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Theorem mvrfval 18203
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    y,
f, D, x    y, W    f, h, I, x, y    R, f, x
Allowed substitution hints:    ph( x, y, f, h)    D( 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 3118 . . . 4  |-  ( I  e.  W  ->  I  e.  _V )
42, 3syl 16 . . 3  |-  ( ph  ->  I  e.  _V )
5 mvrfval.r . . . 4  |-  ( ph  ->  R  e.  Y )
6 elex 3118 . . . 4  |-  ( R  e.  Y  ->  R  e.  _V )
75, 6syl 16 . . 3  |-  ( ph  ->  R  e.  _V )
8 mptexg 6143 . . . 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 16 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  )
) )  e.  _V )
10 simpl 457 . . . . 5  |-  ( ( i  =  I  /\  r  =  R )  ->  i  =  I )
1110oveq2d 6312 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( NN0  ^m  i
)  =  ( NN0 
^m  I ) )
12 rabeq 3103 . . . . . . . 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 16 . . . . . . 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 2516 . . . . . 6  |-  ( ( i  =  I  /\  r  =  R )  ->  { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  =  D )
16 mpteq1 4537 . . . . . . . . 9  |-  ( i  =  I  ->  (
y  e.  i  |->  if ( y  =  x ,  1 ,  0 ) )  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) )
1716adantr 465 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( y  e.  i 
|->  if ( y  =  x ,  1 ,  0 ) )  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) )
1817eqeq2d 2471 . . . . . . 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 461 . . . . . . . . 9  |-  ( ( i  =  I  /\  r  =  R )  ->  r  =  R )
2019fveq2d 5876 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( 1r `  r
)  =  ( 1r
`  R ) )
21 mvrfval.o . . . . . . . 8  |-  .1.  =  ( 1r `  R )
2220, 21syl6eqr 2516 . . . . . . 7  |-  ( ( i  =  I  /\  r  =  R )  ->  ( 1r `  r
)  =  .1.  )
2319fveq2d 5876 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( 0g `  r
)  =  ( 0g
`  R ) )
24 mvrfval.z . . . . . . . 8  |-  .0.  =  ( 0g `  R )
2523, 24syl6eqr 2516 . . . . . . 7  |-  ( ( i  =  I  /\  r  =  R )  ->  ( 0g `  r
)  =  .0.  )
2618, 22, 25ifbieq12d 3971 . . . . . 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 4535 . . . . 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 4535 . . . 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 18133 . . . 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 6431 . . 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 1228 . 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 2510 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   {crab 2811   _Vcvv 3109   ifcif 3944    |-> cmpt 4515   `'ccnv 5007   "cima 5011   ` cfv 5594  (class class class)co 6296    ^m cmap 7438   Fincfn 7535   0cc0 9509   1c1 9510   NNcn 10556   NN0cn0 10816   0gc0g 14857   1rcur 17280   mVar cmvr 18128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-mvr 18133
This theorem is referenced by:  mvrval  18204  mvrf  18207  subrgmvr  18250
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