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Theorem lshpkrlem3 35234
Description: Lemma for lshpkrex 35240. Defining property of  G `  X. (Contributed by NM, 15-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
lshpkrlem3  |-  ( ph  ->  E. z  e.  U  X  =  ( z  .+  ( ( G `  X )  .x.  Z
) ) )
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   
z,  .+    z, G    z, U    z, X    z, Z, k, x, y    z,  .x.
Allowed substitution hints:    ph( x, y, z, k)    D( x, y, z, k)    .(+) ( x, y, z, k)    G( x, y, k)    H( x, y, z, k)    K( y, z)    N( x, y, z, k)    V( y, z, k)    W( x, y, z, k)    .0. ( x, y, z)

Proof of Theorem lshpkrlem3
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 lshpkrlem.v . . . . 5  |-  V  =  ( Base `  W
)
2 lshpkrlem.a . . . . 5  |-  .+  =  ( +g  `  W )
3 lshpkrlem.n . . . . 5  |-  N  =  ( LSpan `  W )
4 lshpkrlem.p . . . . 5  |-  .(+)  =  (
LSSum `  W )
5 lshpkrlem.h . . . . 5  |-  H  =  (LSHyp `  W )
6 lshpkrlem.w . . . . 5  |-  ( ph  ->  W  e.  LVec )
7 lshpkrlem.u . . . . 5  |-  ( ph  ->  U  e.  H )
8 lshpkrlem.z . . . . 5  |-  ( ph  ->  Z  e.  V )
9 lshpkrlem.x . . . . 5  |-  ( ph  ->  X  e.  V )
10 lshpkrlem.e . . . . 5  |-  ( ph  ->  ( U  .(+)  ( N `
 { Z }
) )  =  V )
11 lshpkrlem.d . . . . 5  |-  D  =  (Scalar `  W )
12 lshpkrlem.k . . . . 5  |-  K  =  ( Base `  D
)
13 lshpkrlem.t . . . . 5  |-  .x.  =  ( .s `  W )
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13lshpsmreu 35231 . . . 4  |-  ( ph  ->  E! l  e.  K  E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) ) )
15 riotasbc 6247 . . . 4  |-  ( E! l  e.  K  E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) )  ->  [. ( iota_ l  e.  K  E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) ) )  / 
l ]. E. z  e.  U  X  =  ( z  .+  ( l 
.x.  Z ) ) )
1614, 15syl 16 . . 3  |-  ( ph  ->  [. ( iota_ l  e.  K  E. z  e.  U  X  =  ( z  .+  ( l 
.x.  Z ) ) )  /  l ]. E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) ) )
17 eqeq1 2458 . . . . . . 7  |-  ( x  =  X  ->  (
x  =  ( z 
.+  ( l  .x.  Z ) )  <->  X  =  ( z  .+  (
l  .x.  Z )
) ) )
1817rexbidv 2965 . . . . . 6  |-  ( x  =  X  ->  ( E. z  e.  U  x  =  ( z  .+  ( l  .x.  Z
) )  <->  E. z  e.  U  X  =  ( z  .+  (
l  .x.  Z )
) ) )
1918riotabidv 6234 . . . . 5  |-  ( x  =  X  ->  ( iota_ l  e.  K  E. z  e.  U  x  =  ( z  .+  ( l  .x.  Z
) ) )  =  ( iota_ l  e.  K  E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) ) ) )
20 lshpkrlem.g . . . . . 6  |-  G  =  ( x  e.  V  |->  ( iota_ k  e.  K  E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z
) ) ) )
21 oveq1 6277 . . . . . . . . . . . 12  |-  ( k  =  l  ->  (
k  .x.  Z )  =  ( l  .x.  Z ) )
2221oveq2d 6286 . . . . . . . . . . 11  |-  ( k  =  l  ->  (
y  .+  ( k  .x.  Z ) )  =  ( y  .+  (
l  .x.  Z )
) )
2322eqeq2d 2468 . . . . . . . . . 10  |-  ( k  =  l  ->  (
x  =  ( y 
.+  ( k  .x.  Z ) )  <->  x  =  ( y  .+  (
l  .x.  Z )
) ) )
2423rexbidv 2965 . . . . . . . . 9  |-  ( k  =  l  ->  ( E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z
) )  <->  E. y  e.  U  x  =  ( y  .+  (
l  .x.  Z )
) ) )
25 oveq1 6277 . . . . . . . . . . 11  |-  ( y  =  z  ->  (
y  .+  ( l  .x.  Z ) )  =  ( z  .+  (
l  .x.  Z )
) )
2625eqeq2d 2468 . . . . . . . . . 10  |-  ( y  =  z  ->  (
x  =  ( y 
.+  ( l  .x.  Z ) )  <->  x  =  ( z  .+  (
l  .x.  Z )
) ) )
