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 ∈ ( ℂ ∖ ( ℤ ∖ ℕ ) ) → ( 1/Γ ‘ 𝐴 ) = ( exp ‘ - ( log Γ ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eflgam	⊢ ( 𝐴 ∈ ( ℂ ∖ ( ℤ ∖ ℕ ) ) → ( exp ‘ ( log Γ ‘ 𝐴 ) ) = ( Γ ‘ 𝐴 ) )
2	1	oveq2d	⊢ ( 𝐴 ∈ ( ℂ ∖ ( ℤ ∖ ℕ ) ) → ( 1 / ( exp ‘ ( log Γ ‘ 𝐴 ) ) ) = ( 1 / ( Γ ‘ 𝐴 ) ) )
3		lgamcl	⊢ ( 𝐴 ∈ ( ℂ ∖ ( ℤ ∖ ℕ ) ) → ( log Γ ‘ 𝐴 ) ∈ ℂ )
4		efneg	⊢ ( ( log Γ ‘ 𝐴 ) ∈ ℂ → ( exp ‘ - ( log Γ ‘ 𝐴 ) ) = ( 1 / ( exp ‘ ( log Γ ‘ 𝐴 ) ) ) )
5	3 4	syl	⊢ ( 𝐴 ∈ ( ℂ ∖ ( ℤ ∖ ℕ ) ) → ( exp ‘ - ( log Γ ‘ 𝐴 ) ) = ( 1 / ( exp ‘ ( log Γ ‘ 𝐴 ) ) ) )
6		igamgam	⊢ ( 𝐴 ∈ ( ℂ ∖ ( ℤ ∖ ℕ ) ) → ( 1/Γ ‘ 𝐴 ) = ( 1 / ( Γ ‘ 𝐴 ) ) )
7	2 5 6	3eqtr4rd	⊢ ( 𝐴 ∈ ( ℂ ∖ ( ℤ ∖ ℕ ) ) → ( 1/Γ ‘ 𝐴 ) = ( exp ‘ - ( log Γ ‘ 𝐴 ) ) )