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Theorem fprodmul 11554

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 𝑝 𝑓 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1t1e1 9030	. . . . 5 ⊢ (1 · 1) = 1
2		prod0 11548	. . . . . 6 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1
3		prod0 11548	. . . . . 6 ⊢ ∏𝑘 ∈ ∅ 𝐶 = 1
4	2, 3	oveq12i 5865	. . . . 5 ⊢ (∏𝑘 ∈ ∅ 𝐵 · ∏𝑘 ∈ ∅ 𝐶) = (1 · 1)
5		prod0 11548	. . . . 5 ⊢ ∏𝑘 ∈ ∅ (𝐵 · 𝐶) = 1
6	1, 4, 5	3eqtr4ri 2202	. . . 4 ⊢ ∏𝑘 ∈ ∅ (𝐵 · 𝐶) = (∏𝑘 ∈ ∅ 𝐵 · ∏𝑘 ∈ ∅ 𝐶)
7		prodeq1 11516	. . . 4 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = ∏𝑘 ∈ ∅ (𝐵 · 𝐶))
8		prodeq1 11516	. . . . 5 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
9		prodeq1 11516	. . . . 5 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
10	8, 9	oveq12d 5871	. . . 4 ⊢ (𝐴 = ∅ → (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶) = (∏𝑘 ∈ ∅ 𝐵 · ∏𝑘 ∈ ∅ 𝐶))
11	6, 7, 10	3eqtr4a 2229	. . 3 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶))
12	11	a1i 9	. 2 ⊢ (𝜑 → (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶)))
13		simprl 526	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ)
14		nnuz 9522	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
15	13, 14	eleqtrdi 2263	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ (ℤ_≥‘1))
16		elnnuz 9523	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℕ ↔ 𝑝 ∈ (ℤ_≥‘1))
17	16	biimpri 132	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (ℤ_≥‘1) → 𝑝 ∈ ℕ)
18	17	adantl 275	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) → 𝑝 ∈ ℕ)
19		fprodmul.2	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
20	19	fmpttd 5651	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
21	20	adantr 274	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
22		f1of 5442	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))⟶𝐴)
23	22	ad2antll 488	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
24		fco 5363	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
25	21, 23, 24	syl2anc 409	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
26	25	ad2antrr 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
27		simpr 109	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → 𝑝 ≤ (♯‘𝐴))
28		simplr 525	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → 𝑝 ∈ (ℤ_≥‘1))
29	13	ad2antrr 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℕ)
30	29	nnzd 9333	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℤ)
31		elfz5 9973	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ (ℤ_≥‘1) ∧ (♯‘𝐴) ∈ ℤ) → (𝑝 ∈ (1...(♯‘𝐴)) ↔ 𝑝 ≤ (♯‘𝐴)))
32	28, 30, 31	syl2anc 409	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (𝑝 ∈ (1...(♯‘𝐴)) ↔ 𝑝 ≤ (♯‘𝐴)))
33	27, 32	mpbird 166	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → 𝑝 ∈ (1...(♯‘𝐴)))
34	26, 33	ffvelrnd 5632	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝) ∈ ℂ)
35		1cnd 7936	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑝 ≤ (♯‘𝐴)) → 1 ∈ ℂ)
36	18	nnzd 9333	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) → 𝑝 ∈ ℤ)
37	13	adantr 274	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) → (♯‘𝐴) ∈ ℕ)
38	37	nnzd 9333	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) → (♯‘𝐴) ∈ ℤ)
39		zdcle 9288	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑝 ≤ (♯‘𝐴))
40	36, 38, 39	syl2anc 409	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) → DECID 𝑝 ≤ (♯‘𝐴))
41	34, 35, 40	ifcldadc 3555	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝), 1) ∈ ℂ)
42		breq1 3992	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑝 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑝 ≤ (♯‘𝐴)))
43		fveq2 5496	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑝 → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝))
44	42, 43	ifbieq1d 3548	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑝 → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 1) = if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝), 1))
45		eqid 2170	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 1))
46	44, 45	fvmptg 5572	. . . . . . . . . 10 ⊢ ((𝑝 ∈ ℕ ∧ if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝), 1) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 1))‘𝑝) = if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝), 1))
47	18, 41, 46	syl2anc 409	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 1))‘𝑝) = if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝), 1))
48	47, 41	eqeltrd 2247	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 1))‘𝑝) ∈ ℂ)
49		fprodmul.3	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
50	49	fmpttd 5651	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℂ)
51	50	adantr 274	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℂ)
52		fco 5363	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
53	51, 23, 52	syl2anc 409	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
54	53	ad2antrr 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
55	54, 33	ffvelrnd 5632	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝) ∈ ℂ)
56	55, 35, 40	ifcldadc 3555	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝), 1) ∈ ℂ)
