset of (rng F) ; let L be Linear_Combination of V; ::_thesis: ( Carrier L c= rng F implies Sum (L (#) F) = Sum L ) assume A2: Carrier L c= rng F ; ::_thesis: Sum (L (#) F) = Sum L consider G being FinSequence of the carrier of V such that A3: G is one-to-one and A4: rng G = Carrier L and A5: Sum L = Sum (L (#) G) by VECTSP_6:def_6; reconsider A = rng G as Subset of (rng F) by A2, A4; set F1 = F - (A `); X \ (A `) = X /\ ((A `) `) by SUBSET_1:13 .= A by XBOOLE_1:28 ; then A6: rng (F - (A `)) = rng G by FINSEQ_3:65; F - (A `) is one-to-one by A1, FINSEQ_3:87; then F - (A `),G are_fiberwise_equipotent by A3, A6, RFINSEQ:26; then A7: ex Q being Permutation of (dom G) st F - (A `) = G * Q by RFINSEQ:4; reconsider F1 = F - (A `), F2 = F - A as FinSequence of the carrier of V by FINSEQ_3:86; rng F2 = (rng F) \ (rng G) by FINSEQ_3:65; then A8: rng F2 misses rng G by XBOOLE_1:79; ex P being Permutation of (dom F) st (F - (A `)) ^ (F - A) = F * P by FINSEQ_3:115; then Sum (L (#) F) = Sum (L (#) (F1 ^ F2)) by Th1 .= Sum ((L (#) F1) ^ (L (#) F2)) by VECTSP_6:13 .= (Sum (L (#) F1)) + (Sum (L (#) F2)) by RLVECT_1:41 .= (Sum (L (#) F1)) + (0. V) by A4, A8, Th2 .= (Sum (L (#) G)) + (0. V) by A7, Th1 .= Sum L by A5, RLVECT_1:4 ; hence Sum (L (#) F) = Sum L ; ::_thesis: verum end; theorem Th4: :: VECTSP_9:4 for GF being Field for V being VectSp of GF for L being Linear_Combination of V for F being FinSequence of the carrier of V ex K being Linear_Combination of V st ( Carrier K = (rng F) /\ (Carrier L) & L (#) F = K (#) F ) proof let GF be Field; ::_thesis: for V being VectSp of GF for L being Linear_Combination of V for F being FinSequence of the carrier of V ex K being Linear_Combination of V st ( Carrier K = (rng F) /\ (Carrier L) & L (#) F = K (#) F ) let V be VectSp of GF; ::_thesis: for L being Linear_Combination of V for F being FinSequence of the carrier of V ex K being Linear_Combination of V st ( Carrier K = (rng F) /\ (Carrier L) & L (#) F = K (#) F ) let L be Linear_Combination of V; ::_thesis: for F being FinSequence of the carrier of V ex K being Linear_Combination of V st ( Carrier K = (rng F) /\ (Carrier L) & L (#) F = K (#) F ) let F be FinSequence of the carrier of V; ::_thesis: ex K being Linear_Combination of V st ( Carrier K = (rng F) /\ (Carrier L) & L (#) F = K (#) F ) defpred S1[ set , set ] means ( not $1 is Vector of V or ( $1 in rng F & $2 = L . $1 ) or ( not $1 in rng F & $2 = 0. GF ) ); reconsider R = rng F as finite Subset of V by FINSEQ_1:def_4; A1: for x being set st x in the carrier of V holds ex y being set st ( y in the carrier of GF & S1[x,y] ) proof let x be set ; ::_thesis: ( x in the carrier of V implies ex y being set st ( y in the carrier of GF & S1[x,y] ) ) assume x in the carrier of V ; ::_thesis: ex y being set st ( y in the carrier of GF & S1[x,y] ) then reconsider x9 = x as Vector of V ; percases ( x in rng F or not x in rng F ) ; suppose x in rng F ; ::_thesis: ex y being set st ( y in the carrier of GF & S1[x,y] ) then S1[x,L . x9] ; hence ex y being set st ( y in the carrier of GF & S1[x,y] ) ; ::_thesis: verum end; suppose not x in rng F ; ::_thesis: ex y being set st ( y in the carrier of GF & S1[x,y] ) hence ex y being set st ( y in the carrier of GF & S1[x,y] ) ; ::_thesis: verum end; end; end; ex K being Function of the carrier of V, the carrier of GF st for x being set st x in the carrier of V holds S1[x,K . x] from FUNCT_2:sch_1(A1); then consider K being Function of the carrier of V, the carrier of GF such that A2: for x being set st x in the carrier of V holds S1[x,K . x] ; A3: now__::_thesis:_for_v_being_Vector_of_V_st_not_v_in_R_/\_(Carrier_L)_holds_ K_._v_=_0._GF let v be Vector of V; ::_thesis: ( not v in R /\ (Carrier L) implies K . b1 = 0. GF ) assume A4: not v in R /\ (Carrier L) ; ::_thesis: K . b1 = 0. GF percases ( not v in R or not v in Carrier L ) by A4, XBOOLE_0:def_4; suppose not v in R ; ::_thesis: K . b1 = 0. GF hence K . v = 0. GF by A2; ::_thesis: verum end; supposeA5: not v in Carrier L ; ::_thesis: K . b1 = 0. GF ( ( S1[v,K . v] & S1[v,L . v] ) or ( S1[v,K . v] & S1[v, 0. GF] ) ) by A2; hence K . v = 0. GF by A5, VECTSP_6:2; ::_thesis: verum end; end; end; reconsider K = K as Element of Funcs ( the carrier of V, the carrier of GF) by FUNCT_2:8; reconsider K = K as Linear_Combination of V by A3, VECTSP_6:def_1; now__::_thesis:_for_v_being_set_st_v_in_(rng_F)_/\_(Carrier_L)_holds_ v_in_Carrier_K let v be set ; ::_thesis: ( v in (rng F) /\ (Carrier L) implies v in Carrier K ) assume A6: v in (rng F) /\ (Carrier L) ; ::_thesis: v in Carrier K then reconsider v9 = v as Vector of V ; v in Carrier L by A6, XBOOLE_0:def_4; then A7: L . v9 <> 0. GF by VECTSP_6:2; v in R by A6, XBOOLE_0:def_4; then K . v9 = L . v9 by A2; hence v in Carrier K by A7, VECTSP_6:1; ::_thesis: verum end; then A8: (rng F) /\ (Carrier L) c= Carrier K by TARSKI:def_3; take K ; ::_thesis: ( Carrier K = (rng F) /\ (Carrier L) & L (#) F = K (#) F ) A9: L (#) F = K (#) F proof set p = L (#) F; set q = K (#) F; A10: len (L (#) F) = len F by VECTSP_6:def_5; len (K (#) F) = len F by VECTSP_6:def_5; then A11: dom (L (#) F) = dom (K (#) F) by A10, FINSEQ_3:29; A12: dom (L (#) F) = dom F by A10, FINSEQ_3:29; now__::_thesis:_for_k_being_Nat_st_k_in_dom_(L_(#)_F)_holds_ (L_(#)_F)_._k_=_(K_(#)_F)_._k let k be Nat; ::_thesis: ( k in dom (L (#) F) implies (L (#) F) . k = (K (#) F) . k ) set u = F /. k; A13: S1[F /. k,K . (F /. k)] by A2; assume A14: k in dom (L (#) F) ; ::_thesis: (L (#) F) . k = (K (#) F) . k then ( F /. k = F . k & (L (#) F) . k = (L . (F /. k)) * (F /. k) ) by A12, PARTFUN1:def_6, VECTSP_6:def_5; hence (L (#) F) . k = (K (#) F) . k by A11, A12, A14, A13, FUNCT_1:def_3, VECTSP_6:def_5; ::_thesis: verum end; hence L (#) F = K (#) F by A11, FINSEQ_1:13; ::_thesis: verum end; now__::_thesis:_for_v_being_set_st_v_in_Carrier_K_holds_ v_in_(rng_F)_/\_(Carrier_L) let v be set ; ::_thesis: ( v in Carrier K implies v in (rng F) /\ (Carrier L) ) assume v in Carrier K ; ::_thesis: v in (rng F) /\ (Carrier L) then ex v9 being Vector of V st ( v9 = v & K . v9 <> 0. GF ) by VECTSP_6:1; hence v in (rng F) /\ (Carrier L) by A3; ::_thesis: verum end; then Carrier K c= (rng F) /\ (Carrier L) by TARSKI:def_3; hence ( Carrier K = (rng F) /\ (Carrier L) & L (#) F = K (#) F ) by A8, A9, XBOOLE_0:def_10; ::_thesis: verum end; theorem Th5: :: VECTSP_9:5 for GF being Field for V being VectSp of GF for L being Linear_Combination of V for A being Subset of V for F being FinSequence of the carrier of V st rng F c= the carrier of (Lin A) holds ex K being Linear_Combination of A st Sum (L (#) F) = Sum K proof let GF be Field; ::_thesis: for V being VectSp of GF for L being Linear_Combination of V for A being Subset of V for F being FinSequence of the carrier of V st rng F c= the carrier of (Lin A) holds ex K being Linear_Combination of A st Sum (L (#) F) = Sum K let V be VectSp of GF; ::_thesis: for L being Linear_Combination of V for A being Subset of V for F being FinSequence of the carrier of V st rng F c= the carrier of (Lin A) holds ex K being Linear_Combination of A st Sum (L (#) F) = Sum K let L be Linear_Combination of V; ::_thesis: for A being Subset of V for F being FinSequence of the carrier of V st rng F c= the carrier of (Lin A) holds ex K being Linear_Combination of A st Sum (L (#) F) = Sum K let A be Subset of V; ::_thesis: for F being FinSequence of the carrier of V st rng F c= the carrier of (Lin A) holds ex K being Linear_Combination of A st Sum (L (#) F) = Sum K defpred S1[ Nat] means for F being FinSequence of the carrier of V st rng F c= the carrier of (Lin A) & len F = $1 holds ex K being Linear_Combination of A st Sum (L (#) F) = Sum K; A1: for n being Nat st S1[n] holds S1[n + 1] proof let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] ) assume A2: S1[n] ; ::_thesis: S1[n + 1] let F be FinSequence of the carrier of V; ::_thesis: ( rng F c= the carrier of (Lin A) & len F = n + 1 implies ex K being Linear_Combination of A st Sum (L (#) F) = Sum K ) assume that A3: rng F c= the carrier of (Lin A) and A4: len F = n + 1 ; ::_thesis: ex K being Linear_Combination of A st Sum (L (#) F) = Sum K consider G being FinSequence, x being set such that A5: F = G ^ <*x*> by A4, FINSEQ_2:18; reconsider G = G, x9 = <*x*> as FinSequence of the carrier of V by A5, FINSEQ_1:36; A6: rng (G ^ <*x*>) = (rng G) \/ (rng <*x*>) by FINSEQ_1:31 .= (rng G) \/ {x} by FINSEQ_1:38 ; then A7: rng G c= rng F by A5, XBOOLE_1:7; {x} c= rng F by A5, A6, XBOOLE_1:7; then {x} c= the carrier of (Lin A) by A3, XBOOLE_1:1; then ( x in {x} implies x in the carrier of (Lin A) ) ; then A8: x in Lin A by STRUCT_0:def_5, TARSKI:def_1; then consider LA1 being Linear_Combination of A such that A9: x = Sum LA1 by VECTSP_7:7; x in V by A8, VECTSP_4:9; then reconsider x = x as Vector of V by STRUCT_0:def_5; len (G ^ <*x*>) = (len G) + (len <*x*>) by FINSEQ_1:22 .= (len G) + 1 by FINSEQ_1:39 ; then consider LA2 being Linear_Combination of A such that A10: Sum (L (#) G) = Sum LA2 by A2, A3, A4, A5, A7, XBOOLE_1:1; (L . x) * LA1 is Linear_Combination of A by VECTSP_6:31; then A11: LA2 + ((L . x) * LA1) is Linear_Combination of A by VECTSP_6:24; Sum (L (#) F) = Sum ((L (#) G) ^ (L (#) x9)) by A5, VECTSP_6:13 .= (Sum LA2) + (Sum (L (#) x9)) by A10, RLVECT_1:41 .= (Sum LA2) + (Sum <*((L . x) * x)*>) by VECTSP_6:10 .= (Sum LA2) + ((L . x) * (Sum LA1)) by A9, RLVECT_1:44 .= (Sum LA2) + (Sum ((L . x) * LA1)) by VECTSP_6:45 .= Sum (LA2 + ((L . x) * LA1)) by VECTSP_6:44 ; hence ex K being Linear_Combination of A st Sum (L (#) F) = Sum K by A11; ::_thesis: verum end; let F be FinSequence of the carrier of V; ::_thesis: ( rng F c= the carrier of (Lin A) implies ex K being Linear_Combination of A st Sum (L (#) F) = Sum K ) assume A12: rng F c= the carrier of (Lin A) ; ::_thesis: ex K being Linear_Combination of A st Sum (L (#) F) = Sum K A13: len F is Nat ; A14: S1[ 0 ] proof let F be FinSequence of the carrier of V; ::_thesis: ( rng F c= the carrier of (Lin A) & len F = 0 implies ex K being Linear_Combination of A st Sum (L (#) F) = Sum K ) assume that rng F c= the carrier of (Lin A) and A15: len F = 0 ; ::_thesis: ex K being Linear_Combination of A st Sum (L (#) F) = Sum K F = <*> the carrier of V by A15; then L (#) F = <*> the carrier of V by VECTSP_6:9; then A16: Sum (L (#) F) = 0. V by RLVECT_1:43 .