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Theorem lshpkrlem3 33786
Description: Lemma for lshpkrex 33792. 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 33783 . . . 4  |-  ( ph  ->  E! l  e.  K  E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) ) )
15 riotasbc 6254 . . . 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 2466 . . . . . . 7  |-  ( x  =  X  ->  (
x  =  ( z 
.+  ( l  .x.  Z ) )  <->  X  =  ( z  .+  (
l  .x.  Z )
) ) )
1817rexbidv 2968 . . . . . 6  |-  ( x  =  X  ->  ( E. z  e.  U  x  =  ( z  .+  ( l  .x.  Z
) )  <->  E. z  e.  U  X  =  ( z  .+  (
l  .x.  Z )
) ) )
1918riotabidv 6240 . . . . 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 6284 . . . . . . . . . . . 12  |-  ( k  =  l  ->  (
k  .x.  Z )  =  ( l  .x.  Z ) )
2221oveq2d 6293 . . . . . . . . . . 11  |-  ( k  =  l  ->  (
y  .+  ( k  .x.  Z ) )  =  ( y  .+  (
l  .x.  Z )
) )
2322eqeq2d 2476 . . . . . . . . . 10  |-  ( k  =  l  ->  (
x  =  ( y 
.+  ( k  .x.  Z ) )  <->  x  =  ( y  .+  (
l  .x.  Z )
) ) )
2423rexbidv 2968 . . . . . . . . 9  |-  ( k  =  l  ->  ( E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z
) )  <->  E. y  e.  U  x  =  ( y  .+  (
l  .x.  Z )
) ) )
25 oveq1 6284 . . . . . . . . . . 11  |-  ( y  =  z  ->  (
y  .+  ( l  .x.  Z ) )  =  ( z  .+  (
l  .x.  Z )
) )
2625eqeq2d 2476 . . . . . . . . . 10  |-  ( y  =  z  ->  (
x  =  ( y 
.+  ( l  .x.  Z ) )  <->  x  =  ( z  .+  (
l  .x.  Z )
) ) )
2726cbvrexv 3084 . . . . . . . . 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 6249 . . . . . . 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 4525 . . . . . 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 2491 . . . . 5  |-  G  =  ( x  e.  V  |->  ( iota_ l  e.  K  E. z  e.  U  x  =  ( z  .+  ( l  .x.  Z
) ) ) )
32 riotaex 6242 . . . . 5  |-  ( iota_ l  e.  K  E. z  e.  U  X  =  ( z  .+  (
l  .x.  Z )
) )  e.  _V
3319, 31, 32fvmpt 5943 . . . 4  |-  ( X  e.  V  ->  ( G `  X )  =  ( iota_ l  e.  K  E. z  e.  U  X  =  ( z  .+  ( l 
.x.  Z ) ) ) )
34 dfsbcq 3328 . . . 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 5869 . . 3  |-  ( G `
 X )  e. 
_V
38 oveq1 6284 . . . . . 6  |-  ( l  =  ( G `  X )  ->  (
l  .x.  Z )  =  ( ( G `
 X )  .x.  Z ) )
3938oveq2d 6293 . . . . 5  |-  ( l  =  ( G `  X )  ->  (
z  .+  ( l  .x.  Z ) )  =  ( z  .+  (
( G `  X
)  .x.  Z )
) )
4039eqeq2d 2476 . . . 4  |-  ( l  =  ( G `  X )  ->  ( X  =  ( z  .+  ( l  .x.  Z
) )  <->  X  =  ( z  .+  (
( G `  X
)  .x.  Z )
) ) )
4140rexbidv 2968 . . 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 3361 . 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 1374    e. wcel 1762   E.wrex 2810   E!wreu 2811   [.wsbc 3326   {csn 4022    |-> cmpt 4500   ` cfv 5581   iota_crio 6237  (class class class)co 6277   Basecbs 14481   +g cplusg 14546  Scalarcsca 14549   .scvsca 14550   0gc0g 14686   LSSumclsm 16445   LSpanclspn 17395   LVecclvec 17526  LSHypclsh 33649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-tpos 6947  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-0g 14688  df-mnd 15723  df-submnd 15773  df-grp 15853  df-minusg 15854  df-sbg 15855  df-subg 15988  df-cntz 16145  df-lsm 16447  df-cmn 16591  df-abl 16592  df-mgp 16927  df-ur 16939  df-rng 16983  df-oppr 17051  df-dvdsr 17069  df-unit 17070  df-invr 17100  df-drng 17176  df-lmod 17292  df-lss 17357  df-lsp 17396  df-lvec 17527  df-lshyp 33651
This theorem is referenced by:  lshpkrlem6  33789
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