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Theorem lshpkrlem6 34402
Description: Lemma for lshpkrex 34405. Show linearlity of G. (Contributed by NM, 17-Jul-2014.)
Hypotheses
Ref Expression
lshpkrlem.v |- V = ( Base ` W )
lshpkrlem.a |- .+ = ( +g ` W )
lshpkrlem.n |- N = ( LSpan ` W )
lshpkrlem.p |- .(+) = ( LSSum ` W )
lshpkrlem.h |- H = (LSHyp ` W )
lshpkrlem.w |- ( ph -> W e. LVec )
lshpkrlem.u |- ( ph -> U e. H )
lshpkrlem.z |- ( ph -> Z e. V )
lshpkrlem.x |- ( ph -> X e. V )
lshpkrlem.e |- ( ph -> ( U .(+) ( N ` { Z } ) ) = V )
lshpkrlem.d |- D = (Scalar ` W )
lshpkrlem.k |- K = ( Base ` D )
lshpkrlem.t |- .x. = ( .s ` W )
lshpkrlem.o |- .0. = ( 0g ` D )
lshpkrlem.g |- G = ( x e. V |-> ( iota_ k e. K E. y e. U x = ( y .+ ( k .x. Z ) ) ) )
Assertion
Ref Expression
lshpkrlem6 |- ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) -> ( G ` ( ( l .x. u ) .+ v ) ) = ( ( l ( .r ` D ) ( G ` u ) ) ( +g ` D ) ( G ` v ) ) )
Distinct variable groups:   x, k, y, .+   k, K, x   .0. , k   .x. , k, x, y   U, k, x, y   x, V   k, X, x, y   k, Z, x, y   .+ , l   G, l   K, l   U, l   X, l   Z, l, k, x, y   .x. , l   u, k, v, x, y, l
Allowed substitution hints:   ph( x, y, v, u, k, l)   D( x, y, v, u, k, l)   .+ ( v, u)   .(+) ( x, y, v, u, k, l)   .x. ( v, u)   U( v, u)   G( x, y, v, u, k)   H( x, y, v, u, k, l)   K( y, v, u)   N( x, y, v, u, k, l)   V( y, v, u, k, l)   W( x, y, v, u, k, l)   X( v, u)   .0. ( x, y, v, u, l)   Z( v, u)
Proof of Theorem lshpkrlem6
Dummy variables z s r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lshpkrlem.v . . 3 |- V = ( Base ` W )
2 lshpkrlem.a . . 3 |- .+ = ( +g ` W )
3 lshpkrlem.n . . 3 |- N = ( LSpan ` W )
4 lshpkrlem.p . . 3 |- .(+) = ( LSSum ` W )
5 lshpkrlem.h . . 3 |- H = (LSHyp ` W )
6 lshpkrlem.w . . . 4 |- ( ph -> W e. LVec )
76adantr 481 . . 3 |- ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) -> W e. LVec )
8 lshpkrlem.u . . . 4 |- ( ph -> U e. H )
98adantr 481 . . 3 |- ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) -> U e. H )
10 lshpkrlem.z . . . 4 |- ( ph -> Z e. V )
1110adantr 481 . . 3 |- ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) -> Z e. V )
12 simpr2 1068 . . 3 |- ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) -> u e. V )
13 lshpkrlem.e . . . 4 |- ( ph -> ( U .(+) ( N ` { Z } ) ) = V )
1413adantr 481 . . 3 |- ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) -> ( U .(+) ( N ` { Z } ) ) = V )
15 lshpkrlem.d . . 3 |- D = (Scalar ` W )
16 lshpkrlem.k . . 3 |- K = ( Base ` D )
17 lshpkrlem.t . . 3 |- .x. = ( .s ` W )
18 lshpkrlem.o . . 3 |- .0. = ( 0g ` D )
19 lshpkrlem.g . . 3 |- G = ( x e. V |-> ( iota_ k e. K E. y e. U x = ( y .+ ( k .x. Z ) ) ) )
201, 2, 3, 4, 5, 7, 9, 11, 12, 14, 15, 16, 17, 18, 19lshpkrlem3 34399 . 2 |- ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) -> E. r e. U u = ( r .+ ( ( G ` u ) .x. Z ) ) )
21 simpr3 1069 . . 3 |- ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) -> v e. V )
221, 2, 3, 4, 5, 7, 9, 11, 21, 14, 15, 16, 17, 18, 19lshpkrlem3 34399 . 2 |- ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) -> E. s e. U v = ( s .+ ( ( G ` v ) .x. Z ) ) )
23 lveclmod 19106 . . . . 5 |- ( W e. LVec -> W e. LMod )
247, 23syl 17 . . . 4 |- ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) -> W e. LMod )
25 simpr1 1067 . . . . 5 |- ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) -> l e. K )
261, 15, 17, 16lmodvscl 18880 . . . . 5 |- ( ( W e. LMod /\ l e. K /\ u e. V ) -> ( l .x. u ) e. V )
2724, 25, 12, 26syl3anc 1326 . . . 4 |- ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) -> ( l .x. u ) e. V )