57		fveq2 5496	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑝 → (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛) = (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝))
58	42, 57	ifbieq1d 3548	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑝 → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 1) = if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝), 1))
59		eqid 2170	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 1))
60	58, 59	fvmptg 5572	. . . . . . . . . 10 ⊢ ((𝑝 ∈ ℕ ∧ if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝), 1) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 1))‘𝑝) = if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝), 1))
61	18, 56, 60	syl2anc 409	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 1))‘𝑝) = if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝), 1))
62	61, 56	eqeltrd 2247	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 1))‘𝑝) ∈ ℂ)
63	23	ad2antrr 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
64	63, 33	ffvelrnd 5632	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (𝑓‘𝑝) ∈ 𝐴)
65		csbov12g 5892	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑝) ∈ 𝐴 → ⦋(𝑓‘𝑝) / 𝑘⦌(𝐵 · 𝐶) = (⦋(𝑓‘𝑝) / 𝑘⦌𝐵 · ⦋(𝑓‘𝑝) / 𝑘⦌𝐶))
66	64, 65	syl 14	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ⦋(𝑓‘𝑝) / 𝑘⦌(𝐵 · 𝐶) = (⦋(𝑓‘𝑝) / 𝑘⦌𝐵 · ⦋(𝑓‘𝑝) / 𝑘⦌𝐶))
67	19, 49	mulcld 7940	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 · 𝐶) ∈ ℂ)
68	67	ralrimiva 2543	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝐵 · 𝐶) ∈ ℂ)
69	68	ad3antrrr 489	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ∀𝑘 ∈ 𝐴 (𝐵 · 𝐶) ∈ ℂ)
70		nfcsb1v 3082	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋(𝑓‘𝑝) / 𝑘⦌(𝐵 · 𝐶)
71	70	nfel1 2323	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘⦋(𝑓‘𝑝) / 𝑘⦌(𝐵 · 𝐶) ∈ ℂ
72		csbeq1a 3058	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑓‘𝑝) → (𝐵 · 𝐶) = ⦋(𝑓‘𝑝) / 𝑘⦌(𝐵 · 𝐶))
73	72	eleq1d 2239	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑓‘𝑝) → ((𝐵 · 𝐶) ∈ ℂ ↔ ⦋(𝑓‘𝑝) / 𝑘⦌(𝐵 · 𝐶) ∈ ℂ))
74	71, 73	rspc 2828	. . . . . . . . . . . . . . 15 ⊢ ((𝑓‘𝑝) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 (𝐵 · 𝐶) ∈ ℂ → ⦋(𝑓‘𝑝) / 𝑘⦌(𝐵 · 𝐶) ∈ ℂ))
75	64, 69, 74	sylc 62	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ⦋(𝑓‘𝑝) / 𝑘⦌(𝐵 · 𝐶) ∈ ℂ)
76		eqid 2170	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) = (𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))
77	76	fvmpts 5574	. . . . . . . . . . . . . 14 ⊢ (((𝑓‘𝑝) ∈ 𝐴 ∧ ⦋(𝑓‘𝑝) / 𝑘⦌(𝐵 · 𝐶) ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑝)) = ⦋(𝑓‘𝑝) / 𝑘⦌(𝐵 · 𝐶))
78	64, 75, 77	syl2anc 409	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑝)) = ⦋(𝑓‘𝑝) / 𝑘⦌(𝐵 · 𝐶))
79	19	ralrimiva 2543	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
80	79	ad3antrrr 489	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
81		nfcsb1v 3082	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘⦋(𝑓‘𝑝) / 𝑘⦌𝐵
82	81	nfel1 2323	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋(𝑓‘𝑝) / 𝑘⦌𝐵 ∈ ℂ
83		csbeq1a 3058	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑓‘𝑝) → 𝐵 = ⦋(𝑓‘𝑝) / 𝑘⦌𝐵)
84	83	eleq1d 2239	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑓‘𝑝) → (𝐵 ∈ ℂ ↔ ⦋(𝑓‘𝑝) / 𝑘⦌𝐵 ∈ ℂ))
85	82, 84	rspc 2828	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘𝑝) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝑓‘𝑝) / 𝑘⦌𝐵 ∈ ℂ))
86	64, 80, 85	sylc 62	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ⦋(𝑓‘𝑝) / 𝑘⦌𝐵 ∈ ℂ)
87		eqid 2170	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
88	87	fvmpts 5574	. . . . . . . . . . . . . . 15 ⊢ (((𝑓‘𝑝) ∈ 𝐴 ∧ ⦋(𝑓‘𝑝) / 𝑘⦌𝐵 ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑝)) = ⦋(𝑓‘𝑝) / 𝑘⦌𝐵)
89	64, 86, 88	syl2anc 409	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑝)) = ⦋(𝑓‘𝑝) / 𝑘⦌𝐵)
90	49	ralrimiva 2543	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ)
91	90	ad3antrrr 489	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ)
92		nfcsb1v 3082	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘⦋(𝑓‘𝑝) / 𝑘⦌𝐶
93	92	nfel1 2323	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋(𝑓‘𝑝) / 𝑘⦌𝐶 ∈ ℂ
94		csbeq1a 3058	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑓‘𝑝) → 𝐶 = ⦋(𝑓‘𝑝) / 𝑘⦌𝐶)
95	94	eleq1d 2239	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑓‘𝑝) → (𝐶 ∈ ℂ ↔ ⦋(𝑓‘𝑝) / 𝑘⦌𝐶 ∈ ℂ))
96	93, 95	rspc 2828	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘𝑝) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ → ⦋(𝑓‘𝑝) / 𝑘⦌𝐶 ∈ ℂ))
97	64, 91, 96	sylc 62	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ⦋(𝑓‘𝑝) / 𝑘⦌𝐶 ∈ ℂ)
98		eqid 2170	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ 𝐶)
99	98	fvmpts 5574	. . . . . . . . . . . . . . 15 ⊢ (((𝑓‘𝑝) ∈ 𝐴 ∧ ⦋(𝑓‘𝑝) / 𝑘⦌𝐶 ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑝)) = ⦋(𝑓‘𝑝) / 𝑘⦌𝐶)
100	64, 97, 99	syl2anc 409	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑝)) = ⦋(𝑓‘𝑝) / 𝑘⦌𝐶)
101	89, 100	oveq12d 5871	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑝)) · ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑝))) = (⦋(𝑓‘𝑝) / 𝑘⦌𝐵 · ⦋(𝑓‘𝑝) / 𝑘⦌𝐶))
