Theorem fallfacp1d

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

	Ref	Expression
Hypotheses	rffacp1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	rffacp1d.2	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
Assertion	fallfacp1d	⊢ ( 𝜑 → ( 𝐴 FallFac ( 𝑁 + 1 ) ) = ( ( 𝐴 FallFac 𝑁 ) · ( 𝐴 − 𝑁 ) ) )

Proof

Step	Hyp	Ref	Expression
1		rffacp1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		rffacp1d.2	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
3		fallfacp1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 FallFac ( 𝑁 + 1 ) ) = ( ( 𝐴 FallFac 𝑁 ) · ( 𝐴 − 𝑁 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 FallFac ( 𝑁 + 1 ) ) = ( ( 𝐴 FallFac 𝑁 ) · ( 𝐴 − 𝑁 ) ) )