281, 2lmodvacl 18877 . . . 4 |- ( ( W e. LMod /\ ( l .x. u ) e. V /\ v e. V ) -> ( ( l .x. u ) .+ v ) e. V )
2924, 27, 21, 28syl3anc 1326 . . 3 |- ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) -> ( ( l .x. u ) .+ v ) e. V )
301, 2, 3, 4, 5, 7, 9, 11, 29, 14, 15, 16, 17, 18, 19lshpkrlem3 34399 . 2 |- ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) -> E. z e. U ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) )
31 3reeanv 3108 . . 3 |- ( E. r e. U E. s e. U E. z e. U ( u = ( r .+ ( ( G ` u ) .x. Z ) ) /\ v = ( s .+ ( ( G ` v ) .x. Z ) ) /\ ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) ) <-> ( E. r e. U u = ( r .+ ( ( G ` u ) .x. Z ) ) /\ E. s e. U v = ( s .+ ( ( G ` v ) .x. Z ) ) /\ E. z e. U ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) ) )
32 simp1l 1085 . . . . . . . 8 |- ( ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) /\ ( ( r e. U /\ s e. U ) /\ z e. U ) /\ ( u = ( r .+ ( ( G ` u ) .x. Z ) ) /\ v = ( s .+ ( ( G ` v ) .x. Z ) ) /\ ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) ) ) -> ph )
33 simp1r1 1157 . . . . . . . 8 |- ( ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) /\ ( ( r e. U /\ s e. U ) /\ z e. U ) /\ ( u = ( r .+ ( ( G ` u ) .x. Z ) ) /\ v = ( s .+ ( ( G ` v ) .x. Z ) ) /\ ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) ) ) -> l e. K )
34 simp1r2 1158 . . . . . . . 8 |- ( ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) /\ ( ( r e. U /\ s e. U ) /\ z e. U ) /\ ( u = ( r .+ ( ( G ` u ) .x. Z ) ) /\ v = ( s .+ ( ( G ` v ) .x. Z ) ) /\ ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) ) ) -> u e. V )
35 simp1r3 1159 . . . . . . . 8 |- ( ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) /\ ( ( r e. U /\ s e. U ) /\ z e. U ) /\ ( u = ( r .+ ( ( G ` u ) .x. Z ) ) /\ v = ( s .+ ( ( G ` v ) .x. Z ) ) /\ ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) ) ) -> v e. V )
36 simp2ll 1128 . . . . . . . 8 |- ( ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) /\ ( ( r e. U /\ s e. U ) /\ z e. U ) /\ ( u = ( r .+ ( ( G ` u ) .x. Z ) ) /\ v = ( s .+ ( ( G ` v ) .x. Z ) ) /\ ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) ) ) -> r e. U )
37 simp2lr 1129 . . . . . . . . 9 |- ( ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) /\ ( ( r e. U /\ s e. U ) /\ z e. U ) /\ ( u = ( r .+ ( ( G ` u ) .x. Z ) ) /\ v = ( s .+ ( ( G ` v ) .x. Z ) ) /\ ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) ) ) -> s e. U )
38 simp2r 1088 . . . . . . . . 9 |- ( ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) /\ ( ( r e. U /\ s e. U ) /\ z e. U ) /\ ( u = ( r .+ ( ( G ` u ) .x. Z ) ) /\ v = ( s .+ ( ( G ` v ) .x. Z ) ) /\ ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) ) ) -> z e. U )
3937, 38jca 554 . . . . . . . 8 |- ( ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) /\ ( ( r e. U /\ s e. U ) /\ z e. U ) /\ ( u = ( r .+ ( ( G ` u ) .x. Z ) ) /\ v = ( s .+ ( ( G ` v ) .x. Z ) ) /\ ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) ) ) -> ( s e. U /\ z e. U ) )
40 simp31 1097 . . . . . . . 8 |- ( ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) /\ ( ( r e. U /\ s e. U ) /\ z e. U ) /\ ( u = ( r .+ ( ( G ` u ) .x. Z ) ) /\ v = ( s .+ ( ( G ` v ) .x. Z ) ) /\ ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) ) ) -> u = ( r .+ ( ( G ` u ) .x. Z ) ) )
41 simp32 1098 . . . . . . . 8 |- ( ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) /\ ( ( r e. U /\ s e. U ) /\ z e. U ) /\ ( u = ( r .+ ( ( G ` u ) .x. Z ) ) /\ v = ( s .+ ( ( G ` v ) .x. Z ) ) /\ ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) ) ) -> v = ( s .+ ( ( G ` v ) .x. Z ) ) )
