Theorem igamgam

Description: Value of the inverse Gamma function in terms of the Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017)

	Ref	Expression
Assertion	igamgam	|- ( A e. ( CC \ ( ZZ \ NN ) ) -> ( 1/_G ` A ) = ( 1 / ( _G ` A ) ) )

Proof

Step	Hyp	Ref	Expression
1		eldif	|- ( A e. ( CC \ ( ZZ \ NN ) ) <-> ( A e. CC /\ -. A e. ( ZZ \ NN ) ) )
2		igamval	|- ( A e. CC -> ( 1/_G ` A ) = if ( A e. ( ZZ \ NN ) , 0 , ( 1 / ( _G ` A ) ) ) )
3		iffalse	|- ( -. A e. ( ZZ \ NN ) -> if ( A e. ( ZZ \ NN ) , 0 , ( 1 / ( _G ` A ) ) ) = ( 1 / ( _G ` A ) ) )
4	2 3	sylan9eq	|- ( ( A e. CC /\ -. A e. ( ZZ \ NN ) ) -> ( 1/_G ` A ) = ( 1 / ( _G ` A ) ) )
5	1 4	sylbi	|- ( A e. ( CC \ ( ZZ \ NN ) ) -> ( 1/_G ` A ) = ( 1 / ( _G ` A ) ) )