= Sum (ZeroLC V) by VECTSP_6:15 ; ZeroLC V is Linear_Combination of A by VECTSP_6:5; hence ex K being Linear_Combination of A st Sum (L (#) F) = Sum K by A16; ::_thesis: verum end; for n being Nat holds S1[n] from NAT_1:sch_2(A14, A1); hence ex K being Linear_Combination of A st Sum (L (#) F) = Sum K by A12, A13; ::_thesis: verum end; theorem Th6: :: VECTSP_9:6 for GF being Field for V being VectSp of GF for L being Linear_Combination of V for A being Subset of V st Carrier L c= the carrier of (Lin A) holds ex K being Linear_Combination of A st Sum L = Sum K proof let GF be Field; ::_thesis: for V being VectSp of GF for L being Linear_Combination of V for A being Subset of V st Carrier L c= the carrier of (Lin A) holds ex K being Linear_Combination of A st Sum L = Sum K let V be VectSp of GF; ::_thesis: for L being Linear_Combination of V for A being Subset of V st Carrier L c= the carrier of (Lin A) holds ex K being Linear_Combination of A st Sum L = Sum K let L be Linear_Combination of V; ::_thesis: for A being Subset of V st Carrier L c= the carrier of (Lin A) holds ex K being Linear_Combination of A st Sum L = Sum K let A be Subset of V; ::_thesis: ( Carrier L c= the carrier of (Lin A) implies ex K being Linear_Combination of A st Sum L = Sum K ) consider F being FinSequence of the carrier of V such that F is one-to-one and A1: rng F = Carrier L and A2: Sum L = Sum (L (#) F) by VECTSP_6:def_6; assume Carrier L c= the carrier of (Lin A) ; ::_thesis: ex K being Linear_Combination of A st Sum L = Sum K then consider K being Linear_Combination of A such that A3: Sum (L (#) F) = Sum K by A1, Th5; take K ; ::_thesis: Sum L = Sum K thus Sum L = Sum K by A2, A3; ::_thesis: verum end; theorem Th7: :: VECTSP_9:7 for GF being Field for V being VectSp of GF for W being Subspace of V for L being Linear_Combination of V st Carrier L c= the carrier of W holds for K being Linear_Combination of W st K = L | the carrier of W holds ( Carrier L = Carrier K & Sum L = Sum K ) proof let GF be Field; ::_thesis: for V being VectSp of GF for W being Subspace of V for L being Linear_Combination of V st Carrier L c= the carrier of W holds for K being Linear_Combination of W st K = L | the carrier of W holds ( Carrier L = Carrier K & Sum L = Sum K ) let V be VectSp of GF; ::_thesis: for W being Subspace of V for L being Linear_Combination of V st Carrier L c= the carrier of W holds for K being Linear_Combination of W st K = L | the carrier of W holds ( Carrier L = Carrier K & Sum L = Sum K ) let W be Subspace of V; ::_thesis: for L being Linear_Combination of V st Carrier L c= the carrier of W holds for K being Linear_Combination of W st K = L | the carrier of W holds ( Carrier L = Carrier K & Sum L = Sum K ) let L be Linear_Combination of V; ::_thesis: ( Carrier L c= the carrier of W implies for K being Linear_Combination of W st K = L | the carrier of W holds ( Carrier L = Carrier K & Sum L = Sum K ) ) assume A1: Carrier L c= the carrier of W ; ::_thesis: for K being Linear_Combination of W st K = L | the carrier of W holds ( Carrier L = Carrier K & Sum L = Sum K ) let K be Linear_Combination of W; ::_thesis: ( K = L | the carrier of W implies ( Carrier L = Carrier K & Sum L = Sum K ) ) assume A2: K = L | the carrier of W ; ::_thesis: ( Carrier L = Carrier K & Sum L = Sum K ) A3: dom K = the carrier of W by FUNCT_2:def_1; now__::_thesis:_for_x_being_set_st_x_in_Carrier_K_holds_ x_in_Carrier_L let x be set ; ::_thesis: ( x in Carrier K implies x in Carrier L ) assume x in Carrier K ; ::_thesis: x in Carrier L then consider w being Vector of W such that A4: x = w and A5: K . w <> 0. GF by VECTSP_6:1; A6: w is Vector of V by VECTSP_4:10; L . w <> 0. GF by A2, A3, A5, FUNCT_1:47; hence x in Carrier L by A4, A6, VECTSP_6:1; ::_thesis: verum end; then A7: Carrier K c= Carrier L by TARSKI:def_3; consider G being FinSequence of the carrier of W such that A8: ( G is one-to-one & rng G = Carrier K ) and A9: Sum K = Sum (K (#) G) by VECTSP_6:def_6; 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 VECTSP_6:def_6; now__::_thesis:_for_x_being_set_st_x_in_Carrier_L_holds_ x_in_Carrier_K let x be set ; ::_thesis: ( x in Carrier L implies x in Carrier K ) assume A13: x in Carrier L ; ::_thesis: x in Carrier K then consider v being Vector of V such that A14: x = v and A15: L . v <> 0. GF by VECTSP_6:1; K . v <> 0. GF by A1, A2, A3, A13, A14, A15, FUNCT_1:47; hence x in Carrier K by A1, A13, A14, VECTSP_6:1; ::_thesis: verum end; then A16: Carrier L c= Carrier K by TARSKI:def_3; then A17: Carrier K = Carrier L by A7, XBOOLE_0:def_10; then F,G are_fiberwise_equipotent by A10, A11, A8, RFINSEQ:26; then consider P being Permutation of (dom G) such that A18: F = G * P by RFINSEQ:4; len (K (#) G) = len G by VECTSP_6:def_5; then A19: dom (K (#) G) = dom G by FINSEQ_3:29; then reconsider q = (K (#) G) * P as FinSequence of the carrier of W by FINSEQ_2:47; A20: len q = len (K (#) G) by A19, FINSEQ_2:44; then len q = len G by VECTSP_6:def_5; then A21: dom q = dom G by FINSEQ_3:29; set p = L (#) F; A22: the carrier of W c= the carrier of V by VECTSP_4:def_2; rng q c= the carrier of W by FINSEQ_1:def_4; then rng q c= the carrier of V by A22, XBOOLE_1:1; then reconsider q9 = q as FinSequence of the carrier of V by FINSEQ_1:def_4; consider f being Function of NAT, the carrier of W such that A23: Sum q = f . (len q) and A24: f . 0 = 0. W and A25: for i being Element of NAT for w being Vector of W st i < len q & w = q . (i + 1) holds f . (i + 1) = (f . i) + w by RLVECT_1:def_12; ( dom f = NAT & rng f c= the carrier of W ) by FUNCT_2:def_1, RELAT_1:def_19; then reconsider f9 = f as Function of NAT, the carrier of V by A22, FUNCT_2:2, XBOOLE_1:1; A26: for i being Element of NAT for v being Vector of V st i < len q9 & v = q9 . (i + 1) holds f9 . (i + 1) = (f9 . i) + v proof let i be Element of NAT ; ::_thesis: for v being Vector of V st i < len q9 & v = q9 . (i + 1) holds f9 . (i + 1) = (f9 . i) + v let v be Vector of V; ::_thesis: ( i < len q9 & v = q9 . (i + 1) implies f9 . (i + 1) = (f9 . i) + v ) assume that A27: i < len q9 and A28: v = q9 . (i + 1) ; ::_thesis: f9 . (i + 1) = (f9 . i) + v ( 1 <= i + 1 & i + 1 <= len q ) by A27, NAT_1:11, NAT_1:13; then i + 1 in dom q by FINSEQ_3:25; then reconsider v9 = v as Vector of W by A28, FINSEQ_2:11; f . (i + 1) = (f . i) + v9 by A25, A27, A28; hence f9 . (i + 1) = (f9 . i) + v by VECTSP_4:13; ::_thesis: verum end; A29: len G = len F by A18, FINSEQ_2:44; then A30: dom G = dom F by FINSEQ_3:29; len G = len (L (#) F) by A29, VECTSP_6:def_5; then A31: dom (L (#) F) = dom G by FINSEQ_3:29; A32: dom q = dom (K (#) G) by A20, FINSEQ_3:29; now__::_thesis:_for_i_being_Nat_st_i_in_dom_(L_(#)_F)_holds_ (L_(#)_F)_._i_=_q_._i let i be Nat; ::_thesis: ( i in dom (L (#) F) implies (L (#) F) . i = q . i ) set v = F /. i; set j = P . i; assume A33: i in dom (L (#) F) ; ::_thesis: (L (#) F) . i = q . i then A34: F /. i = F . i by A31, A30, PARTFUN1:def_6; then F /. i in rng F by A31, A30, A33, FUNCT_1:def_3; then reconsider w = F /. i as Vector of W by A17, A11; ( dom P = dom G & rng P = dom G ) by FUNCT_2:52, FUNCT_2:def_3; then A35: P . i in dom G by A31, A33, FUNCT_1:def_3; then reconsider j = P . i as Element of NAT ; A36: G /. j = G . (P . i) by A35, PARTFUN1:def_6 .= F /. i by A18, A31, A30, A33, A34, FUNCT_1:12 ; q . i = (K (#) G) . j by A31, A21, A33, FUNCT_1:12 .= (K . w) * w by A32, A21, A35, A36, VECTSP_6:def_5 .= (L . (F /. i)) * w by A2, A3, FUNCT_1:47 .= (L . (F /. i)) * (F /. i) by VECTSP_4:14 ; hence (L (#) F) . i = q . i by A33, VECTSP_6:def_5; ::_thesis: verum end; then A37: L (#) F = (K (#) G) * P by A31, A21, FINSEQ_1:13; len G = len (K (#) G) by VECTSP_6:def_5; then dom G = dom (K (#) G) by FINSEQ_3:29; then reconsider P = P as Permutation of (dom (K (#) G)) ; q = (K (#) G) * P ; then A38: Sum (K (#) G) = Sum q by RLVECT_2:7; A39: f9 . (len q9) is Element of V ; f9 . 0 = 0. V by A24, VECTSP_4:11; hence ( Carrier L = Carrier K & Sum L = Sum K ) by A7, A16, A12, A9, A37, A38, A23, A26, A39, RLVECT_1:def_12, XBOOLE_0:def_10; ::_thesis: verum end; theorem Th8: :: VECTSP_9:8 for GF being Field for V being VectSp of GF for W being Subspace of V for K being Linear_Combination of W ex L being Linear_Combination of V st ( Carrier K = Carrier L & Sum K = Sum L ) proof let GF be Field; ::_thesis: for V being VectSp of GF for W being Subspace of V for K being Linear_Combination of W ex L being Linear_Combination of V st ( Carrier K = Carrier L & Sum K = Sum L ) let V be VectSp of GF; ::_thesis: for W being Subspace of V for K being Linear_Combination of W ex L being Linear_Combination of V st ( Carrier K = Carrier L & Sum K = Sum L ) let W be Subspace of V; ::_thesis: for K being Linear_Combination of W ex L being Linear_Combination of V st ( Carrier K = Carrier L & Sum K = Sum L ) let K be Linear_Combination of W; ::_thesis: ex L being Linear_Combination of V st ( Carrier K = Carrier L & Sum K = Sum L ) defpred S1[ set , set ] means ( ( $1 in W & $2 = K . $1 ) or ( not $1 in W & $2 = 0. GF ) ); reconsider K9 = K as Function of the carrier of W, the carrier of GF ; A1: the carrier of W c= the carrier of V by VECTSP_4:def_2; then reconsider C = Carrier K as finite Subset of V by XBOOLE_1:1; A2: for x being set st x in the carrier of V holds ex y being set st ( y in the carrier of GF & S1[x,y] ) proof let x be set ; ::_thesis: ( x in the carrier of V implies ex y being set st ( y in the carrier of GF & S1[x,y] ) ) assume x in the carrier of V ; ::_thesis: ex y being set st ( y in the carrier of GF & S1[x,y] ) then reconsider x = x as Vector of V ; percases ( x in W or not x in W ) ; supposeA3: x in W ; ::_thesis: ex y being set st ( y in the carrier of GF & S1[x,y] ) then reconsider x = x as Vector of W by STRUCT_0:def_5; S1[x,K . x] by A3; hence ex y being set st ( y in the carrier of GF & S1[x,y] ) ; ::_thesis: verum end; suppose not x in W ; ::_thesis: ex y being set st ( y in the carrier of GF & S1[x,y] ) hence ex y being set st ( y in the carrier of GF & S1[x,y] ) ; ::_thesis: verum end; end; end; ex L being Function of the carrier of V, the carrier of GF st for x being set st x in the carrier of V holds S1[x,L . x] from FUNCT_2:sch_1(A2); then consider L being Function of the carrier of V, the carrier of GF such that A4: for x being set st x in the carrier of V holds S1[x,L . x] ; A5: now__::_thesis:_for_v_being_Vector_of_V_st_not_v_in_C_holds_ L_._v_=_0._GF let v be Vector of V; ::_thesis: ( not v in C implies L . v = 0. GF ) assume not v in C ; ::_thesis: L . v = 0. GF then ( ( S1[v,K . v] & not v in C & v in the carrier of W ) or S1[v, 0. GF] ) by STRUCT_0:def_5; then ( ( S1[v,K . v] & K . v = 0. GF ) or S1[v, 0. GF] ) by VECTSP_6:2; hence L . v = 0. GF by A4; ::_thesis: verum end; L is Element of Funcs ( the carrier of V, the carrier of GF) by FUNCT_2:8; then reconsider L = L as Linear_Combination of V by A5, VECTSP_6:def_1; reconsider L9 = L | the carrier of W as Function of the carrier of W, the carrier of GF by A1, FUNCT_2:32; take L ; ::_thesis: ( Carrier K = Carrier L & Sum K = Sum L ) now__::_thesis:_for_x_being_set_st_x_in_Carrier_L_holds_ x_in_the_carrier_of_W let x be set ; ::_thesis: ( x in Carrier L implies x in the carrier of W ) assume that A6: x in Carrier L and A7: not x in the carrier of W ; ::_thesis: contradiction consider v being Vector of V such that A8: x = v and A9: L . v <> 0. GF by A6, VECTSP_6:1; S1[v, 0. GF] by A7, A8, STRUCT_0:def_5; hence contradiction by A4, A9; ::_thesis: verum end; then A10: Carrier L c= the carrier of W by TARSKI:def_3; now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_W_holds_ K9_._x_=_L9_._x let x be set ; ::_thesis: ( x in the carrier of W implies K9 . x = L9 . x ) assume A11: x in the carrier of W ; ::_thesis: K9 . x = L9 . x then S1[x,L . x] by A4, A1; hence K9 . x = L9 . x by A11, FUNCT_1:49, STRUCT_0:def_5; ::_thesis: verum end; then K9 = L9 by FUNCT_2:12; hence ( Carrier K = Carrier L & Sum K = Sum L ) by A10, Th7; ::_thesis: verum end; theorem Th9: :: VECTSP_9:9 for GF being Field for V being VectSp of GF for W being Subspace of V for L being Linear_Combination of V st Carrier L c= the carrier of W holds ex K being Linear_Combination of W st ( Carrier K = Carrier L & Sum K = Sum L ) proof let GF be Field; ::_thesis: for V being VectSp of GF for W being Subspace of V for L being Linear_Combination of V st Carrier L c= the carrier of W holds ex K being Linear_Combination of W st ( Carrier K = Carrier L & Sum K = Sum L ) let V be VectSp of GF; ::_thesis: for W being Subspace of V for L being Linear_Combination of V st Carrier L c= the carrier of W holds ex K being Linear_Combination of W st ( Carrier K = Carrier L & Sum K = Sum L ) let W be Subspace of V; ::_thesis: for L being Linear_Combination of V st Carrier L c= the carrier of W holds ex K being Linear_Combination of W st ( Carrier K = Carrier L & Sum K = Sum L ) let L be Linear_Combination of V; ::_thesis: ( Carrier L c= the carrier of W implies ex K being Linear_Combination of W st ( Carrier K = Carrier L & Sum K = Sum L ) ) assume A1: Carrier L c= the carrier of W ; ::_thesis: ex K being Linear_Combination of W st ( Carrier K = Carrier L & Sum K = Sum L ) then reconsider C = Carrier L as finite Subset of W ; the carrier of W c= the carrier of V by VECTSP_4:def_2; then reconsider K = L | the carrier of W as Function of the carrier of W, the carrier of GF by FUNCT_2:32; A2: K is Element of Funcs ( the carrier of W, the carrier of GF) by FUNCT_2:8; A3: dom K = the carrier of W by FUNCT_2:def_1; now__::_thesis:_for_w_being_Vector_of_W_st_not_w_in_C_holds_ K_._w_=_0._GF let w be Vector of W; ::_thesis: ( not w in C implies K . w = 0. GF ) A4: w is Vector of V by VECTSP_4:10; assume not w in C ; ::_thesis: K . w = 0. GF then L . w = 0. GF by A4, VECTSP_6:2; hence K . w = 0. GF by A3, FUNCT_1:47; ::_thesis: