Theorem igamgam

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

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

Proof

Step	Hyp	Ref	Expression
1		eldif	⊢ ( 𝐴 ∈ ( ℂ ∖ ( ℤ ∖ ℕ ) ) ↔ ( 𝐴 ∈ ℂ ∧ ¬ 𝐴 ∈ ( ℤ ∖ ℕ ) ) )
2		igamval	⊢ ( 𝐴 ∈ ℂ → ( 1/Γ ‘ 𝐴 ) = if ( 𝐴 ∈ ( ℤ ∖ ℕ ) , 0 , ( 1 / ( Γ ‘ 𝐴 ) ) ) )
3		iffalse	⊢ ( ¬ 𝐴 ∈ ( ℤ ∖ ℕ ) → if ( 𝐴 ∈ ( ℤ ∖ ℕ ) , 0 , ( 1 / ( Γ ‘ 𝐴 ) ) ) = ( 1 / ( Γ ‘ 𝐴 ) ) )
4	2 3	sylan9eq	⊢ ( ( 𝐴 ∈ ℂ ∧ ¬ 𝐴 ∈ ( ℤ ∖ ℕ ) ) → ( 1/Γ ‘ 𝐴 ) = ( 1 / ( Γ ‘ 𝐴 ) ) )
5	1 4	sylbi	⊢ ( 𝐴 ∈ ( ℂ ∖ ( ℤ ∖ ℕ ) ) → ( 1/Γ ‘ 𝐴 ) = ( 1 / ( Γ ‘ 𝐴 ) ) )