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Theorem lshpkrlem1 35283
Description: Lemma for lshpkrex 35291. 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 17884 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 16 . . . 4  |-  ( ph  ->  W  e.  LMod )
4 lshpkrlem.d . . . . 5  |-  D  =  (Scalar `  W )
54lmodfgrp 17653 . . . 4  |-  ( W  e.  LMod  ->  D  e. 
Grp )
6 lshpkrlem.k . . . . 5  |-  K  =  ( Base `  D
)
7 lshpkrlem.o . . . . 5  |-  .0.  =  ( 0g `  D )
86, 7grpidcl 16214 . . . 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 35282 . . 3  |-  ( ph  ->  E! k  e.  K  E. b  e.  U  X  =  ( b  .+  ( k  .x.  Z
) ) )
21 oveq1 6221 . . . . . . 7  |-  ( k  =  .0.  ->  (
k  .x.  Z )  =  (  .0.  .x.  Z
) )
2221oveq2d 6230 . . . . . 6  |-  ( k  =  .0.  ->  (
b  .+  ( k  .x.  Z ) )  =  ( b  .+  (  .0.  .x.  Z ) ) )
2322eqeq2d 2406 . . . . 5  |-  ( k  =  .0.  ->  ( X  =  ( b  .+  ( k  .x.  Z
) )  <->  X  =  ( b  .+  (  .0.  .x.  Z ) ) ) )
2423rexbidv 2906 . . . 4  |-  ( k  =  .0.  ->  ( E. b  e.  U  X  =  ( b  .+  ( k  .x.  Z
) )  <->  E. b  e.  U  X  =  ( b  .+  (  .0.  .x.  Z ) ) ) )
2524riota2 6198 . . 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 659 . 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 459 . . . . . 6  |-  ( (
ph  /\  X  e.  U )  ->  X  e.  U )
28 eqidd 2393 . . . . . 6  |-  ( (
ph  /\  X  e.  U )  ->  X  =  X )
29 eqeq2 2407 . . . . . . 7  |-  ( b  =  X  ->  ( X  =  b  <->  X  =  X ) )
3029rspcev 3148 . . . . . 6  |-  ( ( X  e.  U  /\  X  =  X )  ->  E. b  e.  U  X  =  b )
3127, 28, 30syl2anc 659 . . . . 5  |-  ( (
ph  /\  X  e.  U )  ->  E. b  e.  U  X  =  b )
3231ex 432 . . . 4  |-  ( ph  ->  ( X  e.  U  ->  E. b  e.  U  X  =  b )
)
33 eleq1a 2475 . . . . . 6  |-  ( b  e.  U  ->  ( X  =  b  ->  X  e.  U ) )
3433a1i 11 . . . . 5  |-  ( ph  ->  ( b  e.  U  ->  ( X  =  b  ->  X  e.  U
) ) )
3534rexlimdv 2882 . . . 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 2392 . . . . . . . . . . 11  |-  ( 0g
`  W )  =  ( 0g `  W
)
3810, 4, 19, 7, 37lmod0vs 17677 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  (  .0.  .x.  Z )  =  ( 0g `  W
) )
393, 16, 38syl2anc 659 . . . . . . . . 9  |-  ( ph  ->  (  .0.  .x.  Z
)  =  ( 0g
`  W ) )
4039adantr 463 . . . . . . . 8  |-  ( (
ph  /\  b  e.  U )  ->  (  .0.  .x.  Z )  =  ( 0g `  W
) )
4140oveq2d 6230 . . . . . . 7  |-  ( (
ph  /\  b  e.  U )  ->  (
b  .+  (  .0.  .x. 
Z ) )  =  ( b  .+  ( 0g `  W ) ) )
421adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  U )  ->  W  e.  LVec )
4342, 2syl 16 . . . . . . . 8  |-  ( (
ph  /\  b  e.  U )  ->  W  e.  LMod )
44 eqid 2392 . . . . . . . . . 10  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
4544, 14, 3, 15lshplss 35154 . . . . . . . . 9  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
4610, 44lssel 17716 . . . . . . . . 9  |-  ( ( U  e.  ( LSubSp `  W )  /\  b  e.  U )  ->  b  e.  V )
4745, 46sylan 469 . . . . . . . 8  |-  ( (
ph  /\  b  e.  U )  ->  b  e.  V )
4810, 11, 37lmod0vrid 17675 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  b  e.  V )  ->  (
b  .+  ( 0g `  W ) )  =  b )
4943, 47, 48syl2anc 659 . . . . . . 7  |-  ( (
ph  /\  b  e.  U )  ->  (
b  .+  ( 0g `  W ) )  =  b )
5041, 49eqtrd 2433 . . . . . 6  |-  ( (
ph  /\  b  e.  U )  ->  (
b  .+  (  .0.  .x. 