verum end; then reconsider K = K as Linear_Combination of W by A2, VECTSP_6:def_1; take K ; ::_thesis: ( Carrier K = Carrier L & Sum K = Sum L ) thus ( Carrier K = Carrier L & Sum K = Sum L ) by A1, Th7; ::_thesis: verum end; theorem Th10: :: VECTSP_9:10 for GF being Field for V being VectSp of GF for I being Basis of V for v being Vector of V holds v in Lin I proof let GF be Field; ::_thesis: for V being VectSp of GF for I being Basis of V for v being Vector of V holds v in Lin I let V be VectSp of GF; ::_thesis: for I being Basis of V for v being Vector of V holds v in Lin I let I be Basis of V; ::_thesis: for v being Vector of V holds v in Lin I let v be Vector of V; ::_thesis: v in Lin I v in VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) by STRUCT_0:def_5; hence v in Lin I by VECTSP_7:def_3; ::_thesis: verum end; registration let GF be Field; let V be VectSp of GF; cluster linearly-independent for Element of K27( the carrier of V); existence ex b1 being Subset of V st b1 is linearly-independent proof take {} V ; ::_thesis: {} V is linearly-independent thus {} V is linearly-independent ; ::_thesis: verum end; end; theorem Th11: :: VECTSP_9:11 for GF being Field for V being VectSp of GF for W being Subspace of V for A being Subset of W st A is linearly-independent holds A is linearly-independent Subset of V proof let GF be Field; ::_thesis: for V being VectSp of GF for W being Subspace of V for A being Subset of W st A is linearly-independent holds A is linearly-independent Subset of V let V be VectSp of GF; ::_thesis: for W being Subspace of V for A being Subset of W st A is linearly-independent holds A is linearly-independent Subset of V let W be Subspace of V; ::_thesis: for A being Subset of W st A is linearly-independent holds A is linearly-independent Subset of V let A be Subset of W; ::_thesis: ( A is linearly-independent implies A is linearly-independent Subset of V ) the carrier of W c= the carrier of V by VECTSP_4:def_2; then reconsider A9 = A as Subset of V by XBOOLE_1:1; assume A1: A is linearly-independent ; ::_thesis: A is linearly-independent Subset of V now__::_thesis:_not_A9_is_linearly-dependent assume A9 is linearly-dependent ; ::_thesis: contradiction then consider L being Linear_Combination of A9 such that A2: Sum L = 0. V and A3: Carrier L <> {} by VECTSP_7:def_1; Carrier L c= A9 by VECTSP_6:def_4; then consider LW being Linear_Combination of W such that A4: Carrier LW = Carrier L and A5: Sum LW = Sum L by Th9, XBOOLE_1:1; reconsider LW = LW as Linear_Combination of A by A4, VECTSP_6:def_4; Sum LW = 0. W by A2, A5, VECTSP_4:11; hence contradiction by A1, A3, A4, VECTSP_7:def_1; ::_thesis: verum end; hence A is linearly-independent Subset of V ; ::_thesis: verum end; theorem Th12: :: VECTSP_9:12 for GF being Field for V being VectSp of GF for W being Subspace of V for A being Subset of V st A is linearly-independent & A c= the carrier of W holds A is linearly-independent Subset of W proof let GF be Field; ::_thesis: for V being VectSp of GF for W being Subspace of V for A being Subset of V st A is linearly-independent & A c= the carrier of W holds A is linearly-independent Subset of W let V be VectSp of GF; ::_thesis: for W being Subspace of V for A being Subset of V st A is linearly-independent & A c= the carrier of W holds A is linearly-independent Subset of W let W be Subspace of V; ::_thesis: for A being Subset of V st A is linearly-independent & A c= the carrier of W holds A is linearly-independent Subset of W let A be Subset of V; ::_thesis: ( A is linearly-independent & A c= the carrier of W implies A is linearly-independent Subset of W ) assume that A1: A is linearly-independent and A2: A c= the carrier of W ; ::_thesis: A is linearly-independent Subset of W reconsider A9 = A as Subset of W by A2; now__::_thesis:_not_A9_is_linearly-dependent assume A9 is linearly-dependent ; ::_thesis: contradiction then consider K being Linear_Combination of A9 such that A3: Sum K = 0. W and A4: Carrier K <> {} by VECTSP_7:def_1; consider L being Linear_Combination of V such that A5: Carrier L = Carrier K and A6: Sum L = Sum K by Th8; reconsider L = L as Linear_Combination of A by A5, VECTSP_6:def_4; Sum L = 0. V by A3, A6, VECTSP_4:11; hence contradiction by A1, A4, A5, VECTSP_7:def_1; ::_thesis: verum end; hence A is linearly-independent Subset of W ; ::_thesis: verum end; theorem Th13: :: VECTSP_9:13 for GF being Field for V being VectSp of GF for W being Subspace of V for A being Basis of W ex B being Basis of V st A c= B proof let GF be Field; ::_thesis: for V being VectSp of GF for W being Subspace of V for A being Basis of W ex B being Basis of V st A c= B let V be VectSp of GF; ::_thesis: for W being Subspace of V for A being Basis of W ex B being Basis of V st A c= B let W be Subspace of V; ::_thesis: for A being Basis of W ex B being Basis of V st A c= B let A be Basis of W; ::_thesis: ex B being Basis of V st A c= B A is linearly-independent by VECTSP_7:def_3; then reconsider B = A as linearly-independent Subset of V by Th11; consider I being Basis of V such that A1: B c= I by VECTSP_7:19; take I ; ::_thesis: A c= I thus A c= I by A1; ::_thesis: verum end; theorem Th14: :: VECTSP_9:14 for GF being Field for V being VectSp of GF for A being Subset of V st A is linearly-independent holds for v being Vector of V st v in A holds for B being Subset of V st B = A \ {v} holds not v in Lin B proof let GF be Field; ::_thesis: for V being VectSp of GF for A being Subset of V st A is linearly-independent holds for v being Vector of V st v in A holds for B being Subset of V st B = A \ {v} holds not v in Lin B let V be VectSp of GF; ::_thesis: for A being Subset of V st A is linearly-independent holds for v being Vector of V st v in A holds for B being Subset of V st B = A \ {v} holds not v in Lin B let A be Subset of V; ::_thesis: ( A is linearly-independent implies for v being Vector of V st v in A holds for B being Subset of V st B = A \ {v} holds not v in Lin B ) assume A1: A is linearly-independent ; ::_thesis: for v being Vector of V st v in A holds for B being Subset of V st B = A \ {v} holds not v in Lin B let v be Vector of V; ::_thesis: ( v in A implies for B being Subset of V st B = A \ {v} holds not v in Lin B ) assume v in A ; ::_thesis: for B being Subset of V st B = A \ {v} holds not v in Lin B then A2: {v} is Subset of A by SUBSET_1:41; v in {v} by TARSKI:def_1; then v in Lin {v} by VECTSP_7:8; then consider K being Linear_Combination of {v} such that A3: v = Sum K by VECTSP_7:7; let B be Subset of V; ::_thesis: ( B = A \ {v} implies not v in Lin B ) assume A4: B = A \ {v} ; ::_thesis: not v in Lin B B c= A by A4, XBOOLE_1:36; then A5: B \/ {v} c= A \/ A by A2, XBOOLE_1:13; assume v in Lin B ; ::_thesis: contradiction then consider L being Linear_Combination of B such that A6: v = Sum L by VECTSP_7:7; A7: ( Carrier L c= B & Carrier K c= {v} ) by VECTSP_6:def_4; then (Carrier L) \/ (Carrier K) c= B \/ {v} by XBOOLE_1:13; then ( Carrier (L - K) c= (Carrier L) \/ (Carrier K) & (Carrier L) \/ (Carrier K) c= A ) by A5, VECTSP_6:41, XBOOLE_1:1; then Carrier (L - K) c= A by XBOOLE_1:1; then A8: L - K is Linear_Combination of A by VECTSP_6:def_4; A9: for x being Vector of V st x in Carrier L holds K . x = 0. GF proof let x be Vector of V; ::_thesis: ( x in Carrier L implies K . x = 0. GF ) assume x in Carrier L ; ::_thesis: K . x = 0. GF then not x in Carrier K by A4, A7, XBOOLE_0:def_5; hence K . x = 0. GF by VECTSP_6:2; ::_thesis: verum end; A10: now__::_thesis:_for_x_being_set_st_x_in_Carrier_L_holds_ x_in_Carrier_(L_-_K) let x be set ; ::_thesis: ( x in Carrier L implies x in Carrier (L - K) ) assume that A11: x in Carrier L and A12: not x in Carrier (L - K) ; ::_thesis: contradiction reconsider x = x as Vector of V by A11; A13: L . x <> 0. GF by A11, VECTSP_6:2; (L - K) . x = (L . x) - (K . x) by VECTSP_6:40 .= (L . x) - (0. GF) by A9, A11 .= (L . x) + (- (0. GF)) by RLVECT_1:def_11 .= (L . x) + (0. GF) by RLVECT_1:12 .= L . x by RLVECT_1:4 ; hence contradiction by A12, A13, VECTSP_6:2; ::_thesis: verum end; {v} is linearly-independent by A1, A2, VECTSP_7:1; then v <> 0. V by VECTSP_7:3; then Carrier L <> {} by A6, VECTSP_6:19; then ex w being set st w in Carrier L by XBOOLE_0:def_1; then A14: not Carrier (L - K) is empty by A10; 0. V = (Sum L) + (- (Sum K)) by A6, A3, RLVECT_1:def_10 .= (Sum L) + (Sum (- K)) by VECTSP_6:46 .= Sum (L + (- K)) by VECTSP_6:44 .= Sum (L - K) by VECTSP_6:def_11 ; hence contradiction by A1, A8, A14, VECTSP_7:def_1; ::_thesis: verum end; theorem Th15: :: VECTSP_9:15 for GF being Field for V being VectSp of GF for I being Basis of V for A being non empty Subset of V st A misses I holds for B being Subset of V st B = I \/ A holds B is linearly-dependent proof let GF be Field; ::_thesis: for V being VectSp of GF for I being Basis of V for A being non empty Subset of V st A misses I holds for B being Subset of V st B = I \/ A holds B is linearly-dependent let V be VectSp of GF; ::_thesis: for I being Basis of V for A being non empty Subset of V st A misses I holds for B being Subset of V st B = I \/ A holds B is linearly-dependent let I be Basis of V; ::_thesis: for A being non empty Subset of V st A misses I holds for B being Subset of V st B = I \/ A holds B is linearly-dependent let A be non empty Subset of V; ::_thesis: ( A misses I implies for B being Subset of V st B = I \/ A holds B is linearly-dependent ) assume A1: A misses I ; ::_thesis: for B being Subset of V st B = I \/ A holds B is linearly-dependent consider v being set such that A2: v in A by XBOOLE_0:def_1; let B be Subset of V; ::_thesis: ( B = I \/ A implies B is linearly-dependent ) assume A3: B = I \/ A ; ::_thesis: B is linearly-dependent A4: A c= B by A3, XBOOLE_1:7; reconsider v = v as Vector of V by A2; reconsider Bv = B \ {v} as Subset of V ; A5: I \ {v} c= B \ {v} by A3, XBOOLE_1:7, XBOOLE_1:33; not v in I by A1, A2, XBOOLE_0:3; then I c= Bv by A5, ZFMISC_1:57; then A6: Lin I is Subspace of Lin Bv by VECTSP_7:13; assume A7: B is linearly-independent ; ::_thesis: contradiction v in Lin I by Th10; hence contradiction by A7, A2, A4, A6, Th14, VECTSP_4:8; ::_thesis: verum end; theorem Th16: :: VECTSP_9:16 for GF being Field for V being VectSp of GF for W being Subspace of V for A being Subset of V st A c= the carrier of W holds Lin A is Subspace of W proof let GF be Field; ::_thesis: for V being VectSp of GF for W being Subspace of V for A being Subset of V st A c= the carrier of W holds Lin A is Subspace of W let V be VectSp of GF; ::_thesis: for W being Subspace of V for A being Subset of V st A c= the carrier of W holds Lin A is Subspace of W let W be Subspace of V; ::_thesis: for A being Subset of V st A c= the carrier of W holds Lin A is Subspace of W let A be Subset of V; ::_thesis: ( A c= the carrier of W implies Lin A is Subspace of W ) assume A1: A c= the carrier of W ; ::_thesis: Lin A is Subspace of W now__::_thesis:_for_w_being_set_st_w_in_the_carrier_of_(Lin_A)_holds_ w_in_the_carrier_of_W let w be set ; ::_thesis: ( w in the carrier of (Lin A) implies w in the carrier of W ) assume w in the carrier of (Lin A) ; ::_thesis: w in the carrier of W then w in Lin A by STRUCT_0:def_5; then consider L being Linear_Combination of A such that A2: w = Sum L by VECTSP_7:7; Carrier L c= A by VECTSP_6:def_4; then ex K being Linear_Combination of W st ( Carrier K = Carrier L & Sum L = Sum K ) by A1, Th9, XBOOLE_1:1; hence w in the carrier of W by A2; ::_thesis: verum end; then the carrier of (Lin A) c= the carrier of W by TARSKI:def_3; hence Lin A is Subspace of W by VECTSP_4:27; ::_thesis: verum end; theorem Th17: :: VECTSP_9:17 for GF being Field for V being VectSp of GF for W being Subspace of V for A being Subset of V for B being Subset of W st A = B holds Lin A = Lin B proof let GF be Field; ::_thesis: for V being VectSp of GF for W being Subspace of V for A being Subset of V for B being Subset of W st A = B holds Lin A = Lin B let V be VectSp of GF; ::_thesis: for W being Subspace of V for A being Subset of V for B being Subset of W st A = B holds