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

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	1	oveq2d	$$⊢A\in \left(ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)\to \frac{1}{{\mathrm{e}}^{\log \mathrm{\Gamma }\left(A\right)}}=\frac{1}{\mathrm{\Gamma }\left(A\right)}$$
3		lgamcl	$$⊢A\in \left(ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)\to \log \mathrm{\Gamma }\left(A\right)\in ℂ$$
4		efneg	$$⊢\log \mathrm{\Gamma }\left(A\right)\in ℂ\to {\mathrm{e}}^{-\log \mathrm{\Gamma }\left(A\right)}=\frac{1}{{\mathrm{e}}^{\log \mathrm{\Gamma }\left(A\right)}}$$
5	3 4	syl	$$⊢A\in \left(ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)\to {\mathrm{e}}^{-\log \mathrm{\Gamma }\left(A\right)}=\frac{1}{{\mathrm{e}}^{\log \mathrm{\Gamma }\left(A\right)}}$$
6		igamgam	$$⊢A\in \left(ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)\to \frac{1}{\mathrm{\Gamma }}\left(A\right)=\frac{1}{\mathrm{\Gamma }\left(A\right)}$$
7	2 5 6	3eqtr4rd	$$⊢A\in \left(ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)\to \frac{1}{\mathrm{\Gamma }}\left(A\right)={\mathrm{e}}^{-\log \mathrm{\Gamma }\left(A\right)}$$