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Theorem lshpkrlem1 32767
Description: Lemma for lshpkrex 32775. 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 17199 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 16 . . . 4  |-  ( ph  ->  W  e.  LMod )
4 lshpkrlem.d . . . . 5  |-  D  =  (Scalar `  W )
54lmodfgrp 16969 . . . 4  |-  ( W  e.  LMod  ->  D  e. 
Grp )
6 lshpkrlem.k . . . . 5  |-  K  =  ( Base `  D
)
7 lshpkrlem.o . . . . 5  |-  .0.  =  ( 0g `  D )
86, 7grpidcl 15578 . . . 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 32766 . . 3  |-  ( ph  ->  E! k  e.  K  E. b  e.  U  X  =  ( b  .+  ( k  .x.  Z
) ) )
21 oveq1 6110 . . . . . . 7  |-  ( k  =  .0.  ->  (
k  .x.  Z )  =  (  .0.  .x.  Z
) )
2221oveq2d 6119 . . . . . 6  |-  ( k  =  .0.  ->  (
b  .+  ( k  .x.  Z ) )  =  ( b  .+  (  .0.  .x.  Z ) ) )
2322eqeq2d 2454 . . . . 5  |-  ( k  =  .0.  ->  ( X  =  ( b  .+  ( k  .x.  Z
) )  <->  X  =  ( b  .+  (  .0.  .x.  Z ) ) ) )
2423rexbidv 2748 . . . 4  |-  ( k  =  .0.  ->  ( E. b  e.  U  X  =  ( b  .+  ( k  .x.  Z
) )  <->  E. b  e.  U  X  =  ( b  .+  (  .0.  .x.  Z ) ) ) )
2524riota2 6087 . . 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 2444 . . . . . 6  |-  ( (
ph  /\  X  e.  U )  ->  X  =  X )
29 eqeq2 2452 . . . . . . 7  |-  ( b  =  X  ->  ( X  =  b  <->  X  =  X ) )
3029rspcev 3085 . . . . . 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 2512 . . . . . 6  |-  ( b  e.  U  ->  ( X  =  b  ->  X  e.  U ) )
3433a1i 11 . . . . 5  |-  ( ph  ->  ( b  e.  U  ->  ( X  =  b  ->  X  e.  U
) ) )
3534rexlimdv 2852 . . . 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 2443 . . . . . . . . . . 11  |-  ( 0g
`  W )  =  ( 0g `  W
)
3810, 4, 19, 7, 37lmod0vs 16993 . . . . . . . . . 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 6119 . . . . . . 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 2443 . . . . . . . . . 10  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
4544, 14, 3, 15lshplss 32638 . . . . . . . . 9  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
4610, 44lssel 17031 . . . . . . . . 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 16991 . . . . . . . 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 2475 . . . . . 6  |-  ( (
ph  /\  b  e.  U )  ->  (
b  .+  (  .0.  .x. 
Z ) )  =  b )
5150eqeq2d 2454 . . . . 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 2744 . . 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 2449 . . . . . . . 8  |-  ( x  =  X  ->  (
x  =  ( y 
.+  ( k  .x.  Z ) )  <->  X  =  ( y  .+  (
k  .x.  Z )
) ) )
5655rexbidv 2748 . . . . . . 7  |-  ( x  =  X  ->  ( E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z
) )  <->  E. y  e.  U  X  =  ( y  .+  (
k  .x.  Z )
) ) )
5756riotabidv 6066 . . . . . 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 6068 . . . . . 6  |-  ( iota_ k  e.  K  E. y  e.  U  X  =  ( y  .+  (
k  .x.  Z )
) )  e.  _V
6057, 58, 59fvmpt 5786 . . . . 5  |-  ( X  e.  V  ->  ( G `  X )  =  ( iota_ k  e.  K  E. y  e.  U  X  =  ( y  .+  ( k 
.x.  Z ) ) ) )
61 oveq1 6110 . . . . . . . . 9  |-  ( y  =  b  ->  (
y  .+  ( k  .x.  Z ) )  =  ( b  .+  (
k  .x.  Z )
) )
6261eqeq2d 2454 . . . . . . . 8  |-  ( y  =  b  ->  ( X  =  ( y  .+  ( k  .x.  Z
) )  <->  X  =  ( b  .+  (
k  .x.  Z )
) ) )
6362cbvrexv 2960 . . . . . . 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 6082 . . . . 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 2491 . . . 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 2451 . 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 1369    e. wcel 1756   E.wrex 2728   E!wreu 2729   {csn 3889    e. cmpt 4362   ` cfv 5430   iota_crio 6063  (class class class)co 6103   Basecbs 14186   +g cplusg 14250  Scalarcsca 14253   .scvsca 14254   0gc0g 14390   Grpcgrp 15422   LSSumclsm 16145   LModclmod 16960   LSubSpclss 17025   LSpanclspn 17064   LVecclvec 17195  LSHypclsh 32632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-tpos 6757  df-recs 6844  df-rdg 6878  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-3 10393  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-0g 14392  df-mnd 15427  df-submnd 15477  df-grp 15557  df-minusg 15558  df-sbg 15559  df-subg 15690  df-cntz 15847  df-lsm 16147  df-cmn 16291  df-abl 16292  df-mgp 16604  df-ur 16616  df-rng 16659  df-oppr 16727  df-dvdsr 16745  df-unit 16746  df-invr 16776  df-drng 16846  df-lmod 16962  df-lss 17026  df-lsp 17065  df-lvec 17196  df-lshyp 32634
This theorem is referenced by:  lshpkr  32774
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