102	66, 78, 101	3eqtr4d 2213	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑝)) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑝)) · ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑝))))
103		fvco3 5567	. . . . . . . . . . . . 13 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑝 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝) = ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑝)))
104	63, 33, 103	syl2anc 409	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝) = ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑝)))
105		fvco3 5567	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑝 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑝)))
106	63, 33, 105	syl2anc 409	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑝)))
107		fvco3 5567	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑝 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑝)))
108	63, 33, 107	syl2anc 409	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑝)))
109	106, 108	oveq12d 5871	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ((((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝) · (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝)) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑝)) · ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑝))))
110	102, 104, 109	3eqtr4d 2213	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝) = ((((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝) · (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝)))
111	27	iftrued 3533	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝), 1) = (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝))
112	27	iftrued 3533	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝), 1) = (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝))
113	27	iftrued 3533	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝), 1) = (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝))
114	112, 113	oveq12d 5871	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝), 1) · if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝), 1)) = ((((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝) · (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝)))
115	110, 111, 114	3eqtr4d 2213	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝), 1) = (if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝), 1) · if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝), 1)))
116	1	eqcomi 2174	. . . . . . . . . . 11 ⊢ 1 = (1 · 1)
117		simpr 109	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑝 ≤ (♯‘𝐴)) → ¬ 𝑝 ≤ (♯‘𝐴))
118	117	iffalsed 3536	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑝 ≤ (♯‘𝐴)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝), 1) = 1)
119	117	iffalsed 3536	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑝 ≤ (♯‘𝐴)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝), 1) = 1)
120	117	iffalsed 3536	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑝 ≤ (♯‘𝐴)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝), 1) = 1)
121	119, 120	oveq12d 5871	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑝 ≤ (♯‘𝐴)) → (if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝), 1) · if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝), 1)) = (1 · 1))
122	116, 118, 121	3eqtr4a 2229	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑝 ≤ (♯‘𝐴)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝), 1) = (if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝), 1) · if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝), 1)))
123		exmiddc 831	. . . . . . . . . . 11 ⊢ (DECID 𝑝 ≤ (♯‘𝐴) → (𝑝 ≤ (♯‘𝐴) ∨ ¬ 𝑝 ≤ (♯‘𝐴)))
124	40, 123	syl 14	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) → (𝑝 ≤ (♯‘𝐴) ∨ ¬ 𝑝 ≤ (♯‘𝐴)))
125	115, 122, 124	mpjaodan 793	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝), 1) = (if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝), 1) · if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝), 1)))
126	78, 75	eqeltrd 2247	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑝)) ∈ ℂ)
127	104, 126	eqeltrd 2247	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝) ∈ ℂ)
128	127, 35, 40	ifcldadc 3555	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝), 1) ∈ ℂ)
129		fveq2 5496	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑝 → (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛) = (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝))
130	42, 129	ifbieq1d 3548	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑝 → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛), 1) = if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝), 1))
131		eqid 2170	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛), 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛), 1))
132	130, 131	fvmptg 5572	. . . . . . . . . 10 ⊢ ((𝑝 ∈ ℕ ∧ if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝), 1) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛), 1))‘𝑝) = if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝), 1))
133	18, 128, 132	syl2anc 409	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛), 1))‘𝑝) = if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝), 1))
