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Theorem fprodmul 15916

Description: The product of two finite products. (Contributed by Scott Fenton, 14-Dec-2017.)

**Hypotheses**
Ref	Expression
fprodmul.1	⊢ (𝜑 → 𝐴 ∈ Fin)
fprodmul.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
fprodmul.3	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)

**Assertion**
Ref	Expression
fprodmul	⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶))

Distinct variable groups: 𝐴,𝑘 𝜑,𝑘 Allowed substitution hints: 𝐵(𝑘) 𝐶(𝑘)

Proof of Theorem fprodmul Dummy variables 𝑓 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1t1e1 12329	. . . . 5 ⊢ (1 · 1) = 1
2		prod0 15899	. . . . . 6 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1
3		prod0 15899	. . . . . 6 ⊢ ∏𝑘 ∈ ∅ 𝐶 = 1
4	2, 3	oveq12i 7372	. . . . 5 ⊢ (∏𝑘 ∈ ∅ 𝐵 · ∏𝑘 ∈ ∅ 𝐶) = (1 · 1)
5		prod0 15899	. . . . 5 ⊢ ∏𝑘 ∈ ∅ (𝐵 · 𝐶) = 1
6	1, 4, 5	3eqtr4ri 2771	. . . 4 ⊢ ∏𝑘 ∈ ∅ (𝐵 · 𝐶) = (∏𝑘 ∈ ∅ 𝐵 · ∏𝑘 ∈ ∅ 𝐶)
7		prodeq1 15863	. . . 4 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = ∏𝑘 ∈ ∅ (𝐵 · 𝐶))
8		prodeq1 15863	. . . . 5 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
9		prodeq1 15863	. . . . 5 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
10	8, 9	oveq12d 7378	. . . 4 ⊢ (𝐴 = ∅ → (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶) = (∏𝑘 ∈ ∅ 𝐵 · ∏𝑘 ∈ ∅ 𝐶))
11	6, 7, 10	3eqtr4a 2798	. . 3 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶))
12	11	a1i 11	. 2 ⊢ (𝜑 → (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶)))
13		simprl 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ)
14		nnuz 12818	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
15	13, 14	eleqtrdi 2847	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ (ℤ_≥‘1))
16		fprodmul.2	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
17	16	fmpttd 7061	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
18	17	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
19		f1of 6774	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))⟶𝐴)
20	19	ad2antll 730	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
21		fco 6686	. . . . . . . . . 10 ⊢ (((𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
22	18, 20, 21	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
23	22	ffvelcdmda 7030	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) ∈ ℂ)
24		fprodmul.3	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
25	24	fmpttd 7061	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℂ)
26	25	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℂ)
27		fco 6686	. . . . . . . . . 10 ⊢ (((𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
28	26, 20, 27	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
29	28	ffvelcdmda 7030	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛) ∈ ℂ)
30	20	ffvelcdmda 7030	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝑓‘𝑛) ∈ 𝐴)
31		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
32	16, 24	mulcld 11156	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 · 𝐶) ∈ ℂ)
33		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) = (𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))
34	33	fvmpt2 6953	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ 𝐴 ∧ (𝐵 · 𝐶) ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑘) = (𝐵 · 𝐶))
35	31, 32, 34	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑘) = (𝐵 · 𝐶))
36		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
37	36	fvmpt2 6953	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐵 ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = 𝐵)
38	31, 16, 37	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = 𝐵)
39		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ 𝐶)
40	39	fvmpt2 6953	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐶 ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = 𝐶)
41	31, 24, 40	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = 𝐶)
42	38, 41	oveq12d 7378	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) · ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘)) = (𝐵 · 𝐶))
43	35, 42	eqtr4d 2775	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑘) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) · ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘)))
44	43	ralrimiva 3130	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑘) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) · ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘)))
45	44	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ∀𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑘) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) · ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘)))
46		nffvmpt1 6845	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑛))
47		nffvmpt1 6845	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛))
48		nfcv 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 ·
49		nffvmpt1 6845	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛))
50	47, 48, 49	nfov 7390	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)) · ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
51	46, 50	nfeq 2913	. . . . . . . . . . 11 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑛)) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)) · ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
52		fveq2 6834	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑘) = ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑛)))
53		fveq2 6834	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
54		fveq2 6834	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
55	53, 54	oveq12d 7378	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑓‘𝑛) → (((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) · ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘)) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)) · ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛))))
