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Theorem lshpkrlem1 34124
Description: Lemma for lshpkrex 34132. The value of tentative functional  G is zero iff its argument belongs to hyperplane  U. (Contributed by NM, 14-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
lshpkrlem1  |-  ( ph  ->  ( X  e.  U  <->  ( G `  X )  =  .0.  ) )
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
Allowed substitution hints:    ph( x, y, k)    D( x, y, k)    .(+) (
x, y, k)    G( x, y, k)    H( x, y, k)    K( y)    N( x, y, k)    V( y, k)    W( x, y, k)    .0. ( x, y)

Proof of Theorem lshpkrlem1
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 lshpkrlem.w . . . . 5  |-  ( ph  ->  W  e.  LVec )
2 lveclmod 17564 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 16 . . . 4  |-  ( ph  ->  W  e.  LMod )
4 lshpkrlem.d . . . . 5  |-  D  =  (Scalar `  W )
54lmodfgrp 17333 . . . 4  |-  ( W  e.  LMod  ->  D  e. 
Grp )
6 lshpkrlem.k . . . . 5  |-  K  =  ( Base `  D
)
7 lshpkrlem.o . . . . 5  |-  .0.  =  ( 0g `  D )
86, 7grpidcl 15892 . . . 4  |-  ( D  e.  Grp  ->  .0.  e.  K )
93, 5, 83syl 20 . . 3  |-  ( ph  ->  .0.  e.  K )
10 lshpkrlem.v . . . 4  |-  V  =  ( Base `  W
)
11 lshpkrlem.a . . . 4  |-  .+  =  ( +g  `  W )
12 lshpkrlem.n . . . 4  |-  N  =  ( LSpan `  W )
13 lshpkrlem.p . . . 4  |-  .(+)  =  (
LSSum `  W )
14 lshpkrlem.h . . . 4  |-  H  =  (LSHyp `  W )
15 lshpkrlem.u . . . 4  |-  ( ph  ->  U  e.  H )
16 lshpkrlem.z . . . 4  |-  ( ph  ->  Z  e.  V )
17 lshpkrlem.x . . . 4  |-  ( ph  ->  X  e.  V )
18 lshpkrlem.e . . . 4  |-  ( ph  ->  ( U  .(+)  ( N `
 { Z }
) )  =  V )
19 lshpkrlem.t . . . 4  |-  .x.  =  ( .s `  W )
2010, 11, 12, 13, 14, 1, 15, 16, 17, 18, 4, 6, 19lshpsmreu 34123 . . 3  |-  ( ph  ->  E! k  e.  K  E. b  e.  U  X  =  ( b  .+  ( k  .x.  Z
) ) )
21 oveq1 6292 . . . . . . 7  |-  ( k  =  .0.  ->  (
k  .x.  Z )  =  (  .0.  .x.  Z
) )
2221oveq2d 6301 . . . . . 6  |-  ( k  =  .0.  ->  (
b  .+  ( k  .x.  Z ) )  =  ( b  .+  (  .0.  .x.  Z ) ) )
2322eqeq2d 2481 . . . . 5  |-  ( k  =  .0.  ->  ( X  =  ( b  .+  ( k  .x.  Z
) )  <->  X  =  ( b  .+  (  .0.  .x.  Z ) ) ) )
2423rexbidv 2973 . . . 4  |-  ( k  =  .0.  ->  ( E. b  e.  U  X  =  ( b  .+  ( k  .x.  Z
) )  <->  E. b  e.  U  X  =  ( b  .+  (  .0.  .x.  Z ) ) ) )
2524riota2 6269 . . 3  |-  ( (  .0.  e.  K  /\  E! k  e.  K  E. b  e.  U  X  =  ( b  .+  ( k  .x.  Z
) ) )  -> 
( E. b  e.  U  X  =  ( b  .+  (  .0. 
.x.  Z ) )  <-> 
( iota_ k  e.  K  E. b  e.  U  X  =  ( b  .+  ( k  .x.  Z
) ) )  =  .0.  ) )
269, 20, 25syl2anc 661 . 2  |-  ( ph  ->  ( E. b  e.  U  X  =  ( b  .+  (  .0. 