Z ) )  =  b )
5150eqeq2d 2406 . . . . 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 2903 . . 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 2396 . . . . . . . 8  |-  ( x  =  X  ->  (
x  =  ( y 
.+  ( k  .x.  Z ) )  <->  X  =  ( y  .+  (
k  .x.  Z )
) ) )
5655rexbidv 2906 . . . . . . 7  |-  ( x  =  X  ->  ( E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z
) )  <->  E. y  e.  U  X  =  ( y  .+  (
k  .x.  Z )
) ) )
5756riotabidv 6178 . . . . . 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 6180 . . . . . 6  |-  ( iota_ k  e.  K  E. y  e.  U  X  =  ( y  .+  (
k  .x.  Z )
) )  e.  _V
6057, 58, 59fvmpt 5870 . . . . 5  |-  ( X  e.  V  ->  ( G `  X )  =  ( iota_ k  e.  K  E. y  e.  U  X  =  ( y  .+  ( k 
.x.  Z ) ) ) )
61 oveq1 6221 . . . . . . . . 9  |-  ( y  =  b  ->  (
y  .+  ( k  .x.  Z ) )  =  ( b  .+  (
k  .x.  Z )
) )
6261eqeq2d 2406 . . . . . . . 8  |-  ( y  =  b  ->  ( X  =  ( y  .+  ( k  .x.  Z
) )  <->  X  =  ( b  .+  (
k  .x.  Z )
) ) )
6362cbvrexv 3023 . . . . . . 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 6193 . . . . 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 2449 . . . 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 2394 . 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 367    = wceq 1399    e. wcel 1836   E.wrex 2743   E!wreu 2744   {csn 3957    |-> cmpt 4438   ` cfv 5509   iota_crio 6175  (class class class)co 6214   Basecbs 14653   +g cplusg 14721  Scalarcsca 14724   .scvsca 14725   0gc0g 14866   Grpcgrp 16189   LSSumclsm 16790   LModclmod 17644   LSubSpclss 17710   LSpanclspn 17749   LVecclvec 17880  LSHypclsh 35148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-rep 4491  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509  ax-cnex 9477  ax-resscn 9478  ax-1cn 9479  ax-icn 9480  ax-addcl 9481  ax-addrcl 9482  ax-mulcl 9483  ax-mulrcl 9484  ax-mulcom 9485  ax-addass 9486  ax-mulass 9487  ax-distr 9488  ax-i2m1 9489  ax-1ne0 9490  ax-1rid 9491  ax-rnegex 9492  ax-rrecex 9493  ax-cnre 9494  ax-pre-lttri 9495  ax-pre-lttrn 9496  ax-pre-ltadd 9497  ax-pre-mulgt0 9498
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-nel 2590  df-ral 2747  df-rex 2748  df-reu 2749  df-rmo 2750  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-pss 3418  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-tp 3962  df-op 3964  df-uni 4177  df-int 4213  df-iun 4258  df-br 4381  df-opab 4439  df-mpt 4440  df-tr 4474  df-eprel 4718  df-id 4722  df-po 4727  df-so 4728  df-fr 4765  df-we 4767  df-ord 4808  df-on 4809  df-lim 4810  df-suc 4811  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-f1 5514  df-fo 5515  df-f1o 5516  df-fv 5517  df-riota 6176  df-ov 6217  df-oprab 6218  df-mpt2 6219  df-om 6618  df-1st 6717  df-2nd 6718  df-tpos 6891  df-recs 6978  df-rdg 7012  df-er 7247  df-en 7454  df-dom 7455  df-sdom 7456  df-pnf 9559  df-mnf 9560  df-xr 9561  df-ltxr 9562  df-le 9563  df-sub 9738  df-neg 9739  df-nn 10471  df-2 10529  df-3 10530  df-ndx 14656  df-slot 14657  df-base 14658  df-sets 14659  df-ress 14660  df-plusg 14734  df-mulr 14735  df-0g 14868  df-mgm 16008  df-sgrp 16047  df-mnd 16057  df-submnd 16103  df-grp 16193  df-minusg 16194  df-sbg 16195  df-subg 16334  df-cntz 16491  df-lsm 16792  df-cmn 16936  df-abl 16937  df-mgp 17274  df-ur 17286  df-ring 17332  df-oppr 17404  df-dvdsr 17422  df-unit 17423  df-invr 17453  df-drng 17530  df-lmod 17646  df-lss 17711  df-lsp 17750  df-lvec 17881  df-lshyp 35150
This theorem is referenced by:  lshpkr  35290
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