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Theorem lshpkrlem1 34537
Description: Lemma for lshpkrex 34545. 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 17620 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 16 . . . 4  |-  ( ph  ->  W  e.  LMod )
4 lshpkrlem.d . . . . 5  |-  D  =  (Scalar `  W )
54lmodfgrp 17389 . . . 4  |-  ( W  e.  LMod  ->  D  e. 
Grp )
6 lshpkrlem.k . . . . 5  |-  K  =  ( Base `  D
)
7 lshpkrlem.o . . . . 5  |-  .0.  =  ( 0g `  D )
86, 7grpidcl 15947 . . . 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 34536 . . 3  |-  ( ph  ->  E! k  e.  K  E. b  e.  U  X  =  ( b  .+  ( k  .x.  Z
) ) )
21 oveq1 6284 . . . . . . 7  |-  ( k  =  .0.  ->  (
k  .x.  Z )  =  (  .0.  .x.  Z
) )
2221oveq2d 6293 . . . . . 6  |-  ( k  =  .0.  ->  (
b  .+  ( k  .x.  Z ) )  =  ( b  .+  (  .0.  .x.  Z ) ) )
2322eqeq2d 2455 . . . . 5  |-  ( k  =  .0.  ->  ( X  =  ( b  .+  ( k  .x.  Z
) )  <->  X  =  ( b  .+  (  .0.  .x.  Z ) ) ) )
2423rexbidv 2952 . . . 4  |-  ( k  =  .0.  ->  ( E. b  e.  U  X  =  ( b  .+  ( k  .x.  Z
) )  <->  E. b  e.  U  X  =  ( b  .+  (  .0.  .x.  Z ) ) ) )
2524riota2 6261 . . 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 2442 . . . . . 6  |-  ( (
ph  /\  X  e.  U )  ->  X  =  X )
29 eqeq2 2456 . . . . . . 7  |-  ( b  =  X  ->  ( X  =  b  <->  X  =  X ) )
3029rspcev 3194 . . . . . 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 2524 . . . . . 6  |-  ( b  e.  U  ->  ( X  =  b  ->  X  e.  U ) )
3433a1i 11 . . . . 5  |-  ( ph  ->  ( b  e.  U  ->  ( X  =  b  ->  X  e.  U
) ) )
3534rexlimdv 2931 . . . 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 2441 . . . . . . . . . . 11  |-  ( 0g
`  W )  =  ( 0g `  W
)
3810, 4, 19, 7, 37lmod0vs 17413 . . . . . . . . . 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 6293 . . . . . . 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 2441 . . . . . . . . . 10  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
4544, 14, 3, 15lshplss 34408 . . . . . . . . 9  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
4610, 44lssel 17452 . . . . . . . . 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 17411 . . . . . . . 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 2482 . . . . . 6  |-  ( (
ph  /\  b  e.  U )  ->  (
b  .+  (  .0.  .x. 
Z ) )  =  b )
5150eqeq2d 2455 . . . . 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 2949 . . 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 2445 . . . . . . . 8  |-  ( x  =  X  ->  (
x  =  ( y 
.+  ( k  .x.  Z ) )  <->  X  =  ( y  .+  (
k  .x.  Z )
) ) )
5655rexbidv 2952 . . . . . . 7  |-  ( x  =  X  ->  ( E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z
) )  <->  E. y  e.  U  X  =  ( y  .+  (
k  .x.  Z )
) ) )
5756riotabidv 6240 . . . . . 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 6242 . . . . . 6  |-  ( iota_ k  e.  K  E. y  e.  U  X  =  ( y  .+  (
k  .x.  Z )
) )  e.  _V
6057, 58, 59fvmpt 5937 . . . . 5  |-  ( X  e.  V  ->  ( G `  X )  =  ( iota_ k  e.  K  E. y  e.  U  X  =  ( y  .+  ( k 
.x.  Z ) ) ) )
61 oveq1 6284 . . . . . . . . 9  |-  ( y  =  b  ->  (
y  .+  ( k  .x.  Z ) )  =  ( b  .+  (
k  .x.  Z )
) )
6261eqeq2d 2455 . . . . . . . 8  |-  ( y  =  b  ->  ( X  =  ( y  .+  ( k  .x.  Z
) )  <->  X  =  ( b  .+  (
k  .x.  Z )
) ) )
6362cbvrexv 3069 . . . . . . 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 6256 . . . . 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 2498 . . . 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 2443 . 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 1381    e. wcel 1802   E.wrex 2792   E!wreu 2793   {csn 4010    |-> cmpt 4491   ` cfv 5574   iota_crio 6237  (class class class)co 6277   Basecbs 14504   +g cplusg 14569  Scalarcsca 14572   .scvsca 14573   0gc0g 14709   Grpcgrp 15922   LSSumclsm 16523   LModclmod 17380   LSubSpclss 17446   LSpanclspn 17485   LVecclvec 17616  LSHypclsh 34402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-tpos 6953  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-0g 14711  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-submnd 15836  df-grp 15926  df-minusg 15927  df-sbg 15928  df-subg 16067  df-cntz 16224  df-lsm 16525  df-cmn 16669  df-abl 16670  df-mgp 17010  df-ur 17022  df-ring 17068  df-oppr 17140  df-dvdsr 17158  df-unit 17159  df-invr 17189  df-drng 17266  df-lmod 17382  df-lss 17447  df-lsp 17486  df-lvec 17617  df-lshyp 34404
This theorem is referenced by:  lshpkr  34544
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