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Theorem lshpkrlem1 29908
Description: Lemma for lshpkrex 29916. 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 16178 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 16 . . . 4  |-  ( ph  ->  W  e.  LMod )
4 lshpkrlem.d . . . . 5  |-  D  =  (Scalar `  W )
54lmodfgrp 15959 . . . 4  |-  ( W  e.  LMod  ->  D  e. 
Grp )
6 lshpkrlem.k . . . . 5  |-  K  =  ( Base `  D
)
7 lshpkrlem.o . . . . 5  |-  .0.  =  ( 0g `  D )
86, 7grpidcl 14833 . . . 4  |-  ( D  e.  Grp  ->  .0.  e.  K )
93, 5, 83syl 19 . . 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 29907 . . 3  |-  ( ph  ->  E! k  e.  K  E. b  e.  U  X  =  ( b  .+  ( k  .x.  Z
) ) )
21 oveq1 6088 . . . . . . 7  |-  ( k  =  .0.  ->  (
k  .x.  Z )  =  (  .0.  .x.  Z
) )
2221oveq2d 6097 . . . . . 6  |-  ( k  =  .0.  ->  (
b  .+  ( k  .x.  Z ) )  =  ( b  .+  (  .0.  .x.  Z ) ) )
2322eqeq2d 2447 . . . . 5  |-  ( k  =  .0.  ->  ( X  =  ( b  .+  ( k  .x.  Z
) )  <->  X  =  ( b  .+  (  .0.  .x.  Z ) ) ) )
2423rexbidv 2726 . . . 4  |-  ( k  =  .0.  ->  ( E. b  e.  U  X  =  ( b  .+  ( k  .x.  Z
) )  <->  E. b  e.  U  X  =  ( b  .+  (  .0.  .x.  Z ) ) ) )
2524riota2 6572 . . 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 643 . 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 448 . . . . . 6  |-  ( (
ph  /\  X  e.  U )  ->  X  e.  U )
28 eqidd 2437 . . . . . 6  |-  ( (
ph  /\  X  e.  U )  ->  X  =  X )
29 eqeq2 2445 . . . . . . 7  |-  ( b  =  X  ->  ( X  =  b  <->  X  =  X ) )
3029rspcev 3052 . . . . . 6  |-  ( ( X  e.  U  /\  X  =  X )  ->  E. b  e.  U  X  =  b )
3127, 28, 30syl2anc 643 . . . . 5  |-  ( (
ph  /\  X  e.  U )  ->  E. b  e.  U  X  =  b )
3231ex 424 . . . 4  |-  ( ph  ->  ( X  e.  U  ->  E. b  e.  U  X  =  b )
)
33 eleq1a 2505 . . . . . 6  |-  ( b  e.  U  ->  ( X  =  b  ->  X  e.  U ) )
3433a1i 11 . . . . 5  |-  ( ph  ->  ( b  e.  U  ->  ( X  =  b  ->  X  e.  U
) ) )
3534rexlimdv 2829 . . . 4  |-  ( ph  ->  ( E. b  e.  U  X  =  b  ->  X  e.  U
) )
3632, 35impbid 184 . . 3  |-  ( ph  ->  ( X  e.  U  <->  E. b  e.  U  X  =  b ) )
37 eqid 2436 . . . . . . . . . . 11  |-  ( 0g
`  W )  =  ( 0g `  W
)
3810, 4, 19, 7, 37lmod0vs 15983 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  (  .0.  .x.  Z )  =  ( 0g `  W
) )
393, 16, 38syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  (  .0.  .x.  Z
)  =  ( 0g
`  W ) )
4039adantr 452 . . . . . . . 8  |-  ( (
ph  /\  b  e.  U )  ->  (  .0.  .x.  Z )  =  ( 0g `  W
) )
4140oveq2d 6097 . . . . . . 7  |-  ( (
ph  /\  b  e.  U )  ->  (
b  .+  (  .0.  .x. 
