Metamath Proof Explorer

Theorem igamcl

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

Ref Expression
Assertion igamcl 
|- ( A e. CC -> ( 1/_G ` A ) e. CC )

Proof

Step Hyp Ref Expression
1 igamval 
 |-  ( A e. CC -> ( 1/_G ` A ) = if ( A e. ( ZZ \ NN ) , 0 , ( 1 / ( _G ` A ) ) ) )
2 0cnd 
 |-  ( ( A e. CC /\ A e. ( ZZ \ NN ) ) -> 0 e. CC )
3 eldif 
 |-  ( A e. ( CC \ ( ZZ \ NN ) ) <-> ( A e. CC /\ -. A e. ( ZZ \ NN ) ) )
4 gamcl 
 |-  ( A e. ( CC \ ( ZZ \ NN ) ) -> ( _G ` A ) e. CC )
5 gamne0 
 |-  ( A e. ( CC \ ( ZZ \ NN ) ) -> ( _G ` A ) =/= 0 )
6 4 5 reccld 
 |-  ( A e. ( CC \ ( ZZ \ NN ) ) -> ( 1 / ( _G ` A ) ) e. CC )
7 3 6 sylbir 
 |-  ( ( A e. CC /\ -. A e. ( ZZ \ NN ) ) -> ( 1 / ( _G ` A ) ) e. CC )
8 2 7 ifclda 
 |-  ( A e. CC -> if ( A e. ( ZZ \ NN ) , 0 , ( 1 / ( _G ` A ) ) ) e. CC )
9 1 8 eqeltrd 
 |-  ( A e. CC -> ( 1/_G ` A ) e. CC )