Theorem fallfacp1d

Description: The value of the falling factorial at a successor. (Contributed by Scott Fenton, 19-Mar-2018)

	Ref	Expression
Hypotheses	rffacp1d.1	$$⊢\phi \to A\in ℂ$$
	rffacp1d.2	$$⊢\phi \to N\in {\mathbb{N}}_0$$
Assertion	fallfacp1d	$$⊢\phi \to A^{\underset{\_}{\left(N+1\right)}}=\left(A^{\underset{\_}{N}}\right)\left(A-N\right)$$

Proof

Step	Hyp	Ref	Expression
1		rffacp1d.1	$$⊢\phi \to A\in ℂ$$
2		rffacp1d.2	$$⊢\phi \to N\in {\mathbb{N}}_0$$
3		fallfacp1	$$⊢\left(A\in ℂ\wedge N\in {\mathbb{N}}_0\right)\to A^{\underset{\_}{\left(N+1\right)}}=\left(A^{\underset{\_}{N}}\right)\left(A-N\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to A^{\underset{\_}{\left(N+1\right)}}=\left(A^{\underset{\_}{N}}\right)\left(A-N\right)$$