Theorem eflgam

Description: The exponential of the log-Gamma function is the Gamma function (by definition). (Contributed by Mario Carneiro, 8-Jul-2017)

	Ref	Expression
Assertion	eflgam	|- ( A e. ( CC \ ( ZZ \ NN ) ) -> ( exp ` ( log_G ` A ) ) = ( _G ` A ) )

Proof

Step	Hyp	Ref	Expression
1		df-gam	|- _G = ( exp o. log_G )
2	1	fveq1i	|- ( _G ` A ) = ( ( exp o. log_G ) ` A )
3		lgamf	|- log_G : ( CC \ ( ZZ \ NN ) ) --> CC
4		fvco3	|- ( ( log_G : ( CC \ ( ZZ \ NN ) ) --> CC /\ A e. ( CC \ ( ZZ \ NN ) ) ) -> ( ( exp o. log_G ) ` A ) = ( exp ` ( log_G ` A ) ) )
5	3 4	mpan	|- ( A e. ( CC \ ( ZZ \ NN ) ) -> ( ( exp o. log_G ) ` A ) = ( exp ` ( log_G ` A ) ) )
6	2 5	eqtr2id	|- ( A e. ( CC \ ( ZZ \ NN ) ) -> ( exp ` ( log_G ` A ) ) = ( _G ` A ) )