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Theorem lshpkrlem3 29984
Description: Lemma for lshpkrex 29990. 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 29981 . . . 4  |-  ( ph  ->  E! l  e.  K  E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) ) )
15 riotasbc 6568 . . . 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 2444 . . . . . . 7  |-  ( x  =  X  ->  (
x  =  ( z 
.+  ( l  .x.  Z ) )  <->  X  =  ( z  .+  (
l  .x.  Z )
) ) )
1817rexbidv 2728 . . . . . 6  |-  ( x  =  X  ->  ( E. z  e.  U  x  =  ( z  .+  ( l  .x.  Z
) )  <->  E. z  e.  U  X  =  ( z  .+  (
l  .x.  Z )
) ) )
1918riotabidv 6554 . . . . 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 6091 . . . . . . . . . . . 12  |-  ( k  =  l  ->  (
k  .x.  Z )  =  ( l  .x.  Z ) )
2221oveq2d 6100 . . . . . . . . . . 11  |-  ( k  =  l  ->  (
y  .+  ( k  .x.  Z ) )  =  ( y  .+  (
l  .x.  Z )
) )
2322eqeq2d 2449 . . . . . . . . . 10  |-  ( k  =  l  ->  (
x  =  ( y 
.+  ( k  .x.  Z ) )  <->  x  =  ( y  .+  (
l  .x.  Z )
) ) )
2423rexbidv 2728 . . . . . . . . 9  |-  ( k  =  l  ->  ( E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z
) )  <->  E. y  e.  U  x  =  ( y  .+  (
l  .x.  Z )
) ) )
25 oveq1 6091 . . . . . . . . . . 11  |-  ( y  =  z  ->  (
y  .+  ( l  .x.  Z ) )  =  ( z  .+  (
l  .x.  Z )
) )
2625eqeq2d 2449 . . . . . . . . . 10  |-  ( y  =  z  ->  (
x  =  ( y 
.+  ( l  .x.  Z ) )  <->  x  =  ( z  .+  (
l  .x.  Z )
) ) )
2726cbvrexv 2935 . . . . . . . . 9  |-  ( E. y  e.  U  x  =  ( y  .+  ( l  .x.  Z
) )  <->  E. z  e.  U  x  =  ( z  .+  (
l  .x.  Z )
) )
2824, 27syl6bb 254 . . . . . . . 8  |-  ( k  =  l  ->  ( E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z
) )  <->  E. z  e.  U  x  =  ( z  .+  (
l  .x.  Z )
) ) )
2928cbvriotav 6564 . . . . . . 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 4295 . . . . . 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 2458 . . . . 5  |-  G  =  ( x  e.  V  |->  ( iota_ l  e.  K E. z  e.  U  x  =  ( z  .+  ( l  .x.  Z
) ) ) )
32 riotaex 6556 . . . . 5  |-  ( iota_ l  e.  K E. z  e.  U  X  =  ( z  .+  (
l  .x.  Z )
) )  e.  _V
3319, 31, 32fvmpt 5809 . . . 4  |-  ( X  e.  V  ->  ( G `  X )  =  ( iota_ l  e.  K E. z  e.  U  X  =  ( z  .+  ( l 
.x.  Z ) ) ) )
34 dfsbcq 3165 . . . 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 19 . . 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 225 . 2  |-  ( ph  ->  [. ( G `  X )  /  l ]. E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) ) )
37 fvex 5745 . . 3  |-  ( G `
 X )  e. 
_V
38 oveq1 6091 . . . . . 6  |-  ( l  =  ( G `  X )  ->  (
l  .x.  Z )  =  ( ( G `
 X )  .x.  Z ) )
3938oveq2d 6100 . . . . 5  |-  ( l  =  ( G `  X )  ->  (
z  .+  ( l  .x.  Z ) )  =  ( z  .+  (
( G `  X
)  .x.  Z )
) )
4039eqeq2d 2449 . . . 4  |-  ( l  =  ( G `  X )  ->  ( X  =  ( z  .+  ( l  .x.  Z
) )  <->  X  =  ( z  .+  (
( G `  X
)  .x.  Z )
) ) )
4140rexbidv 2728 . . 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 3197 . 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 190 1  |-  ( ph  ->  E. z  e.  U  X  =  ( z  .+  ( ( G `  X )  .x.  Z
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   E.wrex 2708   E!wreu 2709   [.wsbc 3163   {csn 3816    e. cmpt 4269   ` cfv 5457  (class class class)co 6084   iota_crio 6545   Basecbs 13474   +g cplusg 13534  Scalarcsca 13537   .scvsca 13538   0gc0g 13728   LSSumclsm 15273   LSpanclspn 16052   LVecclvec 16179  LSHypclsh 29847
This theorem is referenced by:  lshpkrlem6  29987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-tpos 6482  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-0g 13732  df-mnd 14695  df-submnd 14744  df-grp 14817  df-minusg 14818  df-sbg 14819  df-subg 14946  df-cntz 15121  df-lsm 15275  df-cmn 15419  df-abl 15420  df-mgp 15654  df-rng 15668  df-ur 15670  df-oppr 15733  df-dvdsr 15751  df-unit 15752  df-invr 15782  df-drng 15842  df-lmod 15957  df-lss 16014  df-lsp 16053  df-lvec 16180  df-lshyp 29849
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