Theorem rpdmgm

Description: A positive real number is in the domain of the Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017)

	Ref	Expression
Assertion	rpdmgm	$$⊢A\in {ℝ}^{+}\to A\in \left(ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		rpcn	$$⊢A\in {ℝ}^{+}\to A\in ℂ$$
2		ax-1	$$⊢A\in {ℝ}^{+}\to \left(A\in ℝ\to A\in {ℝ}^{+}\right)$$
3		eqid	$$⊢ℂ\setminus \left(\mathrm{-\infty },0\right]=ℂ\setminus \left(\mathrm{-\infty },0\right]$$
4	3	ellogdm	$$⊢A\in \left(ℂ\setminus \left(\mathrm{-\infty },0\right]\right)\leftrightarrow \left(A\in ℂ\wedge \left(A\in ℝ\to A\in {ℝ}^{+}\right)\right)$$
5	1 2 4	sylanbrc	$$⊢A\in {ℝ}^{+}\to A\in \left(ℂ\setminus \left(\mathrm{-\infty },0\right]\right)$$
6		dmlogdmgm	$$⊢A\in \left(ℂ\setminus \left(\mathrm{-\infty },0\right]\right)\to A\in \left(ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)$$
7	5 6	syl	$$⊢A\in {ℝ}^{+}\to A\in \left(ℂ\setminus \left(\mathbb{Z}\setminus \mathbb{N}\right)\right)$$