56	52, 55	eqeq12d 2753	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑓‘𝑛) → (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑘) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) · ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘)) ↔ ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑛)) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)) · ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))))
57	51, 56	rspc 3553	. . . . . . . . . 10 ⊢ ((𝑓‘𝑛) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑘) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) · ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘)) → ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑛)) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)) · ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))))
58	30, 45, 57	sylc 65	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑛)) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)) · ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛))))
59		fvco3 6933	. . . . . . . . . 10 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑛)))
60	20, 59	sylan 581	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑛)))
61		fvco3 6933	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
62	20, 61	sylan 581	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
63		fvco3 6933	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
64	20, 63	sylan 581	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
65	62, 64	oveq12d 7378	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) · (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛)) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)) · ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛))))
66	58, 60, 65	3eqtr4d 2782	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛) = ((((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) · (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛)))
67	15, 23, 29, 66	prodfmul 15846	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (seq1( · , ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓))‘(♯‘𝐴)) = ((seq1( · , ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓))‘(♯‘𝐴)) · (seq1( · , ((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓))‘(♯‘𝐴))))
68		fveq2 6834	. . . . . . . 8 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑛)))
69		simprr 773	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
70	32	fmpttd 7061	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)):𝐴⟶ℂ)
71	70	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)):𝐴⟶ℂ)
72	71	ffvelcdmda 7030	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑚) ∈ ℂ)
73	68, 13, 69, 72, 60	fprod 15897	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = (seq1( · , ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓))‘(♯‘𝐴)))
74		fveq2 6834	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
75	18	ffvelcdmda 7030	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) ∈ ℂ)
76	74, 13, 69, 75, 62	fprod 15897	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = (seq1( · , ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓))‘(♯‘𝐴)))
77		fveq2 6834	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
78	26	ffvelcdmda 7030	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) ∈ ℂ)
79	77, 13, 69, 78, 64	fprod 15897	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = (seq1( · , ((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓))‘(♯‘𝐴)))
80	76, 79	oveq12d 7378	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) · ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚)) = ((seq1( · , ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓))‘(♯‘𝐴)) · (seq1( · , ((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓))‘(♯‘𝐴))))
81	67, 73, 80	3eqtr4d 2782	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = (∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) · ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚)))
82		prodfc 15901	. . . . . 6 ⊢ ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶)
83		prodfc 15901	. . . . . . 7 ⊢ ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐵
84		prodfc 15901	. . . . . . 7 ⊢ ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐶
85	83, 84	oveq12i 7372	. . . . . 6 ⊢ (∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) · ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚)) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶)
86	81, 82, 85	3eqtr3g 2795	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶))
87	86	expr 456	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶)))
88	87	exlimdv 1935	. . 3 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶)))
89	88	expimpd 453	. 2 ⊢ (𝜑 → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶)))
90		fprodmul.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
91		fz1f1o 15663	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
92	90, 91	syl 17	. 2 ⊢ (𝜑 → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
93	12, 89, 92	mpjaod 861	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ∀wral 3052 ∅c0 4274 ↦ cmpt 5167 ∘ ccom 5628 ⟶wf 6488 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7360 Fincfn 8886 ℂcc 11027 1c1 11030 · cmul 11034 ℕcn 12165 ℤ_≥cuz 12779 ...cfz 13452 seqcseq 13954 ♯chash 14283 ∏cprod 15859

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9348 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-n0 12429 df-z 12516 df-uz 12780 df-rp 12934 df-fz 13453 df-fzo 13600 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-prod 15860

This theorem is referenced by: fprodsplit 15922 risefallfac 15980 gausslemma2dlem5 27348 gausslemma2dlem6 27349 bcprod 35936

W3C validator