Theorem gamne0

Description: The Gamma function is never zero. (Contributed by Mario Carneiro, 9-Jul-2017)

	Ref	Expression
Assertion	gamne0	$$⊢A\in \left(ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)\to \mathrm{\Gamma }\left(A\right)\ne 0$$

Proof

Step	Hyp	Ref	Expression
1		eflgam	$$⊢A\in \left(ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)\to {\mathrm{e}}^{\log \mathrm{\Gamma }\left(A\right)}=\mathrm{\Gamma }\left(A\right)$$
2		lgamcl	$$⊢A\in \left(ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)\to \log \mathrm{\Gamma }\left(A\right)\in ℂ$$
3		efne0	$$⊢\log \mathrm{\Gamma }\left(A\right)\in ℂ\to {\mathrm{e}}^{\log \mathrm{\Gamma }\left(A\right)}\ne 0$$
4	2 3	syl	$$⊢A\in \left(ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)\to {\mathrm{e}}^{\log \mathrm{\Gamma }\left(A\right)}\ne 0$$
5	1 4	eqnetrrd	$$⊢A\in \left(ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)\to \mathrm{\Gamma }\left(A\right)\ne 0$$