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	$$⊢A\in \left(ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)\to \frac{1}{\mathrm{\Gamma }}\left(A\right)=\frac{1}{\mathrm{\Gamma }\left(A\right)}$$

Proof

Step	Hyp	Ref	Expression
1		eldif	$$⊢A\in \left(ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)\leftrightarrow \left(A\in ℂ\wedge \neg A\in \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)$$
2		igamval	$$⊢A\in ℂ\to \frac{1}{\mathrm{\Gamma }}\left(A\right)=if\left(A\in \left(\mathbb{Z}\setminus \mathbb{N}\right),0,\frac{1}{\mathrm{\Gamma }\left(A\right)}\right)$$
3		iffalse	$$⊢\neg A\in \left(\mathbb{Z}\setminus \mathbb{N}\right)\to if\left(A\in \left(\mathbb{Z}\setminus \mathbb{N}\right),0,\frac{1}{\mathrm{\Gamma }\left(A\right)}\right)=\frac{1}{\mathrm{\Gamma }\left(A\right)}$$
4	2 3	sylan9eq	$$⊢\left(A\in ℂ\wedge \neg A\in \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)\to \frac{1}{\mathrm{\Gamma }}\left(A\right)=\frac{1}{\mathrm{\Gamma }\left(A\right)}$$
5	1 4	sylbi	$$⊢A\in \left(ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)\to \frac{1}{\mathrm{\Gamma }}\left(A\right)=\frac{1}{\mathrm{\Gamma }\left(A\right)}$$