134	47, 61	oveq12d 5871	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) → (((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 1))‘𝑝) · ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 1))‘𝑝)) = (if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝), 1) · if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝), 1)))
135	125, 133, 134	3eqtr4d 2213	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛), 1))‘𝑝) = (((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 1))‘𝑝) · ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 1))‘𝑝)))
136	15, 48, 62, 135	prod3fmul 11504	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛), 1)))‘(♯‘𝐴)) = ((seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 1)))‘(♯‘𝐴)) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 1)))‘(♯‘𝐴))))
137		fveq2 5496	. . . . . . . 8 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑛)))
138		simprr 527	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
139	67	fmpttd 5651	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)):𝐴⟶ℂ)
140	139	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)):𝐴⟶ℂ)
141	140	ffvelrnda 5631	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑚) ∈ ℂ)
142		fvco3 5567	. . . . . . . . 9 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑛)))
143	23, 142	sylan 281	. . . . . . . 8 ⊢ (((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)))
144	137, 13, 138, 141, 143	fprodseq 11546	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛), 1)))‘(♯‘𝐴)))
145		fveq2 5496	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
146	21	ffvelrnda 5631	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) ∈ ℂ)
147		fvco3 5567	. . . . . . . . . 10 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
148	23, 147	sylan 281	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
149	145, 13, 138, 146, 148	fprodseq 11546	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 1)))‘(♯‘𝐴)))
150		fveq2 5496	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
151	51	ffvelrnda 5631	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) ∈ ℂ)
152		fvco3 5567	. . . . . . . . . 10 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
153	23, 152	sylan 281	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
154	150, 13, 138, 151, 153	fprodseq 11546	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 1)))‘(♯‘𝐴)))
155	149, 154	oveq12d 5871	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) · ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚)) = ((seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 1)))‘(♯‘𝐴)) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 1)))‘(♯‘𝐴))))
156	136, 144, 155	3eqtr4d 2213	. . . . . 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)))
157		prodfct 11550	. . . . . . . 8 ⊢ (∀𝑘 ∈ 𝐴 (𝐵 · 𝐶) ∈ ℂ → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶))
158	68, 157	syl 14	. . . . . . 7 ⊢ (𝜑 → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶))
159	158	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶))
160		prodfct 11550	. . . . . . . . 9 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐵)
161	79, 160	syl 14	. . . . . . . 8 ⊢ (𝜑 → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐵)
162		prodfct 11550	. . . . . . . . 9 ⊢ (∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐶)
163	90, 162	syl 14	. . . . . . . 8 ⊢ (𝜑 → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐶)
164	161, 163	oveq12d 5871	. . . . . . 7 ⊢ (𝜑 → (∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) · ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚)) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶))
165	164	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) · ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚)) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶))
166	156, 159, 165	3eqtr3d 2211	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶))
167	166	expr 373	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶)))
168	167	exlimdv 1812	. . 3 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶)))
169	168	expimpd 361	. 2 ⊢ (𝜑 → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶)))
170		fprodmul.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
171		fz1f1o 11338	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
172	170, 171	syl 14	. 2 ⊢ (𝜑 → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
173	12, 169, 172	mpjaod 713	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 703 DECID wdc 829 = wceq 1348 ∃wex 1485 ∈ wcel 2141 ∀wral 2448 ⦋csb 3049 ∅c0 3414 ifcif 3526 class class class wbr 3989 ↦ cmpt 4050 ∘ ccom 4615 ⟶wf 5194 –1-1-onto→wf1o 5197 ‘cfv 5198 (class class class)co 5853 Fincfn 6718 ℂcc 7772 1c1 7775 · cmul 7779 ≤ cle 7955 ℕcn 8878 ℤcz 9212 ℤ_≥cuz 9487 ...cfz 9965 seqcseq 10401 ♯chash 10709 ∏cprod 11513

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-oadd 6399 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fzo 10099 df-seqfrec 10402 df-exp 10476 df-ihash 10710 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-proddc 11514

This theorem is referenced by: fprodsplitdc 11559 fproddivap 11593

W3C validator