.x.  Z ) )  <-> 
( iota_ k  e.  K  E. b  e.  U  X  =  ( b  .+  ( k  .x.  Z
) ) )  =  .0.  ) )
27 simpr 461 . . . . . 6  |-  ( (
ph  /\  X  e.  U )  ->  X  e.  U )
28 eqidd 2468 . . . . . 6  |-  ( (
ph  /\  X  e.  U )  ->  X  =  X )
29 eqeq2 2482 . . . . . . 7  |-  ( b  =  X  ->  ( X  =  b  <->  X  =  X ) )
3029rspcev 3214 . . . . . 6  |-  ( ( X  e.  U  /\  X  =  X )  ->  E. b  e.  U  X  =  b )
3127, 28, 30syl2anc 661 . . . . 5  |-  ( (
ph  /\  X  e.  U )  ->  E. b  e.  U  X  =  b )
3231ex 434 . . . 4  |-  ( ph  ->  ( X  e.  U  ->  E. b  e.  U  X  =  b )
)
33 eleq1a 2550 . . . . . 6  |-  ( b  e.  U  ->  ( X  =  b  ->  X  e.  U ) )
3433a1i 11 . . . . 5  |-  ( ph  ->  ( b  e.  U  ->  ( X  =  b  ->  X  e.  U
) ) )
3534rexlimdv 2953 . . . 4  |-  ( ph  ->  ( E. b  e.  U  X  =  b  ->  X  e.  U
) )
3632, 35impbid 191 . . 3  |-  ( ph  ->  ( X  e.  U  <->  E. b  e.  U  X  =  b ) )
37 eqid 2467 . . . . . . . . . . 11  |-  ( 0g
`  W )  =  ( 0g `  W
)
3810, 4, 19, 7, 37lmod0vs 17357 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  (  .0.  .x.  Z )  =  ( 0g `  W
) )
393, 16, 38syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  (  .0.  .x.  Z
)  =  ( 0g
`  W ) )
4039adantr 465 . . . . . . . 8  |-  ( (
ph  /\  b  e.  U )  ->  (  .0.  .x.  Z )  =  ( 0g `  W
) )
4140oveq2d 6301 . . . . . . 7  |-  ( (
ph  /\  b  e.  U )  ->  (
b  .+  (  .0.  .x. 
Z ) )  =  ( b  .+  ( 0g `  W ) ) )
421adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  U )  ->  W  e.  LVec )
4342, 2syl 16 . . . . . . . 8  |-  ( (
ph  /\  b  e.  U )  ->  W  e.  LMod )
44 eqid 2467 . . . . . . . . . 10  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
4544, 14, 3, 15lshplss 33995 . . . . . . . . 9  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
4610, 44lssel 17396 . . . . . . . . 9  |-  ( ( U  e.  ( LSubSp `  W )  /\  b  e.  U )  ->  b  e.  V )
4745, 46sylan 471 . . . . . . . 8  |-  ( (
ph  /\  b  e.  U )  ->  b  e.  V )
4810, 11, 37lmod0vrid 17355 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  b  e.  V )  ->  (
b  .+  ( 0g `  W ) )  =  b )
4943, 47, 48syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  b  e.  U )  ->  (
b  .+  ( 0g `  W ) )  =  b )
5041, 49eqtrd 2508 . . . . . 6  |-  ( (
ph  /\  b  e.  U )  ->  (
b  .+  (  .0.  .x. 
