Metamath Proof Explorer

Theorem igamlgam

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

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

Proof

Step Hyp Ref Expression
1 eflgam 
 |-  ( A e. ( CC \ ( ZZ \ NN ) ) -> ( exp ` ( log_G ` A ) ) = ( _G ` A ) )
2 1 oveq2d 
 |-  ( A e. ( CC \ ( ZZ \ NN ) ) -> ( 1 / ( exp ` ( log_G ` A ) ) ) = ( 1 / ( _G ` A ) ) )
3 lgamcl 
 |-  ( A e. ( CC \ ( ZZ \ NN ) ) -> ( log_G ` A ) e. CC )
4 efneg 
 |-  ( ( log_G ` A ) e. CC -> ( exp ` -u ( log_G ` A ) ) = ( 1 / ( exp ` ( log_G ` A ) ) ) )
5 3 4 syl 
 |-  ( A e. ( CC \ ( ZZ \ NN ) ) -> ( exp ` -u ( log_G ` A ) ) = ( 1 / ( exp ` ( log_G ` A ) ) ) )
6 igamgam 
 |-  ( A e. ( CC \ ( ZZ \ NN ) ) -> ( 1/_G ` A ) = ( 1 / ( _G ` A ) ) )
7 2 5 6 3eqtr4rd 
 |-  ( A e. ( CC \ ( ZZ \ NN ) ) -> ( 1/_G ` A ) = ( exp ` -u ( log_G ` A ) ) )