Lin A = Lin B let W be Subspace of V; ::_thesis: for A being Subset of V for B being Subset of W st A = B holds Lin A = Lin B let A be Subset of V; ::_thesis: for B being Subset of W st A = B holds Lin A = Lin B let B be Subset of W; ::_thesis: ( A = B implies Lin A = Lin B ) reconsider B9 = Lin B, V9 = V as VectSp of GF ; A1: B9 is Subspace of V9 by VECTSP_4:26; assume A2: A = B ; ::_thesis: Lin A = Lin B now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_(Lin_A)_holds_ x_in_the_carrier_of_(Lin_B) let x be set ; ::_thesis: ( x in the carrier of (Lin A) implies x in the carrier of (Lin B) ) assume x in the carrier of (Lin A) ; ::_thesis: x in the carrier of (Lin B) then x in Lin A by STRUCT_0:def_5; then consider L being Linear_Combination of A such that A3: x = Sum L by VECTSP_7:7; Carrier L c= A by VECTSP_6:def_4; then consider K being Linear_Combination of W such that A4: Carrier K = Carrier L and A5: Sum K = Sum L by A2, Th9, XBOOLE_1:1; reconsider K = K as Linear_Combination of B by A2, A4, VECTSP_6:def_4; x = Sum K by A3, A5; then x in Lin B by VECTSP_7:7; hence x in the carrier of (Lin B) by STRUCT_0:def_5; ::_thesis: verum end; then A6: the carrier of (Lin A) c= the carrier of (Lin B) by TARSKI:def_3; now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_(Lin_B)_holds_ x_in_the_carrier_of_(Lin_A) let x be set ; ::_thesis: ( x in the carrier of (Lin B) implies x in the carrier of (Lin A) ) assume x in the carrier of (Lin B) ; ::_thesis: x in the carrier of (Lin A) then x in Lin B by STRUCT_0:def_5; then consider K being Linear_Combination of B such that A7: x = Sum K by VECTSP_7:7; consider L being Linear_Combination of V such that A8: Carrier L = Carrier K and A9: Sum L = Sum K by Th8; reconsider L = L as Linear_Combination of A by A2, A8, VECTSP_6:def_4; x = Sum L by A7, A9; then x in Lin A by VECTSP_7:7; hence x in the carrier of (Lin A) by STRUCT_0:def_5; ::_thesis: verum end; then the carrier of (Lin B) c= the carrier of (Lin A) by TARSKI:def_3; then the carrier of (Lin A) = the carrier of (Lin B) by A6, XBOOLE_0:def_10; hence Lin A = Lin B by A1, VECTSP_4:29; ::_thesis: verum end; begin theorem Th18: :: VECTSP_9:18 for GF being Field for V being VectSp of GF for A, B being finite Subset of V for 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 GF be Field; ::_thesis: for V being VectSp of GF for A, B being finite Subset of V for 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}) ) let V be VectSp of GF; ::_thesis: for A, B being finite Subset of V for 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}) ) let A, B be finite Subset of V; ::_thesis: for 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}) ) let v be Vector of V; ::_thesis: ( v in Lin (A \/ B) & not v in Lin B implies ex w being Vector of V st ( w in A & w in Lin (((A \/ B) \ {w}) \/ {v}) ) ) assume that A1: v in Lin (A \/ B) and A2: not v in Lin B ; ::_thesis: ex w being Vector of V st ( w in A & w in Lin (((A \/ B) \ {w}) \/ {v}) ) consider L being Linear_Combination of A \/ B such that A3: v = Sum L by A1, VECTSP_7:7; v in {v} by TARSKI:def_1; then v in Lin {v} by VECTSP_7:8; then consider Lv being Linear_Combination of {v} such that A4: v = Sum Lv by VECTSP_7:7; A5: Carrier L c= A \/ B by VECTSP_6:def_4; now__::_thesis:_ex_w_being_Vector_of_V_st_ (_w_in_A_&_not_L_._w_=_0._GF_) assume A6: for w being Vector of V st w in A holds L . w = 0. GF ; ::_thesis: contradiction now__::_thesis:_for_x_being_set_st_x_in_Carrier_L_holds_ not_x_in_A let x be set ; ::_thesis: ( x in Carrier L implies not x in A ) assume that A7: x in Carrier L and A8: x in A ; ::_thesis: contradiction ex u being Vector of V st ( x = u & L . u <> 0. GF ) by A7, VECTSP_6:1; hence contradiction by A6, A8; ::_thesis: verum end; then Carrier L misses A by XBOOLE_0:3; then Carrier L c= B by A5, XBOOLE_1:73; then L is Linear_Combination of B by VECTSP_6:def_4; hence contradiction by A2, A3, VECTSP_7:7; ::_thesis: verum end; then consider w being Vector of V such that A9: w in A and A10: L . w <> 0. GF ; set a = L . w; take w ; ::_thesis: ( w in A & w in Lin (((A \/ B) \ {w}) \/ {v}) ) consider F being FinSequence of the carrier of V such that A11: F is one-to-one and A12: rng F = Carrier L and A13: Sum L = Sum (L (#) F) by VECTSP_6:def_6; A14: w in Carrier L by A10, VECTSP_6:1; then reconsider Fw1 = F -| w as FinSequence of the carrier of V by A12, FINSEQ_4:41; reconsider Fw2 = F |-- w as FinSequence of the carrier of V by A12, A14, FINSEQ_4:50; A15: rng Fw1 misses rng Fw2 by A11, A12, A14, FINSEQ_4:57; set Fw = Fw1 ^ Fw2; F just_once_values w by A11, A12, A14, FINSEQ_4:8; then A16: Fw1 ^ Fw2 = F - {w} by FINSEQ_4:54; then A17: rng (Fw1 ^ Fw2) = (Carrier L) \ {w} by A12, FINSEQ_3:65; F = ((F -| w) ^ <*w*>) ^ (F |-- w) by A12, A14, FINSEQ_4:51; then F = Fw1 ^ (<*w*> ^ Fw2) by FINSEQ_1:32; then L (#) F = (L (#) Fw1) ^ (L (#) (<*w*> ^ Fw2)) by VECTSP_6:13 .= (L (#) Fw1) ^ ((L (#) <*w*>) ^ (L (#) Fw2)) by VECTSP_6:13 .= ((L (#) Fw1) ^ (L (#) <*w*>)) ^ (L (#) Fw2) by FINSEQ_1:32 .= ((L (#) Fw1) ^ <*((L . w) * w)*>) ^ (L (#) Fw2) by VECTSP_6:10 ; then A18: Sum (L (#) F) = Sum ((L (#) Fw1) ^ (<*((L . w) * w)*> ^ (L (#) Fw2))) by FINSEQ_1:32 .= (Sum (L (#) Fw1)) + (Sum (<*((L . w) * w)*> ^ (L (#) Fw2))) by RLVECT_1:41 .= (Sum (L (#) Fw1)) + ((Sum <*((L . w) * w)*>) + (Sum (L (#) Fw2))) by RLVECT_1:41 .= (Sum (L (#) Fw1)) + ((Sum (L (#) Fw2)) + ((L . w) * w)) by RLVECT_1:44 .= ((Sum (L (#) Fw1)) + (Sum (L (#) Fw2))) + ((L . w) * w) by RLVECT_1:def_3 .= (Sum ((L (#) Fw1) ^ (L (#) Fw2))) + ((L . w) * w) by RLVECT_1:41 .= (Sum (L (#) (Fw1 ^ Fw2))) + ((L . w) * w) by VECTSP_6:13 ; consider K being Linear_Combination of V such that A19: Carrier K = (rng (Fw1 ^ Fw2)) /\ (Carrier L) and A20: L (#) (Fw1 ^ Fw2) = K (#) (Fw1 ^ Fw2) by Th4; rng (Fw1 ^ Fw2) = (rng F) \ {w} by A16, FINSEQ_3:65; then A21: Carrier K = rng (Fw1 ^ Fw2) by A12, A19, XBOOLE_1:28; A22: (Carrier L) \ {w} c= (A \/ B) \ {w} by A5, XBOOLE_1:33; then reconsider K = K as Linear_Combination of (A \/ B) \ {w} by A17, A21, VECTSP_6:def_4; set LC = ((L . w) ") * ((- K) + Lv); Carrier ((- K) + Lv) c= (Carrier (- K)) \/ (Carrier Lv) by VECTSP_6:23; then A23: Carrier ((- K) + Lv) c= (Carrier K) \/ (Carrier Lv) by VECTSP_6:38; Carrier Lv c= {v} by VECTSP_6:def_4; then (Carrier K) \/ (Carrier Lv) c= ((A \/ B) \ {w}) \/ {v} by A17, A21, A22, XBOOLE_1:13; then ( Carrier (((L . w) ") * ((- K) + Lv)) c= Carrier ((- K) + Lv) & Carrier ((- K) + Lv) c= ((A \/ B) \ {w}) \/ {v} ) by A23, VECTSP_6:28, XBOOLE_1:1; then Carrier (((L . w) ") * ((- K) + Lv)) c= ((A \/ B) \ {w}) \/ {v} by XBOOLE_1:1; then A24: ((L . w) ") * ((- K) + Lv) is Linear_Combination of ((A \/ B) \ {w}) \/ {v} by VECTSP_6:def_4; ( Fw1 is one-to-one & Fw2 is one-to-one ) by A11, A12, A14, FINSEQ_4:52, FINSEQ_4:53; then Fw1 ^ Fw2 is one-to-one by A15, FINSEQ_3:91; then Sum (K (#) (Fw1 ^ Fw2)) = Sum K by A21, VECTSP_6:def_6; then ((L . w) ") * v = (((L . w) ") * (Sum K)) + (((L . w) ") * ((L . w) * w)) by A3, A13, A18, A20, VECTSP_1:def_14 .= (((L . w) ") * (Sum K)) + w by A10, VECTSP_1:20 ; then w = (((L . w) ") * v) - (((L . w) ") * (Sum K)) by VECTSP_2:2 .= ((L . w) ") * (v - (Sum K)) by VECTSP_1:23 .= ((L . w) ") * ((- (Sum K)) + v) by RLVECT_1:def_11 ; then w = ((L . w) ") * ((Sum (- K)) + (Sum Lv)) by A4, VECTSP_6:46 .= ((L . w) ") * (Sum ((- K) + Lv)) by VECTSP_6:44 .= Sum (((L . w) ") * ((- K) + Lv)) by VECTSP_6:45 ; hence ( w in A & w in Lin (((A \/ B) \ {w}) \/ {v}) ) by A9, A24, VECTSP_7:7; ::_thesis: verum end; theorem Th19: :: VECTSP_9:19 for GF being Field for V being VectSp of GF for A, B being finite Subset of V st VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult 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) & VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin (B \/ C) ) ) proof let GF be Field; ::_thesis: for V being VectSp of GF for A, B being finite Subset of V st VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult 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) & VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin (B \/ C) ) ) let V be VectSp of GF; ::_thesis: for A, B being finite Subset of V st VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult 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) & VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin (B \/ C) ) ) defpred S1[ Nat] means for n being Nat for A, B being finite Subset of V st card A = n & card B = $1 & VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult 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 & VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin (B \/ C) ) ); A1: for m being Nat st S1[m] holds S1[m + 1] proof let m be Nat; ::_thesis: ( S1[m] implies S1[m + 1] ) assume A2: S1[m] ; ::_thesis: S1[m + 1] let n be Nat; ::_thesis: for A, B being finite Subset of V st card A = n & card B = m + 1 & VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin A & B is linearly-independent holds ( m + 1 <= n & ex C being finite Subset of V st ( C c= A & card C = n - (m + 1) & VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin (B \/ C) ) ) let A, B be finite Subset of V; ::_thesis: ( card A = n & card B = m + 1 & VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin A & B is linearly-independent implies ( m + 1 <= n & ex C being finite Subset of V st ( C c= A & card C = n - (m + 1) & VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin (B \/ C) ) ) ) assume that A3: card A = n and A4: card B = m + 1 and A5: VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin A and A6: B is linearly-independent ; ::_thesis: ( m + 1 <= n & ex C being finite Subset of V st ( C c= A & card C = n - (m + 1) & VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin (B \/ C) ) ) consider v being set such that A7: v in B by A4, CARD_1:27, XBOOLE_0:def_1; reconsider v = v as Vector of V by A7; set Bv = B \ {v}; A8: B \ {v} is linearly-independent by A6, VECTSP_7:1, XBOOLE_1:36; {v} is Subset of B by A7, SUBSET_1:41; then A9: card (B \ {v}) = (card B) - (card {v}) by CARD_2:44 .= (m + 1) - 1 by A4, CARD_1:30 .= m ; then consider C being finite Subset of V such that A10: C c= A and A11: card C = n - m and A12: VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin ((B \ {v}) \/ C) by A2, A3, A5, A8; A13: not v in Lin (B \ {v}) by A6, A7, Th14; A14: now__::_thesis:_not_m_=_n assume m = n ; ::_thesis: contradiction then consider C being finite Subset of V such that C c= A and A15: card C = m - m and A16: VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin ((B \ {v}) \/ C) by A2, A3, A5, A9, A8; C = {} by A15; then B \ {v} is Basis of V by A8, A16, VECTSP_7:def_3; hence contradiction by A13, Th10; ::_thesis: verum end; v in Lin ((B \ {v}) \/ C) by A12, STRUCT_0:def_5; then consider w being Vector of V such that A17: w in C and A18: w in Lin (((C \/ (B \ {v})) \ {w}) \/ {v}) by A13, Th18; set Cw = C \ {w}; ((B \ {v}) \ {w}) \/ {v} c= (B \ {v}) \/ {v} by XBOOLE_1:9, XBOOLE_1:36; then (C \ {w}) \/ (((B \ {v}) \ {w}) \/ {v}) c= (C \ {w}) \/ ((B \ {v}) \/ {v}) by XBOOLE_1:9; then A19: (C \ {w}) \/ (((B \ {v}) \ {w}) \/ {v}) c= B \/ (C \ {w}) by A7, Lm1; {w} is Subset of C by A17, SUBSET_1:41; then A20: card (C \ {w}) = (card C) - (card {w}) by CARD_2:44 .= (n - m) - 1 by A11, CARD_1:30 .= n - (m + 1) ; C \ {w} c= C by XBOOLE_1:36; then A21: C \ {w} c= A by A10, XBOOLE_1:1; ((C \/ (B \ {v})) \ {w}) \/ {v} = ((C \ {w}) \/ ((B \ {v}) \ {w})) \/ {v} by XBOOLE_1:42 .