Theorem igamval 24773

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.

Step	Hyp	Ref	Expression
1		eleq1 2689	. . 3 |- ( x = A -> ( x e. ( ZZ \ NN ) <-> A e. ( ZZ \ NN ) ) )
2		fveq2 6191	. . . 4 |- ( x = A -> ( _G ` x ) = ( _G ` A ) )
3	2	oveq2d 6666	. . 3 |- ( x = A -> ( 1 / ( _G ` x ) ) = ( 1 / ( _G ` A ) ) )
4	1, 3	ifbieq2d 4111	. 2 |- ( x = A -> if ( x e. ( ZZ \ NN ) , 0 , ( 1 / ( _G ` x ) ) ) = if ( A e. ( ZZ \ NN ) , 0 , ( 1 / ( _G ` A ) ) ) )
5		df-igam 24747	. 2 |- 1/ _G = ( x e. CC |-> if ( x e. ( ZZ \ NN ) , 0 , ( 1 / ( _G ` x ) ) ) )
6		c0ex 10034	. . 3 |- 0 e. _V
7		ovex 6678	. . 3 |- ( 1 / ( _G ` A ) ) e. _V
8	6, 7	ifex 4156	. 2 |- if ( A e. ( ZZ \ NN ) , 0 , ( 1 / ( _G ` A ) ) ) e. _V
9	4, 5, 8	fvmpt 6282	1 |- ( A e. CC -> (1/ _G ` A ) = if ( A e. ( ZZ \ NN ) , 0 , ( 1 / ( _G ` A ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   \ cdif 3571   ifcif 4086   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   / cdiv 10684   NNcn 11020   ZZcz 11377   _Gcgam 24743  1/ _Gcigam 24744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-mulcl 9998  ax-i2m1 10004

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-igam 24747

This theorem is referenced by:  igamz  24774  igamgam  24775  igamcl  24778