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Theorem igamval 28462
Description: Value of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
Assertion
Ref Expression
igamval  |-  ( A  e.  CC  ->  (1/ _G `  A )  =  if ( A  e.  ( ZZ  \  NN ) ,  0 , 
( 1  /  ( _G `  A ) ) ) )

Proof of Theorem igamval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eleq1 2515 . . 3  |-  ( x  =  A  ->  (
x  e.  ( ZZ 
\  NN )  <->  A  e.  ( ZZ  \  NN ) ) )
2 fveq2 5856 . . . 4  |-  ( x  =  A  ->  ( _G `  x )  =  ( _G `  A
) )
32oveq2d 6297 . . 3  |-  ( x  =  A  ->  (
1  /  ( _G `  x ) )  =  ( 1  /  ( _G `  A ) ) )
41, 3ifbieq2d 3951 . 2  |-  ( x  =  A  ->  if ( x  e.  ( ZZ  \  NN ) ,  0 ,  ( 1  /  ( _G `  x
) ) )  =  if ( A  e.  ( ZZ  \  NN ) ,  0 , 
( 1  /  ( _G `  A ) ) ) )
5 df-igam 28436 . 2  |- 1/ _G  =  ( x  e.  CC  |->  if ( x  e.  ( ZZ  \  NN ) ,  0 ,  ( 1  /  ( _G `  x ) ) ) )
6 c0ex 9593 . . 3  |-  0  e.  _V
7 ovex 6309 . . 3  |-  ( 1  /  ( _G `  A
) )  e.  _V
86, 7ifex 3995 . 2  |-  if ( A  e.  ( ZZ 
\  NN ) ,  0 ,  ( 1  /  ( _G `  A
) ) )  e. 
_V
94, 5, 8fvmpt 5941 1  |-  ( A  e.  CC  ->  (1/ _G `  A )  =  if ( A  e.  ( ZZ  \  NN ) ,  0 , 
( 1  /  ( _G `  A ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804    \ cdif 3458   ifcif 3926   ` cfv 5578  (class class class)co 6281   CCcc 9493   0cc0 9495   1c1 9496    / cdiv 10212   NNcn 10542   ZZcz 10870   _Gcgam 28432  1/ _Gcigam 28433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-mulcl 9557  ax-i2m1 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-iota 5541  df-fun 5580  df-fv 5586  df-ov 6284  df-igam 28436
This theorem is referenced by:  igamz  28463  igamgam  28464  igamcl  28467
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