Metamath Proof Explorer

Theorem igamf

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

Ref Expression
Assertion igamf 
|- 1/_G : CC --> CC

Proof

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