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Theorem mvrfval 17427
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 2971 . . . 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 2971 . . . 4  |-  ( R  e.  Y  ->  R  e.  _V )
75, 6syl 16 . . 3  |-  ( ph  ->  R  e.  _V )
8 mptexg 5934 . . . 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 454 . . . . 5  |-  ( ( i  =  I  /\  r  =  R )  ->  i  =  I )
1110oveq2d 6096 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( NN0  ^m  i
)  =  ( NN0 
^m  I ) )
12 rabeq 2956 . . . . . . . 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 2483 . . . . . 6  |-  ( ( i  =  I  /\  r  =  R )  ->  { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  =  D )
16 mpteq1 4360 . . . . . . . . 9  |-  ( i  =  I  ->  (
y  e.  i  |->  if ( y  =  x ,  1 ,  0 ) )  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) )
1716adantr 462 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( y  e.  i 
|->  if ( y  =  x ,  1 ,  0 ) )  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) )
1817eqeq2d 2444 . . . . . . 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 458 . . . . . . . . 9  |-  ( ( i  =  I  /\  r  =  R )  ->  r  =  R )
2019fveq2d 5683 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( 1r `  r
)  =  ( 1r
`  R ) )
21 mvrfval.o . . . . . . . 8  |-  .1.  =  ( 1r `  R )
2220, 21syl6eqr 2483 . . . . . . 7  |-  ( ( i  =  I  /\  r  =  R )  ->  ( 1r `  r
)  =  .1.  )
2319fveq2d 5683 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( 0g `  r
)  =  ( 0g
`  R ) )
24 mvrfval.z . . . . . . . 8  |-  .0.  =  ( 0g `  R )
2523, 24syl6eqr 2483 . . . . . . 7  |-  ( ( i  =  I  /\  r  =  R )  ->  ( 0g `  r
)  =  .0.  )
2618, 22, 25ifbieq12d 3804 . . . . . 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 4358 . . . . 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 4358 . . . 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 17352 . . . 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 6209 . . 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 1211 . 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 2477 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 1362    e. wcel 1755   {crab 2709   _Vcvv 2962   ifcif 3779    e. cmpt 4338   `'ccnv 4826   "cima 4830   ` cfv 5406  (class class class)co 6080    ^m cmap 7202   Fincfn 7298   0cc0 9270   1c1 9271   NNcn 10310   NN0cn0 10567   0gc0g 14361   1rcur 16579   mVar cmvr 17341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-mvr 17352
This theorem is referenced by:  mvrval  17428  mvrf  17431  subrgmvr  17474
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