= (C \ {w}) \/ (((B \ {v}) \ {w}) \/ {v}) by XBOOLE_1:4 ; then Lin (((C \/ (B \ {v})) \ {w}) \/ {v}) is Subspace of Lin (B \/ (C \ {w})) by A19, VECTSP_7:13; then A22: w in Lin (B \/ (C \ {w})) by A18, VECTSP_4:8; A23: ( B \ {v} c= B & C = (C \ {w}) \/ {w} ) by A17, Lm1, XBOOLE_1:36; now__::_thesis:_for_x_being_set_st_x_in_(B_\_{v})_\/_C_holds_ x_in_the_carrier_of_(Lin_(B_\/_(C_\_{w}))) let x be set ; ::_thesis: ( x in (B \ {v}) \/ C implies x in the carrier of (Lin (B \/ (C \ {w}))) ) assume x in (B \ {v}) \/ C ; ::_thesis: x in the carrier of (Lin (B \/ (C \ {w}))) then ( x in B \ {v} or x in C ) by XBOOLE_0:def_3; then ( x in B or x in C \ {w} or x in {w} ) by A23, XBOOLE_0:def_3; then ( x in B \/ (C \ {w}) or x in {w} ) by XBOOLE_0:def_3; then ( x in Lin (B \/ (C \ {w})) or x = w ) by TARSKI:def_1, VECTSP_7:8; hence x in the carrier of (Lin (B \/ (C \ {w}))) by A22, STRUCT_0:def_5; ::_thesis: verum end; then (B \ {v}) \/ C c= the carrier of (Lin (B \/ (C \ {w}))) by TARSKI:def_3; then Lin ((B \ {v}) \/ C) is Subspace of Lin (B \/ (C \ {w})) by Th16; then A24: the carrier of (Lin ((B \ {v}) \/ C)) c= the carrier of (Lin (B \/ (C \ {w}))) by VECTSP_4:def_2; the carrier of (Lin (B \/ (C \ {w}))) c= the carrier of V by VECTSP_4:def_2; then the carrier of (Lin (B \/ (C \ {w}))) = the carrier of V by A12, A24, XBOOLE_0:def_10; then A25: VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin (B \/ (C \ {w})) by A12, VECTSP_4:29; m <= n by A2, A3, A5, A9, A8; then m < n by A14, XXREAL_0:1; hence ( m + 1 <= n & ex C being finite Subset of V st ( C c= A & card C = n - (m + 1) & VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin (B \/ C) ) ) by A21, A20, A25, NAT_1:13; ::_thesis: verum end; A26: S1[ 0 ] proof let n be Nat; ::_thesis: for A, B being finite Subset of V st card A = n & card B = 0 & VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin A & B is linearly-independent holds ( 0 <= n & ex C being finite Subset of V st ( C c= A & card C = n - 0 & VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin (B \/ C) ) ) let A, B be finite Subset of V; ::_thesis: ( card A = n & card B = 0 & VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin A & B is linearly-independent implies ( 0 <= n & ex C being finite Subset of V st ( C c= A & card C = n - 0 & VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin (B \/ C) ) ) ) assume that A27: card A = n and A28: card B = 0 and A29: VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin A and B is linearly-independent ; ::_thesis: ( 0 <= n & ex C being finite Subset of V st ( C c= A & card C = n - 0 & VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin (B \/ C) ) ) B = {} by A28; then A = B \/ A ; hence ( 0 <= n & ex C being finite Subset of V st ( C c= A & card C = n - 0 & VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin (B \/ C) ) ) by A27, A29; ::_thesis: verum end; A30: for m being Nat holds S1[m] from NAT_1:sch_2(A26, A1); let A, B be finite Subset of V; ::_thesis: ( VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin A & B is linearly-independent implies ( card B <= card A & ex C being finite Subset of V st ( C c= A & card C = (card A) - (card B) & VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin (B \/ C) ) ) ) assume ( VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin A & B is linearly-independent ) ; ::_thesis: ( card B <= card A & ex C being finite Subset of V st ( C c= A & card C = (card A) - (card B) & VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin (B \/ C) ) ) hence ( card B <= card A & ex C being finite Subset of V st ( C c= A & card C = (card A) - (card B) & VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin (B \/ C) ) ) by A30; ::_thesis: verum end; begin theorem Th20: :: VECTSP_9:20 for GF being Field for V being VectSp of GF st V is finite-dimensional holds for I being Basis of V holds I is finite proof let GF be Field; ::_thesis: for V being VectSp of GF st V is finite-dimensional holds for I being Basis of V holds I is finite let V be VectSp of GF; ::_thesis: ( V is finite-dimensional implies for I being Basis of V holds I is finite ) assume V is finite-dimensional ; ::_thesis: for I being Basis of V holds I is finite then consider A being finite Subset of V such that A1: A is Basis of V by MATRLIN:def_1; let B be Basis of V; ::_thesis: B is finite consider p being FinSequence such that A2: rng p = A by FINSEQ_1:52; reconsider p = 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 { (Carrier L) where L is Linear_Combination of B : ex i being Nat st ( i in dom p & Sum L = p . i ) } ; A3: union { (Carrier L) where L is Linear_Combination of B : ex i being Nat st ( i in dom p & Sum L = p . i ) } c= B proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union { (Carrier L) where L is Linear_Combination of B : ex i being Nat st ( i in dom p & Sum L = p . i ) } or x in B ) assume x in union { (Carrier L) where L is Linear_Combination of B : ex i being Nat st ( i in dom p & Sum L = p . i ) } ; ::_thesis: x in B then consider R being set such that A4: x in R and A5: R in { (Carrier L) where L is Linear_Combination of B : ex i being Nat st ( i in dom p & Sum L = p . i ) } by TARSKI:def_4; ex L being Linear_Combination of B st ( R = Carrier L & ex i being Nat st ( i in dom p & Sum L = p . i ) ) by A5; then R c= B by VECTSP_6:def_4; hence x in B by A4; ::_thesis: verum end; then reconsider C = union { (Carrier L) where L is Linear_Combination of B : ex i being Nat st ( i in dom p & Sum L = p . i ) } as Subset of V by XBOOLE_1:1; for v being Vector of V holds ( v in (Omega). V iff v in Lin C ) proof let v be Vector of V; ::_thesis: ( v in (Omega). V iff v in Lin C ) hereby ::_thesis: ( v in Lin C implies v in (Omega). V ) assume v in (Omega). V ; ::_thesis: v in Lin C then v in VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) by VECTSP_4:def_4; then v in Lin A by A1, VECTSP_7:def_3; then consider LA being Linear_Combination of A such that A6: v = Sum LA by VECTSP_7:7; Carrier LA c= the carrier of (Lin C) proof let w be set ; :: according to TARSKI:def_3 ::_thesis: ( not w in Carrier LA or w in the carrier of (Lin C) ) assume A7: w in Carrier LA ; ::_thesis: w in the carrier of (Lin C) then reconsider w9 = w as Vector of V ; w9 in Lin B by Th10; then consider LB being Linear_Combination of B such that A8: w = Sum LB by VECTSP_7:7; A9: Carrier LA c= A by VECTSP_6:def_4; ex i being Nat st ( i in dom p & w = p . i ) proof consider i being set such that A10: i in dom p and A11: w = p . i by A2, A7, A9, FUNCT_1:def_3; reconsider i = i as Element of NAT by A10; take i ; ::_thesis: ( i in dom p & w = p . i ) thus ( i in dom p & w = p . i ) by A10, A11; ::_thesis: verum end; then A12: Carrier LB in { (Carrier L) where L is Linear_Combination of B : ex i being Nat st ( i in dom p & Sum L = p . i ) } by A8; Carrier LB c= C proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Carrier LB or x in C ) assume x in Carrier LB ; ::_thesis: x in C hence x in C by A12, TARSKI:def_4; ::_thesis: verum end; then LB is Linear_Combination of C by VECTSP_6:def_4; then w in Lin C by A8, VECTSP_7:7; hence w in the carrier of (Lin C) by STRUCT_0:def_5; ::_thesis: verum end; then ex LC being Linear_Combination of C st Sum LA = Sum LC by Th6; hence v in Lin C by A6, VECTSP_7:7; ::_thesis: verum end; assume v in Lin C ; ::_thesis: v in (Omega). V v in the carrier of VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) ; then v in the carrier of ((Omega). V) by VECTSP_4:def_4; hence v in (Omega). V by STRUCT_0:def_5; ::_thesis: verum end; then (Omega). V = Lin C by VECTSP_4:30; then A13: VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin C by VECTSP_4:def_4; A14: B is linearly-independent by VECTSP_7:def_3; then C is linearly-independent by A3, VECTSP_7:1; then A15: C is Basis of V by A13, VECTSP_7:def_3; B c= C proof set D = B \ C; assume not B c= C ; ::_thesis: contradiction then ex v being set st ( v in B & not v in C ) by TARSKI:def_3; then reconsider D = B \ C as non empty Subset of V by XBOOLE_0:def_5; reconsider B = B as Subset of V ; C \/ (B \ C) = C \/ B by XBOOLE_1:39 .= B by A3, XBOOLE_1:12 ; then B = C \/ D ; hence contradiction by A14, A15, Th15, XBOOLE_1:79; ::_thesis: verum end; then A16: B = C by A3, XBOOLE_0:def_10; defpred S1[ set , set ] means ex L being Linear_Combination of B st ( $2 = Carrier L & Sum L = p . $1 ); A17: for i being Nat st i in Seg (len p) holds ex x being set st S1[i,x] proof let i be Nat; ::_thesis: ( i in Seg (len p) implies ex x being set st S1[i,x] ) assume i in Seg (len p) ; ::_thesis: ex x being set st S1[i,x] then i in dom p by FINSEQ_1:def_3; then p . i in the carrier of V by FINSEQ_2:11; then p . i in Lin B by Th10; then consider L being Linear_Combination of B such that A18: p . i = Sum L by VECTSP_7:7; S1[i, Carrier L] by A18; hence ex x being set st S1[i,x] ; ::_thesis: verum end; ex q being FinSequence st ( dom q = Seg (len p) & ( for i being Nat st i in Seg (len p) holds S1[i,q . i] ) ) from FINSEQ_1:sch_1(A17); then consider q being FinSequence such that A19: dom q = Seg (len p) and A20: for i being Nat st i in Seg (len p) holds S1[i,q . i] ; A21: dom p = dom q by A19, FINSEQ_1:def_3; A22: for i being Nat for y1, y2 being set st i in Seg (len p) & S1[i,y1] & S1[i,y2] holds y1 = y2 proof let i be Nat; ::_thesis: for y1, y2 being set st i in Seg (len p) & S1[i,y1] & S1[i,y2] holds y1 = y2 let y1, y2 be set ; ::_thesis: ( i in Seg (len p) & S1[i,y1] & S1[i,y2] implies y1 = y2 ) assume that i in Seg (len p) and A23: S1[i,y1] and A24: S1[i,y2] ; ::_thesis: y1 = y2 consider L1 being Linear_Combination of B such that A25: ( y1 = Carrier L1 & Sum L1 = p . i ) by A23; consider L2 being Linear_Combination of B such that A26: ( y2 = Carrier L2 & Sum L2 = p . i ) by A24; ( Carrier L1 c= B & Carrier L2 c= B ) by VECTSP_6:def_4; hence y1 = y2 by A14, A25, A26, MATRLIN:5; ::_thesis: verum end; now__::_thesis:_for_x_being_set_st_x_in__{__(Carrier_L)_where_L_is_Linear_Combination_of_B_:_ex_i_being_Nat_st_ (_i_in_dom_p_&_Sum_L_=_p_._i_)__}__holds_ x_in_rng_q let x be set ; ::_thesis: ( x in { (Carrier L) where L is Linear_Combination of B : ex i being Nat st ( i in dom p & Sum L = p . i ) } implies x in rng q ) assume x in { (Carrier L) where L is Linear_Combination of B : ex i being Nat st ( i in dom p & Sum L = p . i ) } ; ::_thesis: x in rng q then consider L being Linear_Combination of B such that A27: x = Carrier L and A28: ex i being Nat st ( i in dom p & Sum L = p . i ) ; consider i being Nat such that A29: i in dom p and A30: Sum L = p . i by A28; S1[i,q . i] by A19, A20, A21, A29; then x = q . i by A22, A19, A21, A27, A29, A30; hence x in rng q by A21, A29, FUNCT_1:def_3; ::_thesis: verum end; then A31: { (Carrier L) where L is Linear_Combination of B : ex i being Nat st ( i in dom p & Sum L = p . i ) } c= rng q by TARSKI:def_3; for R being set st R in { (Carrier L) where L is Linear_Combination of B : ex i being Nat st ( i in dom p & Sum L = p . i ) } holds R is finite proof let R be set ; ::_thesis: ( R in { (Carrier L) where L is Linear_Combination of B : ex i being Nat st ( i in dom p & Sum L = p . i ) } implies R is finite ) assume R in { (Carrier L) where L is Linear_Combination of B : ex i being Nat st ( i in dom p & Sum L = p . i ) } ; ::_thesis: R is finite then ex L being Linear_Combination of B st ( R = Carrier L & ex i being Nat st ( i in dom p & Sum L = p . i ) ) ; hence R is finite ; ::_thesis: verum end; hence B is finite by A16, A31, FINSET_1:7; ::_thesis: verum end; theorem :: VECTSP_9:21 for GF being Field for V being VectSp of GF st V is finite-dimensional holds for A being Subset of V st A is linearly-independent holds A is finite proof let GF be Field; ::_thesis: for V being VectSp of GF st V is finite-dimensional holds for A being Subset of V st A is linearly-independent holds A is finite let V be VectSp of GF; ::_thesis: ( V is finite-dimensional implies for A being Subset of V st A is linearly-independent holds A is finite ) assume A1: V is finite-dimensional ; ::_thesis: for A being Subset of V st A is linearly-independent holds A is finite let A be Subset of V; ::_thesis: ( A is linearly-independent implies A is finite ) assume A is linearly-independent ; ::_thesis: A is finite then consider B being Basis of V such that A2: A c= B by VECTSP_7:19; B is finite by A1, Th20; hence A is finite by A2; ::_thesis: verum end; theorem Th22: :: VECTSP_9:22 for GF being Field for V being VectSp of GF st V is finite-dimensional holds for A, B being Basis of V holds card A = card B proof let GF be Field; ::_thesis: for V being VectSp of GF st V is finite-dimensional holds for A, B being Basis of V holds card A = card B let V be VectSp of GF; ::_thesis: ( V is finite-dimensional implies for A, B being Basis of V holds card A = card B ) assume A1: V is finite-dimensional ; ::_thesis: for A, B being Basis of V holds card A = card B let A, B be Basis of V; ::_thesis: card A = card B reconsider A9 = A, B9 = B as finite Subset of V by A1, Th20; ( VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin B & A9 is linearly-independent ) by VECTSP_7:def_3; then A2: card A9 <= card B9 by Th19; ( VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = Lin A & B9 is linearly-independent ) by VECTSP_7:def_3; then card B9 <= card A9 by Th19; hence card A = card B by A2, XXREAL_0:1; ::_thesis: verum end; theorem Th23: :: VECTSP_9:23 for GF being Field for V being VectSp of GF holds (0). V is finite-dimensional proof let GF be Field; ::_thesis: for V being VectSp of GF holds (0). V is finite-dimensional let V be VectSp of GF; ::_thesis: (0). V is finite-dimensional reconsider V9 = (0). V as strict VectSp of GF ; reconsider I = {} the carrier of V9 as finite Subset of V9 ; the carrier of V9 = {(0. V)} by VECTSP_4:def_3 .