Z ) )  =  b )
5150eqeq2d 2481 . . . . 5  |-  ( (
ph  /\  b  e.  U )  ->  ( X  =  ( b  .+  (  .0.  .x.  Z
) )  <->  X  =  b ) )
5251bicomd 201 . . . 4  |-  ( (
ph  /\  b  e.  U )  ->  ( X  =  b  <->  X  =  ( b  .+  (  .0.  .x.  Z ) ) ) )
5352rexbidva 2970 . . 3  |-  ( ph  ->  ( E. b  e.  U  X  =  b  <->  E. b  e.  U  X  =  ( b  .+  (  .0.  .x.  Z
) ) ) )
5436, 53bitrd 253 . 2  |-  ( ph  ->  ( X  e.  U  <->  E. b  e.  U  X  =  ( b  .+  (  .0.  .x.  Z )
) ) )
55 eqeq1 2471 . . . . . . . 8  |-  ( x  =  X  ->  (
x  =  ( y 
.+  ( k  .x.  Z ) )  <->  X  =  ( y  .+  (
k  .x.  Z )
) ) )
5655rexbidv 2973 . . . . . . 7  |-  ( x  =  X  ->  ( E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z
) )  <->  E. y  e.  U  X  =  ( y  .+  (
k  .x.  Z )
) ) )
5756riotabidv 6248 . . . . . 6  |-  ( x  =  X  ->  ( iota_ k  e.  K  E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z
) ) )  =  ( iota_ k  e.  K  E. y  e.  U  X  =  ( y  .+  ( k  .x.  Z
) ) ) )
58 lshpkrlem.g . . . . . 6  |-  G  =  ( x  e.  V  |->  ( iota_ k  e.  K  E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z
) ) ) )
59 riotaex 6250 . . . . . 6  |-  ( iota_ k  e.  K  E. y  e.  U  X  =  ( y  .+  (
k  .x.  Z )
) )  e.  _V
6057, 58, 59fvmpt 5951 . . . . 5  |-  ( X  e.  V  ->  ( G `  X )  =  ( iota_ k  e.  K  E. y  e.  U  X  =  ( y  .+  ( k 
.x.  Z ) ) ) )
61 oveq1 6292 . . . . . . . . 9  |-  ( y  =  b  ->  (
y  .+  ( k  .x.  Z ) )  =  ( b  .+  (
k  .x.  Z )
) )
6261eqeq2d 2481 . . . . . . . 8  |-  ( y  =  b  ->  ( X  =  ( y  .+  ( k  .x.  Z
) )  <->  X  =  ( b  .+  (
k  .x.  Z )
) ) )
6362cbvrexv 3089 . . . . . . 7  |-  ( E. y  e.  U  X  =  ( y  .+  ( k  .x.  Z
) )  <->  E. b  e.  U  X  =  ( b  .+  (
k  .x.  Z )
) )
6463a1i 11 . . . . . 6  |-  ( k  e.  K  ->  ( E. y  e.  U  X  =  ( y  .+  ( k  .x.  Z
) )  <->  E. b  e.  U  X  =  ( b  .+  (
k  .x.  Z )
) ) )
6564riotabiia 6264 . . . . 5  |-  ( iota_ k  e.  K  E. y  e.  U  X  =  ( y  .+  (
k  .x.  Z )
) )  =  (
iota_ k  e.  K  E. b  e.  U  X  =  ( b  .+  ( k  .x.  Z
) ) )
6660, 65syl6eq 2524 . . . 4  |-  ( X  e.  V  ->  ( G `  X )  =  ( iota_ k  e.  K  E. b  e.  U  X  =  ( b  .+  ( k 
.x.  Z ) ) ) )
6717, 66syl 16 . . 3  |-  ( ph  ->  ( G `  X
)  =  ( iota_ k  e.  K  E. b  e.  U  X  =  ( b  .+  (
k  .x.  Z )
) ) )
6867eqeq1d 2469 . 2  |-  ( ph  ->  ( ( G `  X )  =  .0.  <->  (
iota_ k  e.  K  E. b  e.  U  X  =  ( b  .+  ( k  .x.  Z
) ) )  =  .0.  ) )
6926, 54, 683bitr4d 285 1  |-  ( ph  ->  ( X  e.  U  <->  ( G `  X )  =  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   E!wreu 2816   {csn 4027    |-> cmpt 4505   ` cfv 5588   iota_crio 6245  (class class class)co 6285   Basecbs 14493   +g cplusg 14558  Scalarcsca 14561   .scvsca 14562   0gc0g 14698   Grpcgrp 15730   LSSumclsm 16469   LModclmod 17324   LSubSpclss 17390   LSpanclspn 17429   LVecclvec 17560  LSHypclsh 33989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6956  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-0g 14700  df-mnd 15735  df-submnd 15790  df-grp 15871  df-minusg 15872  df-sbg 15873  df-subg 16012  df-cntz 16169  df-lsm 16471  df-cmn 16615  df-abl 16616  df-mgp 16956  df-ur 16968  df-rng 17014  df-oppr 17085  df-dvdsr 17103  df-unit 17104  df-invr 17134  df-drng 17210  df-lmod 17326  df-lss 17391  df-lsp 17430  df-lvec 17561  df-lshyp 33991
This theorem is referenced by:  lshpkr  34131
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