42 simp33 1099 . . . . . . . 8 |- ( ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) /\ ( ( r e. U /\ s e. U ) /\ z e. U ) /\ ( u = ( r .+ ( ( G ` u ) .x. Z ) ) /\ v = ( s .+ ( ( G ` v ) .x. Z ) ) /\ ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) ) ) -> ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) )
431, 2, 3, 4, 5, 6, 8, 10, 10, 13, 15, 16, 17, 18, 19lshpkrlem5 34401 . . . . . . . 8 |- ( ( ( ph /\ l e. K /\ u e. V ) /\ ( v e. V /\ r e. U /\ ( s e. U /\ z e. U ) ) /\ ( u = ( r .+ ( ( G ` u ) .x. Z ) ) /\ v = ( s .+ ( ( G ` v ) .x. Z ) ) /\ ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) ) ) -> ( G ` ( ( l .x. u ) .+ v ) ) = ( ( l ( .r ` D ) ( G ` u ) ) ( +g ` D ) ( G ` v ) ) )
4432, 33, 34, 35, 36, 39, 40, 41, 42, 43syl333anc 1358 . . . . . . 7 |- ( ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) /\ ( ( r e. U /\ s e. U ) /\ z e. U ) /\ ( u = ( r .+ ( ( G ` u ) .x. Z ) ) /\ v = ( s .+ ( ( G ` v ) .x. Z ) ) /\ ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) ) ) -> ( G ` ( ( l .x. u ) .+ v ) ) = ( ( l ( .r ` D ) ( G ` u ) ) ( +g ` D ) ( G ` v ) ) )
45443exp 1264 . . . . . 6 |- ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) -> ( ( ( r e. U /\ s e. U ) /\ z e. U ) -> ( ( u = ( r .+ ( ( G ` u ) .x. Z ) ) /\ v = ( s .+ ( ( G ` v ) .x. Z ) ) /\ ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) ) -> ( G ` ( ( l .x. u ) .+ v ) ) = ( ( l ( .r ` D ) ( G ` u ) ) ( +g ` D ) ( G ` v ) ) ) ) )
4645expdimp 453 . . . . 5 |- ( ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) /\ ( r e. U /\ s e. U ) ) -> ( z e. U -> ( ( u = ( r .+ ( ( G ` u ) .x. Z ) ) /\ v = ( s .+ ( ( G ` v ) .x. Z ) ) /\ ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) ) -> ( G ` ( ( l .x. u ) .+ v ) ) = ( ( l ( .r ` D ) ( G ` u ) ) ( +g ` D ) ( G ` v ) ) ) ) )
4746rexlimdv 3030 . . . 4 |- ( ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) /\ ( r e. U /\ s e. U ) ) -> ( E. z e. U ( u = ( r .+ ( ( G ` u ) .x. Z ) ) /\ v = ( s .+ ( ( G ` v ) .x. Z ) ) /\ ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) ) -> ( G ` ( ( l .x. u ) .+ v ) ) = ( ( l ( .r ` D ) ( G ` u ) ) ( +g ` D ) ( G ` v ) ) ) )
4847rexlimdvva 3038 . . 3 |- ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) -> ( E. r e. U E. s e. U E. z e. U ( u = ( r .+ ( ( G ` u ) .x. Z ) ) /\ v = ( s .+ ( ( G ` v ) .x. Z ) ) /\ ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) ) -> ( G ` ( ( l .x. u ) .+ v ) ) = ( ( l ( .r ` D ) ( G ` u ) ) ( +g ` D ) ( G ` v ) ) ) )
4931, 48syl5bir 233 . 2 |- ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) -> ( ( E. r e. U u = ( r .+ ( ( G ` u ) .x. Z ) ) /\ E. s e. U v = ( s .+ ( ( G ` v ) .x. Z ) ) /\ E. z e. U ( ( l .x. u ) .+ v ) = ( z .+ ( ( G ` ( ( l .x. u ) .+ v ) ) .x. Z ) ) ) -> ( G ` ( ( l .x. u ) .+ v ) ) = ( ( l ( .r ` D ) ( G ` u ) ) ( +g ` D ) ( G ` v ) ) ) )
5020, 22, 30, 49mp3and 1427 1 |- ( ( ph /\ ( l e. K /\ u e. V /\ v e. V ) ) -> ( G ` ( ( l .x. u ) .+ v ) ) = ( ( l ( .r ` D ) ( G ` u ) ) ( +g ` D ) ( G ` v ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   {csn 4177   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   LSSumclsm 18049   LModclmod 18863   LSpanclspn 18971   LVecclvec 19102  LSHypclsh 34262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-cntz 17750  df-lsm 18051  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-drng 18749  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lvec 19103  df-lshyp 34264
This theorem is referenced by:  lshpkrcl  34403
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