= {(0. V9)} by VECTSP_4:11 .= the carrier of ((0). V9) by VECTSP_4:def_3 ; then A1: V9 = (0). V9 by VECTSP_4:31; Lin I = (0). V9 by VECTSP_7:9; then I is Basis of V9 by A1, VECTSP_7:def_3; hence (0). V is finite-dimensional by MATRLIN:def_1; ::_thesis: verum end; theorem Th24: :: VECTSP_9:24 for GF being Field for V being VectSp of GF for W being Subspace of V st V is finite-dimensional holds W is finite-dimensional proof let GF be Field; ::_thesis: for V being VectSp of GF for W being Subspace of V st V is finite-dimensional holds W is finite-dimensional let V be VectSp of GF; ::_thesis: for W being Subspace of V st V is finite-dimensional holds W is finite-dimensional let W be Subspace of V; ::_thesis: ( V is finite-dimensional implies W is finite-dimensional ) set A = the Basis of W; consider I being Basis of V such that A1: the Basis of W c= I by Th13; assume V is finite-dimensional ; ::_thesis: W is finite-dimensional then I is finite by Th20; hence W is finite-dimensional by A1, MATRLIN:def_1; ::_thesis: verum end; registration let GF be Field; let V be VectSp of GF; cluster non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional for Subspace of V; existence ex b1 being Subspace of V st ( b1 is strict & b1 is finite-dimensional ) proof take (0). V ; ::_thesis: ( (0). V is strict & (0). V is finite-dimensional ) thus ( (0). V is strict & (0). V is finite-dimensional ) by Th23; ::_thesis: verum end; end; registration let GF be Field; let V be finite-dimensional VectSp of GF; cluster -> finite-dimensional for Subspace of V; correctness coherence for b1 being Subspace of V holds b1 is finite-dimensional ; by Th24; end; registration let GF be Field; let V be finite-dimensional VectSp of GF; cluster non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional for Subspace of V; existence ex b1 being Subspace of V st b1 is strict proof (0). V is strict finite-dimensional Subspace of V ; hence ex b1 being Subspace of V st b1 is strict ; ::_thesis: verum end; end; begin definition let GF be Field; let V be VectSp of GF; assume A1: V is finite-dimensional ; func dim V -> Nat means :Def1: :: VECTSP_9:def 1 for I being Basis of V holds it = card I; existence ex b1 being Nat st for I being Basis of V holds b1 = card I proof consider A being finite Subset of V such that A2: A is Basis of V by A1, MATRLIN:def_1; consider n being Nat such that A3: n = card A ; for I being Basis of V holds card I = n by A1, A2, A3, Th22; hence ex b1 being Nat st for I being Basis of V holds b1 = card I ; ::_thesis: verum end; uniqueness for b1, b2 being Nat st ( for I being Basis of V holds b1 = card I ) & ( for I being Basis of V holds b2 = card I ) holds b1 = b2 proof let n, m be Nat; ::_thesis: ( ( for I being Basis of V holds n = card I ) & ( for I being Basis of V holds m = card I ) implies n = m ) 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 ; ::_thesis: n = m consider A being finite Subset of V such that A6: A is Basis of V by A1, MATRLIN:def_1; card A = n by A4, A6; hence n = m by A5, A6; ::_thesis: verum end; end; :: deftheorem Def1 defines dim VECTSP_9:def_1_:_ for GF being Field for V being VectSp of GF st V is finite-dimensional holds for b3 being Nat holds ( b3 = dim V iff for I being Basis of V holds b3 = card I ); theorem Th25: :: VECTSP_9:25 for GF being Field for V being finite-dimensional VectSp of GF for W being Subspace of V holds dim W <= dim V proof let GF be Field; ::_thesis: for V being finite-dimensional VectSp of GF for W being Subspace of V holds dim W <= dim V let V be finite-dimensional VectSp of GF; ::_thesis: for W being Subspace of V holds dim W <= dim V let W be Subspace of V; ::_thesis: dim W <= dim V set A = the Basis of W; reconsider A = the Basis of W as Subset of W ; A1: dim W = card A by Def1; A is linearly-independent by VECTSP_7:def_3; then reconsider B = A as linearly-independent Subset of V by Th11; reconsider A9 = B as finite Subset of V by Th20; reconsider V9 = V as VectSp of GF ; set I = the Basis of V9; A2: Lin the Basis of V9 = VectSpStr(# the carrier of V9, the U5 of V9, the ZeroF of V9, the lmult of V9 #) by VECTSP_7:def_3; reconsider I = the Basis of V9 as finite Subset of V by Th20; card A9 <= card I by A2, Th19; hence dim W <= dim V by A1, Def1; ::_thesis: verum end; theorem Th26: :: VECTSP_9:26 for GF being Field for V being finite-dimensional VectSp of GF for A being Subset of V st A is linearly-independent holds card A = dim (Lin A) proof let GF be Field; ::_thesis: for V being finite-dimensional VectSp of GF for A being Subset of V st A is linearly-independent holds card A = dim (Lin A) let V be finite-dimensional VectSp of GF; ::_thesis: for A being Subset of V st A is linearly-independent holds card A = dim (Lin A) let A be Subset of V; ::_thesis: ( A is linearly-independent implies card A = dim (Lin A) ) assume A1: A is linearly-independent ; ::_thesis: card A = dim (Lin A) set W = Lin A; now__::_thesis:_for_x_being_set_st_x_in_A_holds_ x_in_the_carrier_of_(Lin_A) let x be set ; ::_thesis: ( x in A implies x in the carrier of (Lin A) ) assume x in A ; ::_thesis: x in the carrier of (Lin A) then x in Lin A by VECTSP_7:8; hence x in the carrier of (Lin A) by STRUCT_0:def_5; ::_thesis: verum end; then reconsider B = A as linearly-independent Subset of (Lin A) by A1, Th12, TARSKI:def_3; Lin A = Lin B by Th17; then reconsider B = B as Basis of Lin A by VECTSP_7:def_3; card B = dim (Lin A) by Def1; hence card A = dim (Lin A) ; ::_thesis: verum end; theorem Th27: :: VECTSP_9:27 for GF being Field for V being finite-dimensional VectSp of GF holds dim V = dim ((Omega). V) proof let GF be Field; ::_thesis: for V being finite-dimensional VectSp of GF holds dim V = dim ((Omega). V) let V be finite-dimensional VectSp of GF; ::_thesis: dim V = dim ((Omega). V) consider I being finite Subset of V such that A1: I is Basis of V by MATRLIN:def_1; A2: (Omega). V = VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) by VECTSP_4:def_4 .= Lin I by A1, VECTSP_7:def_3 ; ( card I = dim V & I is linearly-independent ) by A1, Def1, VECTSP_7:def_3; hence dim V = dim ((Omega). V) by A2, Th26; ::_thesis: verum end; theorem :: VECTSP_9:28 for GF being Field for V being finite-dimensional VectSp of GF for W being Subspace of V holds ( dim V = dim W iff (Omega). V = (Omega). W ) proof let GF be Field; ::_thesis: for V being finite-dimensional VectSp of GF for W being Subspace of V holds ( dim V = dim W iff (Omega). V = (Omega). W ) let V be finite-dimensional VectSp of GF; ::_thesis: for W being Subspace of V holds ( dim V = dim W iff (Omega). V = (Omega). W ) let W be Subspace of V; ::_thesis: ( dim V = dim W iff (Omega). V = (Omega). W ) consider A being finite Subset of V such that A1: A is Basis of V by MATRLIN:def_1; hereby ::_thesis: ( (Omega). V = (Omega). W implies dim V = dim W ) set A = the Basis of W; consider B being Basis of V such that A2: the Basis of W c= B by Th13; the carrier of W c= the carrier of V by VECTSP_4:def_2; then reconsider A9 = the Basis of W as finite Subset of V by Th20, XBOOLE_1:1; reconsider B9 = B as finite Subset of V by Th20; assume dim V = dim W ; ::_thesis: (Omega). V = (Omega). W then A3: card the Basis of W = dim V by Def1 .= card B by Def1 ; A4: now__::_thesis:_not_the_Basis_of_W_<>_B assume the Basis of W <> B ; ::_thesis: contradiction then the Basis of W c< B by A2, XBOOLE_0:def_8; then card A9 < card B9 by CARD_2:48; hence contradiction by A3; ::_thesis: verum end; reconsider B = B as Subset of V ; reconsider A = the Basis of W as Subset of W ; (Omega). V = VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) by VECTSP_4:def_4 .= Lin B by VECTSP_7:def_3 .= Lin A by A4, Th17 .= VectSpStr(# the carrier of W, the U5 of W, the ZeroF of W, the lmult of W #) by VECTSP_7:def_3 .= (Omega). W by VECTSP_4:def_4 ; hence (Omega). V = (Omega). W ; ::_thesis: verum end; consider B being finite Subset of W such that A5: B is Basis of W by MATRLIN:def_1; A6: A is linearly-independent by A1, VECTSP_7:def_3; assume (Omega). V = (Omega). W ; ::_thesis: dim V = dim W then VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = (Omega). W by VECTSP_4:def_4 .= VectSpStr(# the carrier of W, the U5 of W, the ZeroF of W, the lmult of W #) by VECTSP_4:def_4 ; then A7: Lin A = VectSpStr(# the carrier of W, the U5 of W, the ZeroF of W, the lmult of W #) by A1, VECTSP_7:def_3 .= Lin B by A5, VECTSP_7:def_3 ; A8: B is linearly-independent by A5, VECTSP_7:def_3; reconsider B = B as Subset of W ; reconsider A = A as Subset of V ; dim V = card A by A1, Def1 .= dim (Lin B) by A6, A7, Th26 .= card B by A8, Th26 .= dim W by A5, Def1 ; hence dim V = dim W ; ::_thesis: verum end; theorem Th29: :: VECTSP_9:29 for GF being Field for V being finite-dimensional VectSp of GF holds ( dim V = 0 iff (Omega). V = (0). V ) proof let GF be Field; ::_thesis: for V being finite-dimensional VectSp of GF holds ( dim V = 0 iff (Omega). V = (0). V ) let V be finite-dimensional VectSp of GF; ::_thesis: ( dim V = 0 iff (Omega). V = (0). V ) consider I being finite Subset of V such that A1: I is Basis of V by MATRLIN:def_1; hereby ::_thesis: ( (Omega). V = (0). V implies dim V = 0 ) consider I being finite Subset of V such that A2: I is Basis of V by MATRLIN:def_1; assume dim V = 0 ; ::_thesis: (Omega). V = (0). V then card I = 0 by A2, Def1; then A3: I = {} the carrier of V ; (Omega). V = VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) by VECTSP_4:def_4 .= Lin I by A2, VECTSP_7:def_3 .