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Theorem mvrfval 16181
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 2809 . . . 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 2809 . . . 4  |-  ( R  e.  Y  ->  R  e.  _V )
75, 6syl 15 . . 3  |-  ( ph  ->  R  e.  _V )
8 mptexg 5761 . . . 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 5890 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( NN0  ^m  i
)  =  ( NN0 
^m  I ) )
12 rabeq 2795 . . . . . . . 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 2346 . . . . . 6  |-  ( ( i  =  I  /\  r  =  R )  ->  { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  =  D )
16 mpteq1 4116 . . . . . . . . 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 2307 . . . . . . 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 5545 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( 1r `  r
)  =  ( 1r
`  R ) )
21 mvrfval.o . . . . . . . 8  |-  .1.  =  ( 1r `  R )
2220, 21syl6eqr 2346 . . . . . . 7  |-  ( ( i  =  I  /\  r  =  R )  ->  ( 1r `  r
)  =  .1.  )
2319fveq2d 5545 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( 0g `  r
)  =  ( 0g
`  R ) )
24 mvrfval.z . . . . . . . 8  |-  .0.  =  ( 0g `  R )
2523, 24syl6eqr 2346 . . . . . . 7  |-  ( ( i  =  I  /\  r  =  R )  ->  ( 0g `  r
)  =  .0.  )
2618, 22, 25ifbieq12d 3600 . . . . . 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 4114 . . . . 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 4114 . . . 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 16115 . . . 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 5993 . . 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 2340 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 1632    e. wcel 1696   {crab 2560   _Vcvv 2801   ifcif 3578    e. cmpt 4093   `'ccnv 4704   "cima 4708   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Fincfn 6879   0cc0 8753   1c1 8754   NNcn 9762   NN0cn0 9981   0gc0g 13416   1rcur 15355   mVar cmvr 16104
This theorem is referenced by:  mvrval  16182  mvrf  16185  subrgmvr  16221
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-mvr 16115
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