Theorem bccld

Description: A binomial coefficient, in its extended domain, is a nonnegative integer. (Contributed by Glauco Siliprandi, 5-Apr-2020)

	Ref	Expression
Hypotheses	bccld.n	$$⊢\phi \to N\in {\mathbb{N}}_0$$
	bccld.k	$$⊢\phi \to K\in \mathbb{Z}$$
Assertion	bccld	$$⊢\phi \to \left(\genfrac{}{}{0ex}{}{N}{K}\right)\in {\mathbb{N}}_0$$

Proof

Step	Hyp	Ref	Expression
1		bccld.n	$$⊢\phi \to N\in {\mathbb{N}}_0$$
2		bccld.k	$$⊢\phi \to K\in \mathbb{Z}$$
3		bccl	$$⊢\left(N\in {\mathbb{N}}_0\wedge K\in \mathbb{Z}\right)\to \left(\genfrac{}{}{0ex}{}{N}{K}\right)\in {\mathbb{N}}_0$$
4	1 2 3	syl2anc	$$⊢\phi \to \left(\genfrac{}{}{0ex}{}{N}{K}\right)\in {\mathbb{N}}_0$$