Z ) )  =  ( b  .+  ( 0g `  W ) ) )
421adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  U )  ->  W  e.  LVec )
4342, 2syl 16 . . . . . . . 8  |-  ( (
ph  /\  b  e.  U )  ->  W  e.  LMod )
44 eqid 2436 . . . . . . . . . 10  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
4544, 14, 3, 15lshplss 29779 . . . . . . . . 9  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
4610, 44lssel 16014 . . . . . . . . 9  |-  ( ( U  e.  ( LSubSp `  W )  /\  b  e.  U )  ->  b  e.  V )
4745, 46sylan 458 . . . . . . . 8  |-  ( (
ph  /\  b  e.  U )  ->  b  e.  V )
4810, 11, 37lmod0vrid 15981 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  b  e.  V )  ->  (
b  .+  ( 0g `  W ) )  =  b )
4943, 47, 48syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  b  e.  U )  ->  (
b  .+  ( 0g `  W ) )  =  b )
5041, 49eqtrd 2468 . . . . . 6  |-  ( (
ph  /\  b  e.  U )  ->  (
b  .+  (  .0.  .x. 
Z ) )  =  b )
5150eqeq2d 2447 . . . . 5  |-  ( (
ph  /\  b  e.  U )  ->  ( X  =  ( b  .+  (  .0.  .x.  Z
) )  <->  X  =  b ) )
5251bicomd 193 . . . 4  |-  ( (
ph  /\  b  e.  U )  ->  ( X  =  b  <->  X  =  ( b  .+  (  .0.  .x.  Z ) ) ) )
5352rexbidva 2722 . . 3  |-  ( ph  ->  ( E. b  e.  U  X  =  b  <->  E. b  e.  U  X  =  ( b  .+  (  .0.  .x.  Z
) ) ) )
5436, 53bitrd 245 . 2  |-  ( ph  ->  ( X  e.  U  <->  E. b  e.  U  X  =  ( b  .+  (  .0.  .x.  Z )
) ) )
55 eqeq1 2442 . . . . . . . 8  |-  ( x  =  X  ->  (
x  =  ( y 
.+  ( k  .x.  Z ) )  <->  X  =  ( y  .+  (
k  .x.  Z )
) ) )
5655rexbidv 2726 . . . . . . 7  |-  ( x  =  X  ->  ( E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z
) )  <->  E. y  e.  U  X  =  ( y  .+  (
k  .x.  Z )
) ) )
5756riotabidv 6551 . . . . . 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 6553 . . . . . 6  |-  ( iota_ k  e.  K E. y  e.  U  X  =  ( y  .+  (
k  .x.  Z )
) )  e.  _V
6057, 58, 59fvmpt 5806 . . . . 5  |-  ( X  e.  V  ->  ( G `  X )  =  ( iota_ k  e.  K E. y  e.  U  X  =  ( y  .+  ( k 
.x.  Z ) ) ) )
61 oveq1 6088 . . . . . . . . 9  |-  ( y  =  b  ->  (
y  .+  ( k  .x.  Z ) )  =  ( b  .+  (
k  .x.  Z )
) )
6261eqeq2d 2447 . . . . . . . 8  |-  ( y  =  b  ->  ( X  =  ( y  .+  ( k  .x.  Z
) )  <->  X  =  ( b  .+  (
k  .x.  Z )
) ) )
6362cbvrexv 2933 . . . . . . 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 6567 . . . . 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 2484 . . . 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 2444 . 2  |-  ( ph  ->  ( ( G `  X )  =  .0.  <->  (
iota_ k  e.  K E. b  e.  U  X  =  ( b  .+  ( k  .x.  Z
) ) )  =  .0.  ) )
6926, 54, 683bitr4d 277 1  |-  ( ph  ->  ( X  e.  U  <->  ( G `  X )  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706   E!wreu 2707   {csn 3814    e. cmpt 4266   ` cfv 5454  (class class class)co 6081   iota_crio 6542   Basecbs 13469   +g cplusg 13529  Scalarcsca 13532   .scvsca 13533   0gc0g 13723   Grpcgrp 14685   LSSumclsm 15268   LModclmod 15950   LSubSpclss 16008   LSpanclspn 16047   LVecclvec 16174  LSHypclsh 29773
This theorem is referenced by:  lshpkr  29915
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-0g 13727  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-cntz 15116  df-lsm 15270  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-drng 15837  df-lmod 15952  df-lss 16009  df-lsp 16048  df-lvec 16175  df-lshyp 29775
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