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Theorem mvrval 16485
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 )
mvrval.x  |-  ( ph  ->  X  e.  I )
Assertion
Ref Expression
mvrval  |-  ( ph  ->  ( V `  X
)  =  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  X , 
1 ,  0 ) ) ,  .1.  ,  .0.  ) ) )
Distinct variable groups:    .0. , f    .1. , f    D, f    y, W   
f, h, y, I    R, f    f, X, h, y
Allowed substitution hints:    ph( y, f, h)    D( y, h)    R( y, h)    .1. ( y, h)    V( y, f, h)    W( f, h)    Y( y, f, h)    .0. ( y, h)

Proof of Theorem mvrval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 mvrfval.v . . . 4  |-  V  =  ( I mVar  R )
2 mvrfval.d . . . 4  |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
3 mvrfval.z . . . 4  |-  .0.  =  ( 0g `  R )
4 mvrfval.o . . . 4  |-  .1.  =  ( 1r `  R )
5 mvrfval.i . . . 4  |-  ( ph  ->  I  e.  W )
6 mvrfval.r . . . 4  |-  ( ph  ->  R  e.  Y )
71, 2, 3, 4, 5, 6mvrfval 16484 . . 3  |-  ( ph  ->  V  =  ( x  e.  I  |->  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) ) )
87fveq1d 5730 . 2  |-  ( ph  ->  ( V `  X
)  =  ( ( x  e.  I  |->  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) ) `
 X ) )
9 mvrval.x . . 3  |-  ( ph  ->  X  e.  I )
10 eqeq2 2445 . . . . . . . . 9  |-  ( x  =  X  ->  (
y  =  x  <->  y  =  X ) )
1110ifbid 3757 . . . . . . . 8  |-  ( x  =  X  ->  if ( y  =  x ,  1 ,  0 )  =  if ( y  =  X , 
1 ,  0 ) )
1211mpteq2dv 4296 . . . . . . 7  |-  ( x  =  X  ->  (
y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) )  =  ( y  e.  I  |->  if ( y  =  X ,  1 ,  0 ) ) )
1312eqeq2d 2447 . . . . . 6  |-  ( x  =  X  ->  (
f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) )  <->  f  =  ( y  e.  I  |->  if ( y  =  X ,  1 ,  0 ) ) ) )
1413ifbid 3757 . . . . 5  |-  ( x  =  X  ->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  )  =  if ( f  =  ( y  e.  I  |->  if ( y  =  X ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) )
1514mpteq2dv 4296 . . . 4  |-  ( x  =  X  ->  (
f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  X ,  1 ,  0 ) ) ,  .1.  ,  .0.  )
) )
16 eqid 2436 . . . 4  |-  ( x  e.  I  |->  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) )  =  ( x  e.  I  |->  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  )
) )
17 ovex 6106 . . . . . . 7  |-  ( NN0 
^m  I )  e. 
_V
1817rabex 4354 . . . . . 6  |-  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  e.  _V
192, 18eqeltri 2506 . . . . 5  |-  D  e. 
_V
2019mptex 5966 . . . 4  |-  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  X , 
1 ,  0 ) ) ,  .1.  ,  .0.  ) )  e.  _V
2115, 16, 20fvmpt 5806 . . 3  |-  ( X  e.  I  ->  (
( x  e.  I  |->  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  )
) ) `  X
)  =  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  X , 
1 ,  0 ) ) ,  .1.  ,  .0.  ) ) )
229, 21syl 16 . 2  |-  ( ph  ->  ( ( x  e.  I  |->  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) ) `  X )  =  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  X ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) )
238, 22eqtrd 2468 1  |-  ( ph  ->  ( V `  X
)  =  ( 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    = wceq 1652    e. wcel 1725   {crab 2709   _Vcvv 2956   ifcif 3739    e. cmpt 4266   `'ccnv 4877   "cima 4881   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   Fincfn 7109   0cc0 8990   1c1 8991   NNcn 10000   NN0cn0 10221   0gc0g 13723   1rcur 15662   mVar cmvr 16407
This theorem is referenced by:  mvrval2  16486  mplcoe3  16529  evlslem1  19936
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  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-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-mvr 16418
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