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Theorem fprodmul 15895

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 12314	. . . . 5 ⊢ (1 · 1) = 1
2		prod0 15878	. . . . . 6 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1
3		prod0 15878	. . . . . 6 ⊢ ∏𝑘 ∈ ∅ 𝐶 = 1
4	2, 3	oveq12i 7380	. . . . 5 ⊢ (∏𝑘 ∈ ∅ 𝐵 · ∏𝑘 ∈ ∅ 𝐶) = (1 · 1)
5		prod0 15878	. . . . 5 ⊢ ∏𝑘 ∈ ∅ (𝐵 · 𝐶) = 1
6	1, 4, 5	3eqtr4ri 2771	. . . 4 ⊢ ∏𝑘 ∈ ∅ (𝐵 · 𝐶) = (∏𝑘 ∈ ∅ 𝐵 · ∏𝑘 ∈ ∅ 𝐶)
7		prodeq1 15842	. . . 4 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = ∏𝑘 ∈ ∅ (𝐵 · 𝐶))
8		prodeq1 15842	. . . . 5 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
9		prodeq1 15842	. . . . 5 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
10	8, 9	oveq12d 7386	. . . 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 12802	. . . . . . . . 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 7069	. . . . . . . . . . 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 6782	. . . . . . . . . . 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 6694	. . . . . . . . . 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 7038	. . . . . . . 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 7069	. . . . . . . . . . 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 6694	. . . . . . . . . 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 7038	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛) ∈ ℂ)
30	20	ffvelcdmda 7038	. . . . . . . . . 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 11164	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 · 𝐶) ∈ ℂ)
33		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) = (𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))
34	33	fvmpt2 6961	. . . . . . . . . . . . . 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 6961	. . . . . . . . . . . . . . 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 6961	. . . . . . . . . . . . . . 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 7386	. . . . . . . . . . . . 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 6853	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑛))
47		nffvmpt1 6853	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛))
48		nfcv 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 ·
49		nffvmpt1 6853	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛))
50	47, 48, 49	nfov 7398	. . . . . . . . . . . 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 6842	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑘) = ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑛)))
53		fveq2 6842	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
54		fveq2 6842	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
55	53, 54	oveq12d 7386	. . . . . . . . . . . 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 3566	. . . . . . . . . 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 6941	. . . . . . . . . 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 6941	. . . . . . . . . . 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 6941	. . . . . . . . . . 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 7386	. . . . . . . . 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 15825	. . . . . . 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 6842	. . . . . . . 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 7069	. . . . . . . . . 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 7038	. . . . . . . 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 15876	. . . . . . 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 6842	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
75	18	ffvelcdmda 7038	. . . . . . . . 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 15876	. . . . . . . 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 6842	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
78	26	ffvelcdmda 7038	. . . . . . . . 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 15876	. . . . . . . 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 7386	. . . . . . 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 15880	. . . . . 6 ⊢ ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶)
83		prodfc 15880	. . . . . . 7 ⊢ ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐵
84		prodfc 15880	. . . . . . 7 ⊢ ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐶
85	83, 84	oveq12i 7380	. . . . . 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 15645	. . 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 4287 ↦ cmpt 5181 ∘ ccom 5636 ⟶wf 6496 –1-1-onto→wf1o 6499 ‘cfv 6500 (class class class)co 7368 Fincfn 8895 ℂcc 11036 1c1 11039 · cmul 11043 ℕcn 12157 ℤ_≥cuz 12763 ...cfz 13435 seqcseq 13936 ♯chash 14265 ∏cprod 15838

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 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

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 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-n0 12414 df-z 12501 df-uz 12764 df-rp 12918 df-fz 13436 df-fzo 13583 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-prod 15839

This theorem is referenced by: fprodsplit 15901 risefallfac 15959 gausslemma2dlem5 27350 gausslemma2dlem6 27351 bcprod 35951

W3C validator