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Theorem lshpkrlem3 29848
Description: Lemma for lshpkrex 29854. 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 29845 . . . 4  |-  ( ph  ->  E! l  e.  K  E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) ) )
15 riotasbc 6558 . . . 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 2442 . . . . . . 7  |-  ( x  =  X  ->  (
x  =  ( z 
.+  ( l  .x.  Z ) )  <->  X  =  ( z  .+  (
l  .x.  Z )
) ) )
1817rexbidv 2719 . . . . . 6  |-  ( x  =  X  ->  ( E. z  e.  U  x  =  ( z  .+  ( l  .x.  Z
) )  <->  E. z  e.  U  X  =  ( z  .+  (
l  .x.  Z )
) ) )
1918riotabidv 6544 . . . . 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 6081 . . . . . . . . . . . 12  |-  ( k  =  l  ->  (
k  .x.  Z )  =  ( l  .x.  Z ) )
2221oveq2d 6090 . . . . . . . . . . 11  |-  ( k  =  l  ->  (
y  .+  ( k  .x.  Z ) )  =  ( y  .+  (
l  .x.  Z )
) )
2322eqeq2d 2447 . . . . . . . . . 10  |-  ( k  =  l  ->  (
x  =  ( y 
.+  ( k  .x.  Z ) )  <->  x  =  ( y  .+  (
l  .x.  Z )
) ) )
2423rexbidv 2719 . . . . . . . . 9  |-  ( k  =  l  ->  ( E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z
) )  <->  E. y  e.  U  x  =  ( y  .+  (
l  .x.  Z )
) ) )
25 oveq1 6081 . . . . . . . . . . 11  |-  ( y  =  z  ->  (
y  .+  ( l  .x.  Z ) )  =  ( z  .+  (
l  .x.  Z )
) )
2625eqeq2d 2447 . . . . . . . . . 10  |-  ( y  =  z  ->  (
x  =  ( y 
.+  ( l  .x.  Z ) )  <->  x  =  ( z  .+  (
l  .x.  Z )
) ) )
2726cbvrexv 2926 . . . . . . . . 9  |-  ( E. y  e.  U  x  =  ( y  .+  ( l  .x.  Z
) )  <->  E. z  e.  U  x  =  ( z  .+  (
l  .x.  Z )
) )
2824, 27syl6bb 253 . . . . . . . 8  |-  ( k  =  l  ->  ( E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z
) )  <->  E. z  e.  U  x  =  ( z  .+  (
l  .x.  Z )
) ) )
2928cbvriotav 6554 . . . . . . 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 4285 . . . . . 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 2456 . . . . 5  |-  G  =  ( x  e.  V  |->  ( iota_ l  e.  K E. z  e.  U  x  =  ( z  .+  ( l  .x.  Z
) ) ) )
32 riotaex 6546 . . . . 5  |-  ( iota_ l  e.  K E. z  e.  U  X  =  ( z  .+  (
l  .x.  Z )
) )  e.  _V
3319, 31, 32fvmpt 5799 . . . 4  |-  ( X  e.  V  ->  ( G `  X )  =  ( iota_ l  e.  K E. z  e.  U  X  =  ( z  .+  ( l 
.x.  Z ) ) ) )
34 dfsbcq 3156 . . . 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 224 . 2  |-  ( ph  ->  [. ( G `  X )  /  l ]. E. z  e.  U  X  =  ( z  .+  ( l  .x.  Z
) ) )
37 fvex 5735 . . 3  |-  ( G `
 X )  e. 
_V
38 oveq1 6081 . . . . . 6  |-  ( l  =  ( G `  X )  ->  (
l  .x.  Z )  =  ( ( G `
 X )  .x.  Z ) )
3938oveq2d 6090 . . . . 5  |-  ( l  =  ( G `  X )  ->  (
z  .+  ( l  .x.  Z ) )  =  ( z  .+  (
( G `  X
)  .x.  Z )
) )
4039eqeq2d 2447 . . . 4  |-  ( l  =  ( G `  X )  ->  ( X  =  ( z  .+  ( l  .x.  Z
) )  <->  X  =  ( z  .+  (
( G `  X
)  .x.  Z )
) ) )
4140rexbidv 2719 . . 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 3188 . 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 189 1  |-  ( ph  ->  E. z  e.  U  X  =  ( z  .+  ( ( G `  X )  .x.  Z
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   E.wrex 2699   E!wreu 2700   [.wsbc 3154   {csn 3807    e. cmpt 4259   ` cfv 5447  (class class class)co 6074   iota_crio 6535   Basecbs 13462   +g cplusg 13522  Scalarcsca 13525   .scvsca 13526   0gc0g 13716   LSSumclsm 15261   LSpanclspn 16040   LVecclvec 16167  LSHypclsh 29711
This theorem is referenced by:  lshpkrlem6  29851
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 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060
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 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-tpos 6472  df-riota 6542  df-recs 6626  df-rdg 6661  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-nn 9994  df-2 10051  df-3 10052  df-ndx 13465  df-slot 13466  df-base 13467  df-sets 13468  df-ress 13469  df-plusg 13535  df-mulr 13536  df-0g 13720  df-mnd 14683  df-submnd 14732  df-grp 14805  df-minusg 14806  df-sbg 14807  df-subg 14934  df-cntz 15109  df-lsm 15263  df-cmn 15407  df-abl 15408  df-mgp 15642  df-rng 15656  df-ur 15658  df-oppr 15721  df-dvdsr 15739  df-unit 15740  df-invr 15770  df-drng 15830  df-lmod 15945  df-lss 16002  df-lsp 16041  df-lvec 16168  df-lshyp 29713
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