= (0). V by A3, VECTSP_7:9 ; hence (Omega). V = (0). V ; ::_thesis: verum end; A4: now__::_thesis:_not_I_=_{(0._V)} assume I = {(0. V)} ; ::_thesis: contradiction then I is linearly-dependent by VECTSP_7:3; hence contradiction by A1, VECTSP_7:def_3; ::_thesis: verum end; assume (Omega). V = (0). V ; ::_thesis: dim V = 0 then VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = (0). V by VECTSP_4:def_4; then Lin I = (0). V by A1, VECTSP_7:def_3; then ( I = {} or I = {(0. V)} ) by VECTSP_7:10; hence dim V = 0 by A1, A4, Def1, CARD_1:27; ::_thesis: verum end; theorem :: VECTSP_9:30 for GF being Field for V being finite-dimensional VectSp of GF holds ( dim V = 1 iff ex v being Vector of V st ( v <> 0. V & (Omega). V = Lin {v} ) ) proof let GF be Field; ::_thesis: for V being finite-dimensional VectSp of GF holds ( dim V = 1 iff ex v being Vector of V st ( v <> 0. V & (Omega). V = Lin {v} ) ) let V be finite-dimensional VectSp of GF; ::_thesis: ( dim V = 1 iff ex v being Vector of V st ( v <> 0. V & (Omega). V = Lin {v} ) ) hereby ::_thesis: ( ex v being Vector of V st ( v <> 0. V & (Omega). V = Lin {v} ) implies dim V = 1 ) consider I being finite Subset of V such that A1: I is Basis of V by MATRLIN:def_1; assume dim V = 1 ; ::_thesis: ex v being Vector of V st ( v <> 0. V & (Omega). V = Lin {v} ) then card I = 1 by A1, Def1; then consider v being set such that A2: I = {v} by CARD_2:42; v in I by A2, TARSKI:def_1; then reconsider v = v as Vector of V ; {v} is linearly-independent by A1, A2, VECTSP_7:def_3; then A3: v <> 0. V by VECTSP_7:3; Lin {v} = VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) by A1, A2, VECTSP_7:def_3; hence ex v being Vector of V st ( v <> 0. V & (Omega). V = Lin {v} ) by A3, VECTSP_4:def_4; ::_thesis: verum end; given v being Vector of V such that A4: ( v <> 0. V & (Omega). V = Lin {v} ) ; ::_thesis: dim V = 1 ( {v} is linearly-independent & Lin {v} = VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) ) by A4, VECTSP_4:def_4, VECTSP_7:3; then A5: {v} is Basis of V by VECTSP_7:def_3; card {v} = 1 by CARD_1:30; hence dim V = 1 by A5, Def1; ::_thesis: verum end; theorem :: VECTSP_9:31 for GF being Field for V being finite-dimensional VectSp of GF 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 GF be Field; ::_thesis: for V being finite-dimensional VectSp of GF 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} ) ) let V be finite-dimensional VectSp of GF; ::_thesis: ( dim V = 2 iff ex u, v being Vector of V st ( u <> v & {u,v} is linearly-independent & (Omega). V = Lin {u,v} ) ) hereby ::_thesis: ( ex u, v being Vector of V st ( u <> v & {u,v} is linearly-independent & (Omega). V = Lin {u,v} ) implies dim V = 2 ) consider I being finite Subset of V such that A1: I is Basis of V by MATRLIN:def_1; assume dim V = 2 ; ::_thesis: ex u, v being Vector of V st ( u <> v & {u,v} is linearly-independent & (Omega). V = Lin {u,v} ) then A2: card I = 2 by A1, Def1; then consider u being set such that A3: u in I by CARD_1:27, XBOOLE_0:def_1; reconsider u = u as Vector of V by A3; now__::_thesis:_not_I_c=_{u} assume I c= {u} ; ::_thesis: contradiction then card I <= card {u} by NAT_1:43; then 2 <= 1 by A2, CARD_1:30; hence contradiction ; ::_thesis: verum end; then consider v being set such that A4: v in I and A5: not v in {u} by TARSKI:def_3; reconsider v = v as Vector of V by A4; A6: v <> u by A5, TARSKI:def_1; A7: now__::_thesis:_I_c=_{u,v} assume not I c= {u,v} ; ::_thesis: contradiction then consider w being set such that A8: w in I and A9: not w in {u,v} by TARSKI:def_3; for x being set st x in {u,v,w} holds x in I by A3, A4, A8, ENUMSET1:def_1; then {u,v,w} c= I by TARSKI:def_3; then A10: card {u,v,w} <= card I by NAT_1:43; ( w <> u & w <> v ) by A9, TARSKI:def_2; then 3 <= 2 by A2, A6, A10, CARD_2:58; hence contradiction ; ::_thesis: verum end; for x being set st x in {u,v} holds x in I by A3, A4, TARSKI:def_2; then {u,v} c= I by TARSKI:def_3; then A11: I = {u,v} by A7, XBOOLE_0:def_10; then A12: {u,v} is linearly-independent by A1, VECTSP_7:def_3; Lin {u,v} = VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) by A1, A11, VECTSP_7:def_3 .= (Omega). V by VECTSP_4:def_4 ; hence ex u, v being Vector of V st ( u <> v & {u,v} is linearly-independent & (Omega). V = Lin {u,v} ) by A6, A12; ::_thesis: verum end; given u, v being Vector of V such that A13: u <> v and A14: {u,v} is linearly-independent and A15: (Omega). V = Lin {u,v} ; ::_thesis: dim V = 2 Lin {u,v} = VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) by A15, VECTSP_4:def_4; then A16: {u,v} is Basis of V by A14, VECTSP_7:def_3; card {u,v} = 2 by A13, CARD_2:57; hence dim V = 2 by A16, Def1; ::_thesis: verum end; theorem Th32: :: VECTSP_9:32 for GF being Field for V being finite-dimensional VectSp of GF for W1, W2 being Subspace of V holds (dim (W1 + W2)) + (dim (W1 /\ W2)) = (dim W1) + (dim W2) proof let GF be Field; ::_thesis: for V being finite-dimensional VectSp of GF for W1, W2 being Subspace of V holds (dim (W1 + W2)) + (dim (W1 /\ W2)) = (dim W1) + (dim W2) let V be finite-dimensional VectSp of GF; ::_thesis: for W1, W2 being Subspace of V holds (dim (W1 + W2)) + (dim (W1 /\ W2)) = (dim W1) + (dim W2) let W1, W2 be Subspace of V; ::_thesis: (dim (W1 + W2)) + (dim (W1 /\ W2)) = (dim W1) + (dim W2) reconsider V = V as VectSp of GF ; reconsider W1 = W1, W2 = W2 as Subspace of V ; consider I being finite Subset of (W1 /\ W2) such that A1: I is Basis of W1 /\ W2 by MATRLIN:def_1; W1 /\ W2 is Subspace of W2 by VECTSP_5:15; then consider I2 being Basis of W2 such that A2: I c= I2 by A1, Th13; W1 /\ W2 is Subspace of W1 by VECTSP_5:15; then consider I1 being Basis of W1 such that A3: I c= I1 by A1, Th13; reconsider I2 = I2 as finite Subset of W2 by Th20; reconsider I1 = I1 as finite Subset of W1 by Th20; A4: now__::_thesis:_I1_/\_I2_c=_I I1 is linearly-independent by VECTSP_7:def_3; then reconsider I19 = I1 as linearly-independent Subset of V by Th11; assume not I1 /\ I2 c= I ; ::_thesis: contradiction then consider x being set such that A5: x in I1 /\ I2 and A6: not x in I by TARSKI:def_3; x in I1 by A5, XBOOLE_0:def_4; then x in Lin I1 by VECTSP_7:8; then x in VectSpStr(# the carrier of W1, the U5 of W1, the ZeroF of W1, the lmult of W1 #) by VECTSP_7:def_3; then A7: x in the carrier of W1 by STRUCT_0:def_5; A8: the carrier of W1 c= the carrier of V by VECTSP_4:def_2; then reconsider x9 = x as Vector of V by A7; now__::_thesis:_for_y_being_set_st_y_in_I_\/_{x}_holds_ y_in_the_carrier_of_V let y be set ; ::_thesis: ( y in I \/ {x} implies y in the carrier of V ) the carrier of (W1 /\ W2) c= the carrier of V by VECTSP_4:def_2; then A9: I c= the carrier of V by XBOOLE_1:1; assume y in I \/ {x} ; ::_thesis: y in the carrier of V then ( y in I or y in {x} ) by XBOOLE_0:def_3; then ( y in the carrier of V or y = x ) by A9, TARSKI:def_1; hence y in the carrier of V by A7, A8; ::_thesis: verum end; then reconsider Ix = I \/ {x} as Subset of V by TARSKI:def_3; now__::_thesis:_for_y_being_set_st_y_in_I_\/_{x}_holds_ y_in_I19 let y be set ; ::_thesis: ( y in I \/ {x} implies y in I19 ) assume y in I \/ {x} ; ::_thesis: y in I19 then ( y in I or y in {x} ) by XBOOLE_0:def_3; then ( y in I1 or y = x ) by A3, TARSKI:def_1; hence y in I19 by A5, XBOOLE_0:def_4; ::_thesis: verum end; then A10: Ix c= I19 by TARSKI:def_3; x in {x} by TARSKI:def_1; then A11: x9 in Ix by XBOOLE_0:def_3; x in I2 by A5, XBOOLE_0:def_4; then x in Lin I2 by VECTSP_7:8; then x in VectSpStr(# the carrier of W2, the U5 of W2, the ZeroF of W2, the lmult of W2 #) by VECTSP_7:def_3; then x in the carrier of W2 by STRUCT_0:def_5; then x in the carrier of W1 /\ the carrier of W2 by A7, XBOOLE_0:def_4; then x in the carrier of (W1 /\ W2) by VECTSP_5:def_2; then A12: x in VectSpStr(# the carrier of (W1 /\ W2), the U5 of (W1 /\ W2), the ZeroF of (W1 /\ W2), the lmult of (W1 /\ W2) #) by STRUCT_0:def_5; the carrier of (W1 /\ W2) c= the carrier of V by VECTSP_4:def_2; then reconsider I9 = I as Subset of V by XBOOLE_1:1; A13: Lin I = Lin I9 by Th17; Ix \ {x} = I \ {x} by XBOOLE_1:40 .= I by A6, ZFMISC_1:57 ; then not x9 in Lin I9 by A10, A11, Th14, VECTSP_7:1; hence contradiction by A1, A12, A13, VECTSP_7:def_3; ::_thesis: verum end; set A = I1 \/ I2; now__::_thesis:_for_v_being_set_st_v_in_I1_\/_I2_holds_ v_in_the_carrier_of_(W1_+_W2) let v be set ; ::_thesis: ( v in I1 \/ I2 implies v in the carrier of (W1 + W2) ) A14: ( the carrier of W1 c= the carrier of V & the carrier of W2 c= the carrier of V ) by VECTSP_4:def_2; assume v in I1 \/ I2 ; ::_thesis: v in the carrier of (W1 + W2) then A15: ( v in I1 or v in I2 ) by XBOOLE_0:def_3; then ( v in the carrier of W1 or v in the carrier of W2 ) ; then reconsider v9 = v as Vector of V by A14; ( v9 in W1 or v9 in W2 ) by A15, STRUCT_0:def_5; then v9 in W1 + W2 by VECTSP_5:2; hence v in the carrier of (W1 + W2) by STRUCT_0:def_5; ::_thesis: verum end; then reconsider A = I1 \/ I2 as finite Subset of (W1 + W2) by TARSKI:def_3; I c= I1 /\ I2 by A3, A2, XBOOLE_1:19; then I = I1 /\ I2 by A4, XBOOLE_0:def_10; then A16: card A = ((card I1) + (card I2)) - (card I) by CARD_2:45; for L being Linear_Combination of A st Sum L = 0. (W1 + W2) holds Carrier L = {} proof ( W1 is Subspace of W1 + W2 & I1 is linearly-independent ) by VECTSP_5:7, VECTSP_7:def_3; then reconsider I19 = I1 as linearly-independent Subset of (W1 + W2) by Th11; reconsider W29 = W2 as Subspace of W1 + W2 by VECTSP_5:7; reconsider W19 = W1 as Subspace of W1 + W2 by VECTSP_5:7; let L be Linear_Combination of A; ::_thesis: ( Sum L = 0. (W1 + W2) implies Carrier L = {} ) assume A17: Sum L = 0. (W1 + W2) ; ::_thesis: Carrier L = {} A18: I1 misses (Carrier L) \ I1 by XBOOLE_1:79; set B = (Carrier L) /\ I1; consider F being FinSequence of the carrier of (W1 + W2) such that A19: F is one-to-one and A20: rng F = Carrier L and A21: Sum L = Sum (L (#) F) by VECTSP_6:def_6; reconsider B = (Carrier L) /\ I1 as Subset of (rng F) by A20, XBOOLE_1:17; reconsider F1 = F - (B `), F2 = F - B as FinSequence of the carrier of (W1 + W2) by FINSEQ_3:86; consider L1 being Linear_Combination of W1 + W2 such that A22: Carrier L1 = (rng F1) /\ (Carrier L) and A23: L1 (#) F1 = L (#) F1 by Th4; F1 is one-to-one by A19, FINSEQ_3:87; then A24: Sum (L (#) F1) = Sum L1 by A22, A23, Th3, XBOOLE_1:17; rng F c= rng F ; then reconsider X = rng F as Subset of (rng F) ; consider L2 being Linear_Combination of W1 + W2 such that A25: Carrier L2 = (rng F2) /\ (Carrier L) and A26: L2 (#) F2 = L (#) F2 by Th4; F2 is one-to-one by A19, FINSEQ_3:87; then A27: Sum (L (#) F2) = Sum L2 by A25, A26, Th3, XBOOLE_1:17; X \ (B `) = X /\ ((B `) `) by SUBSET_1:13 .= B by XBOOLE_1:28 ; then rng F1 = B by FINSEQ_3:65; then A28: Carrier L1 = I1 /\ ((Carrier L) /\ (Carrier L)) by A22, XBOOLE_1:16 .= (Carrier L) /\ I1 ; then consider K1 being Linear_Combination of W19 such that Carrier K1 = Carrier L1 and A29: Sum K1 = Sum L1 by Th9; rng F2 = (Carrier L) \ ((Carrier L) /\ I1) by A20, FINSEQ_3:65 .= (Carrier L) \ I1 by XBOOLE_1:47 ; then A30: Carrier L2 = (Carrier L) \ I1 by A25, XBOOLE_1:28, XBOOLE_1:36; then (Carrier L1) /\ (Carrier L2) = (Carrier L) /\ (I1 /\ ((Carrier L) \ I1)) by A28, XBOOLE_1:16 .= (Carrier L) /\ {} by A18, XBOOLE_0:def_7 .= {} ; then A31: Carrier L1 misses Carrier L2 by XBOOLE_0:def_7; A32: Carrier L c= I1 \/ I2 by VECTSP_6:def_4; then A33: Carrier L2 c= I2 by A30, XBOOLE_1:43; Carrier L2 c= I2 by A32, A30, XBOOLE_1:43; then consider K2 being Linear_Combination of W29 such that Carrier K2 = Carrier L2 and A34: Sum K2 = Sum L2 by Th9, XBOOLE_1:1; A35: Sum K1 in W1 by STRUCT_0:def_5; ex P being Permutation of (dom F) st (F - (B `)) ^ (F - B) = F * P by FINSEQ_3:115; then A36: 0. (W1 + W2) = Sum (L (#) (F1 ^ F2)) by A17, A21, Th1 .= Sum ((L (#) F1) ^ (L (#) F2)) by VECTSP_6:13 .= (Sum L1) + (Sum L2) by A24, A27, RLVECT_1:41 ; then Sum L1 = - (Sum L2) by VECTSP_1:16 .= - (Sum K2) by A34, VECTSP_4:15 ; then Sum K1 in W2 by A29, STRUCT_0:def_5; then Sum K1 in W1 /\ W2 by A35, VECTSP_5:3; then Sum K1 in Lin I by A1, VECTSP_7:def_3; then consider KI being Linear_Combination of I such that A37: Sum K1 = Sum KI by VECTSP_7:7; A38: Carrier L = (Carrier L1) \/ (Carrier L2) by A28, A30, XBOOLE_1:51; A39: now__::_thesis:_Carrier_L_c=_Carrier_(L1_+_L2) assume not Carrier L c= Carrier (L1 + L2) ; ::_thesis: contradiction then consider x being set such that A40: x in Carrier L and A41: not x in Carrier (L1 + L2) by TARSKI:def_3; reconsider x = x as Vector of (W1 + W2) by A40; A42: 0. GF = (L1 + L2) . x by A41, VECTSP_6:2 .= (L1 . x) + (L2 . x) by VECTSP_6:22 ; percases ( x in Carrier L1 or x in Carrier L2 ) by A38, A40, XBOOLE_0:def_3; supposeA43: x in Carrier L1 ; ::_thesis: contradiction then not x in Carrier L2 by A31, XBOOLE_0:3; then A44: L2 . x = 0. GF by VECTSP_6:2; ex v being Vector of (W1 + W2) st ( x = v & L1 . v <> 0. GF ) by A43, VECTSP_6:1; hence contradiction by A42, A44, RLVECT_1:4; ::_thesis: verum end; supposeA45: x in Carrier L2 ; ::_thesis: contradiction then not x in Carrier L1 by A31, XBOOLE_0:3; then A46: L1 . x = 0. GF by VECTSP_6:2; ex v being Vector of (W1 + W2) st ( x = v & L2 . v <> 0. GF ) by A45, VECTSP_6:1; hence contradiction by A42, A46, RLVECT_1:4; ::_thesis: verum end; end; end; A47: I \/ I2 = I2 by A2, XBOOLE_1:12; A48: I2 is linearly-independent by VECTSP_7:def_3; A49: Carrier L1 c= I1 by A28, XBOOLE_1:17; W1 /\ W2 is Subspace of W1 + W2 by VECTSP_5:23; then consider LI being Linear_Combination of W1 + W2 such that A50: Carrier LI = Carrier KI and A51: Sum LI = Sum KI by Th8; Carrier LI c= I by A50, VECTSP_6:def_4; then Carrier LI c= I19 by A3, XBOOLE_1:1; then A52: LI = L1 by A49, A29, A37, A51, MATRLIN:5; Carrier LI c= I by A50, VECTSP_6:def_4; then ( Carrier (LI + L2) c= (Carrier LI) \/ (Carrier L2) & (Carrier LI) \/ (Carrier L2) c= I2 ) by A47, A33, VECTSP_6:23, XBOOLE_1:13; then A53: Carrier (LI + L2) c= I2 by XBOOLE_1:1; W2 is Subspace of W1 + W2 by VECTSP_5:7; then consider K being Linear_Combination of W2 such that A54: Carrier K = Carrier (LI + L2) and A55: Sum K = Sum (LI + L2) by A53, Th9, XBOOLE_1:1; reconsider K = K as Linear_Combination of I2 by A53, A54, VECTSP_6:def_4; 0. W2 = (Sum LI) + (Sum L2) by A29, A36, A37, A51, VECTSP_4:12 .