2726cbvrexv 3082 . . . . . . . . 9  |-  ( E. y  e.  U  x  =  ( y  .+  ( l  .x.  Z
) )  <->  E. z  e.  U  x  =  ( z  .+  (
l  .x.  Z )
) )
2824, 27syl6bb 261 . . . . . . . 8  |-  ( k  =  l  ->  ( E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z
) )  <->  E. z  e.  U  x  =  ( z  .+  (
l  .x.  Z )
) ) )
2928cbvriotav 6243 . . . . . . 7  |-  ( iota_ k  e.  K  E. y  e.  U  x  =  ( y  .+  (
k  .x.  Z )
) )  =  (
iota_ l  e.  K  E. z  e.  U  x  =  ( z  .+  ( l  .x.  Z
) ) )
3029mpteq2i 4522 . . . . . 6  |-  ( x  e.  V  |->  ( iota_ k  e.  K  E. y  e.  U  x  =  ( y  .+  (
k  .x.  Z )
) ) )  =  ( x  e.  V  |->  ( iota_ l  e.  K  E. z  e.  U  x  =  ( z  .+  ( l  .x.  Z
) ) ) )
3120, 30eqtri 2483 . . . . 5  |-  G  =  ( x  e.  V  |->  ( iota_ l  e.  K  E. z  e.  U  x  =  ( z  .+  ( l  .x.  Z
) ) ) )
32 riotaex 6236 . . . . 5  |-  ( iota_ l  e.  K  E. z  e.  U  X  =  ( z  .+  (
l  .x.  Z )
) )  e.  _V
3319, 31, 32fvmpt 5931 . . . 4  |-  ( X  e.  V  ->  ( G `  X )  =  ( iota_ l  e.  K  E. z  e.  U  X  =  ( z  .+  ( l 
.x.  Z ) ) ) )
34 dfsbcq 3326 . . . 4  |-  ( ( G `  X )  =  ( iota_ l  e.  K  E. z  e.  U  X  =  ( z  .+  ( l 
.x.  Z ) ) )  ->  ( [. ( G `  X )  /  l ]. E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) )  <->  [. ( iota_ l  e.  K  E. z  e.  U  X  =  ( z  .+  (
l  .x.  Z )
) )  /  l ]. E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) ) ) )
359, 33, 343syl 20 . . 3  |-  ( ph  ->  ( [. ( G `
 X )  / 
l ]. E. z  e.  U  X  =  ( z  .+  ( l 
.x.  Z ) )  <->  [. ( iota_ l  e.  K  E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) ) )  / 
l ]. E. z  e.  U  X  =  ( z  .+  ( l 
.x.  Z ) ) ) )
3616, 35mpbird 232 . 2  |-  ( ph  ->  [. ( G `  X )  /  l ]. E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) ) )
37 fvex 5858 . . 3  |-  ( G `
 X )  e. 
_V
38 oveq1 6277 . . . . . 6  |-  ( l  =  ( G `  X )  ->  (
l  .x.  Z )  =  ( ( G `
 X )  .x.  Z ) )
3938oveq2d 6286 . . . . 5  |-  ( l  =  ( G `  X )  ->  (
z  .+  ( l  .x.  Z ) )  =  ( z  .+  (
( G `  X
)  .x.  Z )
) )
4039eqeq2d 2468 . . . 4  |-  ( l  =  ( G `  X )  ->  ( X  =  ( z  .+  ( l  .x.  Z
) )  <->  X  =  ( z  .+  (
( G `  X
)  .x.  Z )
) ) )
4140rexbidv 2965 . . 3  |-  ( l  =  ( G `  X )  ->  ( E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) )  <->  E. z  e.  U  X  =  ( z  .+  (
( G `  X
)  .x.  Z )
) ) )
4237, 41sbcie 3359 . 2  |-  ( [. ( G `  X )  /  l ]. E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) )  <->  E. z  e.  U  X  =  ( z  .+  (
( G `  X
)  .x.  Z )
) )
4336, 42sylib 196 1  |-  ( ph  ->  E. z  e.  U  X  =  ( z  .+  ( ( G `  X )  .x.  Z
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398    e. wcel 1823   E.wrex 2805   E!wreu 2806   [.wsbc 3324   {csn 4016    |-> cmpt 4497   ` cfv 5570   iota_crio 6231  (class class class)co 6270   Basecbs 14716   +g cplusg 14784  Scalarcsca 14787   .scvsca 14788   0gc0g 14929   LSSumclsm 16853   LSpanclspn 17812   LVecclvec 17943  LSHypclsh 35097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-tpos 6947  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-subg 16397  df-cntz 16554  df-lsm 16855  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-oppr 17467  df-dvdsr 17485  df-unit 17486  df-invr 17516  df-drng 17593  df-lmod 17709  df-lss 17774  df-lsp 17813  df-lvec 17944  df-lshyp 35099
This theorem is referenced by:  lshpkrlem6  35237
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