Theorem eflgam

Description: The exponential of the log-Gamma function is the Gamma function (by definition). (Contributed by Mario Carneiro, 8-Jul-2017)

	Ref	Expression
Assertion	eflgam	$$⊢A\in \left(ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)\to {\mathrm{e}}^{\log \mathrm{\Gamma }\left(A\right)}=\mathrm{\Gamma }\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-gam	$$⊢\mathrm{\Gamma }=\exp \circ \log \mathrm{\Gamma }$$
2	1	fveq1i	$$⊢\mathrm{\Gamma }\left(A\right)=\left(\exp \circ \log \mathrm{\Gamma }\right)\left(A\right)$$
3		lgamf	$$⊢\log \mathrm{\Gamma }:ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)⟶ℂ$$
4		fvco3	$$⊢\left(\log \mathrm{\Gamma }:ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)⟶ℂ\wedge A\in \left(ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)\right)\to \left(\exp \circ \log \mathrm{\Gamma }\right)\left(A\right)={\mathrm{e}}^{\log \mathrm{\Gamma }\left(A\right)}$$
5	3 4	mpan	$$⊢A\in \left(ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)\to \left(\exp \circ \log \mathrm{\Gamma }\right)\left(A\right)={\mathrm{e}}^{\log \mathrm{\Gamma }\left(A\right)}$$
6	2 5	eqtr2id	$$⊢A\in \left(ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)\to {\mathrm{e}}^{\log \mathrm{\Gamma }\left(A\right)}=\mathrm{\Gamma }\left(A\right)$$