= Sum K by A55, VECTSP_6:44 ; then {} = Carrier (L1 + L2) by A54, A52, A48, VECTSP_7:def_1; hence Carrier L = {} by A39; ::_thesis: verum end; then A56: A is linearly-independent by VECTSP_7:def_1; the carrier of (W1 + W2) c= the carrier of V by VECTSP_4:def_2; then reconsider A9 = A as Subset of V by XBOOLE_1:1; A57: card I2 = dim W2 by Def1; now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_(W1_+_W2)_holds_ x_in_the_carrier_of_(Lin_A9) let x be set ; ::_thesis: ( x in the carrier of (W1 + W2) implies x in the carrier of (Lin A9) ) assume x in the carrier of (W1 + W2) ; ::_thesis: x in the carrier of (Lin A9) then x in W1 + W2 by STRUCT_0:def_5; then consider w1, w2 being Vector of V such that A58: w1 in W1 and A59: w2 in W2 and A60: x = w1 + w2 by VECTSP_5:1; reconsider w1 = w1 as Vector of W1 by A58, STRUCT_0:def_5; w1 in Lin I1 by Th10; then consider K1 being Linear_Combination of I1 such that A61: w1 = Sum K1 by VECTSP_7:7; reconsider w2 = w2 as Vector of W2 by A59, STRUCT_0:def_5; w2 in Lin I2 by Th10; then consider K2 being Linear_Combination of I2 such that A62: w2 = Sum K2 by VECTSP_7:7; consider L2 being Linear_Combination of V such that A63: Carrier L2 = Carrier K2 and A64: Sum L2 = Sum K2 by Th8; A65: Carrier L2 c= I2 by A63, VECTSP_6:def_4; consider L1 being Linear_Combination of V such that A66: Carrier L1 = Carrier K1 and A67: Sum L1 = Sum K1 by Th8; set L = L1 + L2; Carrier L1 c= I1 by A66, VECTSP_6:def_4; then ( Carrier (L1 + L2) c= (Carrier L1) \/ (Carrier L2) & (Carrier L1) \/ (Carrier L2) c= I1 \/ I2 ) by A65, VECTSP_6:23, XBOOLE_1:13; then Carrier (L1 + L2) c= I1 \/ I2 by XBOOLE_1:1; then reconsider L = L1 + L2 as Linear_Combination of A9 by VECTSP_6:def_4; x = Sum L by A60, A61, A67, A62, A64, VECTSP_6:44; then x in Lin A9 by VECTSP_7:7; hence x in the carrier of (Lin A9) by STRUCT_0:def_5; ::_thesis: verum end; then the carrier of (W1 + W2) c= the carrier of (Lin A9) by TARSKI:def_3; then A68: W1 + W2 is Subspace of Lin A9 by VECTSP_4:27; Lin A9 = Lin A by Th17; then Lin A = W1 + W2 by A68, VECTSP_4:25; then A69: A is Basis of W1 + W2 by A56, VECTSP_7:def_3; card I = dim (W1 /\ W2) by A1, Def1; then (dim (W1 + W2)) + (dim (W1 /\ W2)) = (((card I1) + (card I2)) + (- (card I))) + (card I) by A16, A69, Def1 .= (dim W1) + (dim W2) by A57, Def1 ; hence (dim (W1 + W2)) + (dim (W1 /\ W2)) = (dim W1) + (dim W2) ; ::_thesis: verum end; theorem :: VECTSP_9:33 for GF being Field for V being finite-dimensional VectSp of GF for W1, W2 being Subspace of V holds dim (W1 /\ W2) >= ((dim W1) + (dim W2)) - (dim V) proof let GF be Field; ::_thesis: for V being finite-dimensional VectSp of GF for W1, W2 being Subspace of V holds dim (W1 /\ W2) >= ((dim W1) + (dim W2)) - (dim V) let V be finite-dimensional VectSp of GF; ::_thesis: for W1, W2 being Subspace of V holds dim (W1 /\ W2) >= ((dim W1) + (dim W2)) - (dim V) let W1, W2 be Subspace of V; ::_thesis: dim (W1 /\ W2) >= ((dim W1) + (dim W2)) - (dim V) A1: ( dim (W1 + W2) <= dim V & (dim V) + ((dim (W1 /\ W2)) - (dim V)) = dim (W1 /\ W2) ) by Th25; ((dim W1) + (dim W2)) - (dim V) = ((dim (W1 + W2)) + (dim (W1 /\ W2))) - (dim V) by Th32 .= (dim (W1 + W2)) + ((dim (W1 /\ W2)) - (dim V)) ; hence dim (W1 /\ W2) >= ((dim W1) + (dim W2)) - (dim V) by A1, XREAL_1:6; ::_thesis: verum end; theorem :: VECTSP_9:34 for GF being Field for V being finite-dimensional VectSp of GF for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds dim V = (dim W1) + (dim W2) proof let GF be Field; ::_thesis: for V being finite-dimensional VectSp of GF for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds dim V = (dim W1) + (dim W2) let V be finite-dimensional VectSp of GF; ::_thesis: for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds dim V = (dim W1) + (dim W2) let W1, W2 be Subspace of V; ::_thesis: ( V is_the_direct_sum_of W1,W2 implies dim V = (dim W1) + (dim W2) ) assume A1: V is_the_direct_sum_of W1,W2 ; ::_thesis: dim V = (dim W1) + (dim W2) then A2: VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) = W1 + W2 by VECTSP_5:def_4; W1 /\ W2 = (0). V by A1, VECTSP_5:def_4; then (Omega). (W1 /\ W2) = (0). V by VECTSP_4:def_4 .= (0). (W1 /\ W2) by VECTSP_4:36 ; then dim (W1 /\ W2) = 0 by Th29; then (dim W1) + (dim W2) = (dim (W1 + W2)) + 0 by Th32 .= dim ((Omega). V) by A2, VECTSP_4:def_4 .= dim V by Th27 ; hence dim V = (dim W1) + (dim W2) ; ::_thesis: verum end; Lm2: for GF being Field for n being Nat for V being finite-dimensional VectSp of GF st n <= dim V holds ex W being strict Subspace of V st dim W = n proof let GF be Field; ::_thesis: for n being Nat for V being finite-dimensional VectSp of GF st n <= dim V holds ex W being strict Subspace of V st dim W = n let n be Nat; ::_thesis: for V being finite-dimensional VectSp of GF st n <= dim V holds ex W being strict Subspace of V st dim W = n let V be finite-dimensional VectSp of GF; ::_thesis: ( n <= dim V implies ex W being strict Subspace of V st dim W = n ) consider I being finite Subset of V such that A1: I is Basis of V by MATRLIN:def_1; assume n <= dim V ; ::_thesis: ex W being strict Subspace of V st dim W = n then n <= card I by A1, Def1; then consider A being finite Subset of I such that A2: card A = n by FINSEQ_4:72; reconsider A = A as Subset of V by XBOOLE_1:1; reconsider W = Lin A as strict finite-dimensional Subspace of V ; I is linearly-independent by A1, VECTSP_7:def_3; then dim W = n by A2, Th26, VECTSP_7:1; hence ex W being strict Subspace of V st dim W = n ; ::_thesis: verum end; theorem :: VECTSP_9:35 for GF being Field for n being Nat for V being finite-dimensional VectSp of GF holds ( n <= dim V iff ex W being strict Subspace of V st dim W = n ) by Lm2, Th25; definition let GF be Field; let V be finite-dimensional VectSp of GF; let n be Nat; funcn Subspaces_of V -> set means :Def2: :: VECTSP_9:def 2 for x being set holds ( x in it iff ex W being strict Subspace of V st ( W = x & dim W = n ) ); existence ex b1 being set st for x being set holds ( x in b1 iff ex W being strict Subspace of V st ( W = x & dim W = n ) ) proof set S = { (Lin A) where A is Subset of V : ( A is linearly-independent & card A = n ) } ; take { (Lin A) where A is Subset of V : ( A is linearly-independent & card A = n ) } ; ::_thesis: for x being set holds ( x in { (Lin A) where A is Subset of V : ( A is linearly-independent & card A = n ) } iff ex W being strict Subspace of V st ( W = x & dim W = n ) ) for x being set holds ( x in { (Lin A) where A is Subset of V : ( A is linearly-independent & card A = n ) } iff ex W being strict Subspace of V st ( W = x & dim W = n ) ) proof let x be set ; ::_thesis: ( x in { (Lin A) where A is Subset of V : ( A is linearly-independent & card A = n ) } iff ex W being strict Subspace of V st ( W = x & dim W = n ) ) hereby ::_thesis: ( ex W being strict Subspace of V st ( W = x & dim W = n ) implies x in { (Lin A) where A is Subset of V : ( A is linearly-independent & card A = n ) } ) assume x in { (Lin A) where A is Subset of V : ( A is linearly-independent & card A = n ) } ; ::_thesis: ex W being strict Subspace of V st ( W = x & dim W = n ) then A1: ex A being Subset of V st ( x = Lin A & A is linearly-independent & card A = n ) ; then reconsider W = x as strict Subspace of V ; dim W = n by A1, Th26; hence ex W being strict Subspace of V st ( W = x & dim W = n ) ; ::_thesis: verum end; given W being strict Subspace of V such that A2: W = x and A3: dim W = n ; ::_thesis: x in { (Lin A) where A is Subset of V : ( A is linearly-independent & card A = n ) } consider A being finite Subset of W such that A4: A is Basis of W by MATRLIN:def_1; reconsider A = A as Subset of W ; A is linearly-independent by A4, VECTSP_7:def_3; then reconsider B = A as linearly-independent Subset of V by Th11; A5: x = Lin A by A2, A4, VECTSP_7:def_3 .= Lin B by Th17 ; then card B = n by A2, A3, Th26; hence x in { (Lin A) where A is Subset of V : ( A is linearly-independent & card A = n ) } by A5; ::_thesis: verum end; hence for x being set holds ( x in { (Lin A) where A is Subset of V : ( A is linearly-independent & card A = n ) } iff ex W being strict Subspace of V st ( W = x & dim W = n ) ) ; ::_thesis: verum end; uniqueness for b1, b2 being set st ( for x being set holds ( x in b1 iff ex W being strict Subspace of V st ( W = x & dim W = n ) ) ) & ( for x being set holds ( x in b2 iff ex W being strict Subspace of V st ( W = x & dim W = n ) ) ) holds b1 = b2 proof defpred S1[ set ] means ex W being strict Subspace of V st ( W = $1 & dim W = n ); thus for X1, X2 being set st ( for x being set holds ( x in X1 iff S1[x] ) ) & ( for x being set holds ( x in X2 iff S1[x] ) ) holds X1 = X2 from XBOOLE_0:sch_3(); ::_thesis: verum end; end; :: deftheorem Def2 defines Subspaces_of VECTSP_9:def_2_:_ for GF being Field for V being finite-dimensional VectSp of GF for n being Nat for b4 being set holds ( b4 = n Subspaces_of V iff for x being set holds ( x in b4 iff ex W being strict Subspace of V st ( W = x & dim W = n ) ) ); theorem :: VECTSP_9:36 for GF being Field for n being Nat for V being finite-dimensional VectSp of GF st n <= dim V holds not n Subspaces_of V is empty proof let GF be Field; ::_thesis: for n being Nat for V being finite-dimensional VectSp of GF st n <= dim V holds not n Subspaces_of V is empty let n be Nat; ::_thesis: for V being finite-dimensional VectSp of GF st n <= dim V holds not n Subspaces_of V is empty let V be finite-dimensional VectSp of GF; ::_thesis: ( n <= dim V implies not n Subspaces_of V is empty ) assume n <= dim V ; ::_thesis: not n Subspaces_of V is empty then ex W being strict Subspace of V st dim W = n by Lm2; hence not n Subspaces_of V is empty by Def2; ::_thesis: verum end; theorem :: VECTSP_9:37 for GF being Field for n being Nat for V being finite-dimensional VectSp of GF st dim V < n holds n Subspaces_of V = {} proof let GF be Field; ::_thesis: for n being Nat for V being finite-dimensional VectSp of GF st dim V < n holds n Subspaces_of V = {} let n be Nat; ::_thesis: for V being finite-dimensional VectSp of GF st dim V < n holds n Subspaces_of V = {} let V be finite-dimensional VectSp of GF; ::_thesis: ( dim V < n implies n Subspaces_of V = {} ) assume that A1: dim V < n and A2: n Subspaces_of V <> {} ; ::_thesis: contradiction consider x being set such that A3: x in n Subspaces_of V by A2, XBOOLE_0:def_1; ex W being strict Subspace of V st ( W = x & dim W = n ) by A3, Def2; hence contradiction by A1, Th25; ::_thesis: verum end; theorem :: VECTSP_9:38 for GF being Field for n being Nat for V being finite-dimensional VectSp of GF for W being Subspace of V holds n Subspaces_of W c= n Subspaces_of V proof let GF be Field; ::_thesis: for n being Nat for V being finite-dimensional VectSp of GF for W being Subspace of V holds n Subspaces_of W c= n Subspaces_of V let n be Nat; ::_thesis: for V being finite-dimensional VectSp of GF for W being Subspace of V holds n Subspaces_of W c= n Subspaces_of V let V be finite-dimensional VectSp of GF; ::_thesis: for W being Subspace of V holds n Subspaces_of W c= n Subspaces_of V let W be Subspace of V; ::_thesis: n Subspaces_of W c= n Subspaces_of V let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in n Subspaces_of W or x in n Subspaces_of V ) assume x in n Subspaces_of W ; ::_thesis: x in n Subspaces_of V then consider W1 being strict Subspace of W such that A1: W1 = x and A2: dim W1 = n by Def2; reconsider W1 = W1 as strict Subspace of V by VECTSP_4:26; W1 in n Subspaces_of V by A2, Def2; hence x in n Subspaces_of V by A1; ::_thesis: verum end;