Theorem fprodsplit1

Description: Separate out a term in a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020)

	Ref	Expression
Hypotheses	fprodsplit1.a	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	fprodsplit1.b	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
	fprodsplit1.c	⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )
	fprodsplit1.d	⊢ ( ( 𝜑 ∧ 𝑘 = 𝐶 ) → 𝐵 = 𝐷 )
Assertion	fprodsplit1	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 = ( 𝐷 · ∏ 𝑘 ∈ ( 𝐴 ∖ { 𝐶 } ) 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		fprodsplit1.a	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		fprodsplit1.b	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
3		fprodsplit1.c	⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )
4		fprodsplit1.d	⊢ ( ( 𝜑 ∧ 𝑘 = 𝐶 ) → 𝐵 = 𝐷 )
5		nfv	⊢ Ⅎ 𝑘 𝜑
6		nfcvd	⊢ ( 𝜑 → Ⅎ 𝑘 𝐷 )
7	5 6 1 2 3 4	fprodsplit1f	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 = ( 𝐷 · ∏ 𝑘 ∈ ( 𝐴 ∖ { 𝐶 } ) 𝐵 ) )