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Theorem mvrfval 16165
Description: Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypotheses
Ref Expression
mvrfval.v  |-  V  =  ( I mVar  R )
mvrfval.d  |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
mvrfval.z  |-  .0.  =  ( 0g `  R )
mvrfval.o  |-  .1.  =  ( 1r `  R )
mvrfval.i  |-  ( ph  ->  I  e.  W )
mvrfval.r  |-  ( ph  ->  R  e.  Y )
Assertion
Ref Expression
mvrfval  |-  ( ph  ->  V  =  ( x  e.  I  |->  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) ) )
Distinct variable groups:    x, f,  .0.   
.1. , f, x    D, f, x    y, W    f, h, y, I, x    R, f, x
Allowed substitution hints:    ph( x, y, f, h)    D( y, h)    R( y, h)    .1. ( y, h)    V( x, y, f, h)    W( x, f, h)    Y( x, y, f, h)    .0. ( y, h)

Proof of Theorem mvrfval
Dummy variables  i 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mvrfval.v . 2  |-  V  =  ( I mVar  R )
2 mvrfval.i . . . 4  |-  ( ph  ->  I  e.  W )
3 elex 2796 . . . 4  |-  ( I  e.  W  ->  I  e.  _V )
42, 3syl 15 . . 3  |-  ( ph  ->  I  e.  _V )
5 mvrfval.r . . . 4  |-  ( ph  ->  R  e.  Y )
6 elex 2796 . . . 4  |-  ( R  e.  Y  ->  R  e.  _V )
75, 6syl 15 . . 3  |-  ( ph  ->  R  e.  _V )
8 mptexg 5745 . . . 4  |-  ( I  e.  W  ->  (
x  e.  I  |->  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) )  e.  _V )
92, 8syl 15 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  )
) )  e.  _V )
10 simpl 443 . . . . 5  |-  ( ( i  =  I  /\  r  =  R )  ->  i  =  I )
1110oveq2d 5874 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( NN0  ^m  i
)  =  ( NN0 
^m  I ) )
12 rabeq 2782 . . . . . . . 8  |-  ( ( NN0  ^m  i )  =  ( NN0  ^m  I )  ->  { h  e.  ( NN0  ^m  i
)  |  ( `' h " NN )  e.  Fin }  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin } )
1311, 12syl 15 . . . . . . 7  |-  ( ( i  =  I  /\  r  =  R )  ->  { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  =  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin } )
14 mvrfval.d . . . . . . 7  |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
1513, 14syl6eqr 2333 . . . . . 6  |-  ( ( i  =  I  /\  r  =  R )  ->  { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  =  D )
16 mpteq1 4100 . . . . . . . . 9  |-  ( i  =  I  ->  (
y  e.  i  |->  if ( y  =  x ,  1 ,  0 ) )  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) )
1716adantr 451 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( y  e.  i 
|->  if ( y  =  x ,  1 ,  0 ) )  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) )
1817eqeq2d 2294 . . . . . . 7  |-  ( ( i  =  I  /\  r  =  R )  ->  ( f  =  ( y  e.  i  |->  if ( y  =  x ,  1 ,  0 ) )  <->  f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ) )
19 simpr 447 . . . . . . . . 9  |-  ( ( i  =  I  /\  r  =  R )  ->  r  =  R )
2019fveq2d 5529 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( 1r `  r
)  =  ( 1r
`  R ) )
21 mvrfval.o . . . . . . . 8  |-  .1.  =  ( 1r `  R )
2220, 21syl6eqr 2333 . . . . . . 7  |-  ( ( i  =  I  /\  r  =  R )  ->  ( 1r `  r
)  =  .1.  )
2319fveq2d 5529 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( 0g `  r
)  =  ( 0g
`  R ) )
24 mvrfval.z . . . . . . . 8  |-  .0.  =  ( 0g `  R )
2523, 24syl6eqr 2333 . . . . . . 7  |-  ( ( i  =  I  /\  r  =  R )  ->  ( 0g `  r
)  =  .0.  )
2618, 22, 25ifbieq12d 3587 . . . . . 6  |-  ( ( i  =  I  /\  r  =  R )  ->  if ( f  =  ( y  e.  i 
|->  if ( y  =  x ,  1 ,  0 ) ) ,  ( 1r `  r
) ,  ( 0g
`  r ) )  =  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) )
2715, 26mpteq12dv 4098 . . . . 5  |-  ( ( i  =  I  /\  r  =  R )  ->  ( f  e.  {
h  e.  ( NN0 
^m  i )  |  ( `' h " NN )  e.  Fin } 
|->  if ( f  =  ( y  e.  i 
|->  if ( y  =  x ,  1 ,  0 ) ) ,  ( 1r `  r
) ,  ( 0g
`  r ) ) )  =  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) )
2810, 27mpteq12dv 4098 . . . 4  |-  ( ( i  =  I  /\  r  =  R )  ->  ( x  e.  i 
|->  ( f  e.  {
h  e.  ( NN0 
^m  i )  |  ( `' h " NN )  e.  Fin } 
|->  if ( f  =  ( y  e.  i 
|->  if ( y  =  x ,  1 ,  0 ) ) ,  ( 1r `  r
) ,  ( 0g
`  r ) ) ) )  =  ( x  e.  I  |->  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) ) )
29 df-mvr 16099 . . . 4  |- mVar  =  ( i  e.  _V , 
r  e.  _V  |->  ( x  e.  i  |->  ( f  e.  { h  e.  ( NN0  ^m  i
)  |  ( `' h " NN )  e.  Fin }  |->  if ( f  =  ( y  e.  i  |->  if ( y  =  x ,  1 ,  0 ) ) ,  ( 1r `  r ) ,  ( 0g `  r ) ) ) ) )
3028, 29ovmpt2ga 5977 . . 3  |-  ( ( I  e.  _V  /\  R  e.  _V  /\  (
x  e.  I  |->  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) )  e.  _V )  -> 
( I mVar  R )  =  ( x  e.  I  |->  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) ) )
314, 7, 9, 30syl3anc 1182 . 2  |-  ( ph  ->  ( I mVar  R )  =  ( x  e.  I  |->  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) ) )
321, 31syl5eq 2327 1  |-  ( ph  ->  V  =  ( x  e.  I  |->  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   ifcif 3565    e. cmpt 4077   `'ccnv 4688   "cima 4692   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Fincfn 6863   0cc0 8737   1c1 8738   NNcn 9746   NN0cn0 9965   0gc0g 13400   1rcur 15339   mVar cmvr 16088
This theorem is referenced by:  mvrval  16166  mvrf  16169  subrgmvr  16205
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-mvr 16099
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