Theorem fprod1p

Description: Separate out the first term in a finite product. (Contributed by Scott Fenton, 24-Dec-2017)

	Ref	Expression
Hypotheses	fprod1p.1	$$⊢\phi \to N\in {\mathbb{Z}}_{\ge M}$$
	fprod1p.2	$$⊢\left(\phi \wedge k\in \left(M\dots N\right)\right)\to A\in ℂ$$
	fprod1p.3	$$⊢k=M\to A=B$$
Assertion	fprod1p	$$⊢\phi \to \prod _{k=M}^{N}A=B\prod _{k=M+1}^{N}A$$

Proof

Step	Hyp	Ref	Expression
1		fprod1p.1	$$⊢\phi \to N\in {\mathbb{Z}}_{\ge M}$$
2		fprod1p.2	$$⊢\left(\phi \wedge k\in \left(M\dots N\right)\right)\to A\in ℂ$$
3		fprod1p.3	$$⊢k=M\to A=B$$
4		eluzfz1	$$⊢N\in {\mathbb{Z}}_{\ge M}\to M\in \left(M\dots N\right)$$
5	1 4	syl	$$⊢\phi \to M\in \left(M\dots N\right)$$
6		elfzelz	$$⊢M\in \left(M\dots N\right)\to M\in \mathbb{Z}$$
7	5 6	syl	$$⊢\phi \to M\in \mathbb{Z}$$
8		fzsn	$$⊢M\in \mathbb{Z}\to \left(M\dots M\right)=\left\{M\right\}$$
9	7 8	syl	$$⊢\phi \to \left(M\dots M\right)=\left\{M\right\}$$
10	9	ineq1d	$$⊢\phi \to \left(M\dots M\right)\cap \left(M+1\dots N\right)=\left\{M\right\}\cap \left(M+1\dots N\right)$$
11	7	zred	$$⊢\phi \to M\in ℝ$$
12	11	ltp1d	$$⊢\phi \to M<M+1$$
13		fzdisj	$$⊢M<M+1\to \left(M\dots M\right)\cap \left(M+1\dots N\right)=\varnothing$$
14	12 13	syl	$$⊢\phi \to \left(M\dots M\right)\cap \left(M+1\dots N\right)=\varnothing$$
15	10 14	eqtr3d	$$⊢\phi \to \left\{M\right\}\cap \left(M+1\dots N\right)=\varnothing$$
16		fzsplit	$$⊢M\in \left(M\dots N\right)\to \left(M\dots N\right)=\left(M\dots M\right)\cup \left(M+1\dots N\right)$$
17	5 16	syl	$$⊢\phi \to \left(M\dots N\right)=\left(M\dots M\right)\cup \left(M+1\dots N\right)$$
18	9	uneq1d	$$⊢\phi \to \left(M\dots M\right)\cup \left(M+1\dots N\right)=\left\{M\right\}\cup \left(M+1\dots N\right)$$
19	17 18	eqtrd	$$⊢\phi \to \left(M\dots N\right)=\left\{M\right\}\cup \left(M+1\dots N\right)$$
20		fzfid	$$⊢\phi \to \left(M\dots N\right)\in \mathrm{Fin}$$
21	15 19 20 2	fprodsplit	$$⊢\phi \to \prod _{k=M}^{N}A=\prod _{k\in \left\{M\right\}}A\prod _{k=M+1}^{N}A$$
22	3	eleq1d	$$⊢k=M\to \left(A\in ℂ\leftrightarrow B\in ℂ\right)$$
23	2	ralrimiva	$$⊢\phi \to \forall k\in \left(M\dots N\right)A\in ℂ$$
24	22 23 5	rspcdva	$$⊢\phi \to B\in ℂ$$
25	3	prodsn	$$⊢\left(M\in \left(M\dots N\right)\wedge B\in ℂ\right)\to \prod _{k\in \left\{M\right\}}A=B$$
26	5 24 25	syl2anc	$$⊢\phi \to \prod _{k\in \left\{M\right\}}A=B$$
27	26	oveq1d	$$⊢\phi \to \prod _{k\in \left\{M\right\}}A\prod _{k=M+1}^{N}A=B\prod _{k=M+1}^{N}A$$
28	21 27	eqtrd	$$⊢\phi \to \prod _{k=M}^{N}A=B\prod _{k=M+1}^{N}A$$