Theorem gamigam

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

	Ref	Expression
Assertion	gamigam	⊢ ( 𝐴 ∈ ( ℂ ∖ ( ℤ ∖ ℕ ) ) → ( Γ ‘ 𝐴 ) = ( 1 / ( 1/Γ ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		igamgam	⊢ ( 𝐴 ∈ ( ℂ ∖ ( ℤ ∖ ℕ ) ) → ( 1/Γ ‘ 𝐴 ) = ( 1 / ( Γ ‘ 𝐴 ) ) )
2	1	oveq2d	⊢ ( 𝐴 ∈ ( ℂ ∖ ( ℤ ∖ ℕ ) ) → ( 1 / ( 1/Γ ‘ 𝐴 ) ) = ( 1 / ( 1 / ( Γ ‘ 𝐴 ) ) ) )
3		gamcl	⊢ ( 𝐴 ∈ ( ℂ ∖ ( ℤ ∖ ℕ ) ) → ( Γ ‘ 𝐴 ) ∈ ℂ )
4		gamne0	⊢ ( 𝐴 ∈ ( ℂ ∖ ( ℤ ∖ ℕ ) ) → ( Γ ‘ 𝐴 ) ≠ 0 )
5	3 4	recrecd	⊢ ( 𝐴 ∈ ( ℂ ∖ ( ℤ ∖ ℕ ) ) → ( 1 / ( 1 / ( Γ ‘ 𝐴 ) ) ) = ( Γ ‘ 𝐴 ) )
6	2 5	eqtr2d	⊢ ( 𝐴 ∈ ( ℂ ∖ ( ℤ ∖ ℕ ) ) → ( Γ ‘ 𝐴 ) = ( 1 / ( 1/Γ ‘ 𝐴 ) ) )