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Theorem fprodmul 15900

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 12370	. . . . 5 ⊢ (1 · 1) = 1
2		prod0 15883	. . . . . 6 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1
3		prod0 15883	. . . . . 6 ⊢ ∏𝑘 ∈ ∅ 𝐶 = 1
4	2, 3	oveq12i 7417	. . . . 5 ⊢ (∏𝑘 ∈ ∅ 𝐵 · ∏𝑘 ∈ ∅ 𝐶) = (1 · 1)
5		prod0 15883	. . . . 5 ⊢ ∏𝑘 ∈ ∅ (𝐵 · 𝐶) = 1
6	1, 4, 5	3eqtr4ri 2771	. . . 4 ⊢ ∏𝑘 ∈ ∅ (𝐵 · 𝐶) = (∏𝑘 ∈ ∅ 𝐵 · ∏𝑘 ∈ ∅ 𝐶)
7		prodeq1 15849	. . . 4 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = ∏𝑘 ∈ ∅ (𝐵 · 𝐶))
8		prodeq1 15849	. . . . 5 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
9		prodeq1 15849	. . . . 5 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
10	8, 9	oveq12d 7423	. . . 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 769	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ)
14		nnuz 12861	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
15	13, 14	eleqtrdi 2843	. . . . . . . 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 7111	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
18	17	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
19		f1of 6830	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))⟶𝐴)
20	19	ad2antll 727	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
21		fco 6738	. . . . . . . . . 10 ⊢ (((𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
22	18, 20, 21	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
23	22	ffvelcdmda 7083	. . . . . . . 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 7111	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℂ)
26	25	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℂ)
27		fco 6738	. . . . . . . . . 10 ⊢ (((𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
28	26, 20, 27	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
29	28	ffvelcdmda 7083	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛) ∈ ℂ)
30	20	ffvelcdmda 7083	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝑓‘𝑛) ∈ 𝐴)
31		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
32	16, 24	mulcld 11230	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 · 𝐶) ∈ ℂ)
33		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) = (𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))
34	33	fvmpt2 7006	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ 𝐴 ∧ (𝐵 · 𝐶) ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑘) = (𝐵 · 𝐶))
35	31, 32, 34	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑘) = (𝐵 · 𝐶))
36		eqid 2732	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
37	36	fvmpt2 7006	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐵 ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = 𝐵)
38	31, 16, 37	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = 𝐵)
39		eqid 2732	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ 𝐶)
40	39	fvmpt2 7006	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐶 ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = 𝐶)
41	31, 24, 40	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = 𝐶)
42	38, 41	oveq12d 7423	. . . . . . . . . . . . 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 3146	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑘) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) · ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘)))
45	44	ad2antrr 724	. . . . . . . . . 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 6899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑛))
47		nffvmpt1 6899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛))
48		nfcv 2903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 ·
49		nffvmpt1 6899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛))
50	47, 48, 49	nfov 7435	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)) · ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
51	46, 50	nfeq 2916	. . . . . . . . . . 11 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑛)) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)) · ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
52		fveq2 6888	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑘) = ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑛)))
53		fveq2 6888	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
54		fveq2 6888	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
55	53, 54	oveq12d 7423	. . . . . . . . . . . 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 2748	. . . . . . . . . . 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 3600	. . . . . . . . . 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 6987	. . . . . . . . . 10 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑛)))
60	20, 59	sylan 580	. . . . . . . . 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 6987	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
62	20, 61	sylan 580	. . . . . . . . . 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 6987	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
64	20, 63	sylan 580	. . . . . . . . . 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 7423	. . . . . . . . 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 15832	. . . . . . 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 6888	. . . . . . . 8 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑛)))
69		simprr 771	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
70	32	fmpttd 7111	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)):𝐴⟶ℂ)
71	70	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)):𝐴⟶ℂ)
72	71	ffvelcdmda 7083	. . . . . . . 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 15881	. . . . . . 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 6888	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
75	18	ffvelcdmda 7083	. . . . . . . . 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 15881	. . . . . . . 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 6888	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
78	26	ffvelcdmda 7083	. . . . . . . . 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 15881	. . . . . . . 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 7423	. . . . . . 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 15885	. . . . . 6 ⊢ ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶)
83		prodfc 15885	. . . . . . 7 ⊢ ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐵
84		prodfc 15885	. . . . . . 7 ⊢ ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐶
85	83, 84	oveq12i 7417	. . . . . 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 457	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶)))
88	87	exlimdv 1936	. . 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 454	. 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 15652	. . 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 858	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∨ wo 845 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∀wral 3061 ∅c0 4321 ↦ cmpt 5230 ∘ ccom 5679 ⟶wf 6536 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7405 Fincfn 8935 ℂcc 11104 1c1 11107 · cmul 11111 ℕcn 12208 ℤ_≥cuz 12818 ...cfz 13480 seqcseq 13962 ♯chash 14286 ∏cprod 15845

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-prod 15846

This theorem is referenced by: fprodsplit 15906 risefallfac 15964 gausslemma2dlem5 26863 gausslemma2dlem6 26864 bcprod 34696

W3C validator