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Theorem fprodmul 12285

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 9395	. . . . 5 ⊢ (1 · 1) = 1
2		prod0 12279	. . . . . 6 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1
3		prod0 12279	. . . . . 6 ⊢ ∏𝑘 ∈ ∅ 𝐶 = 1
4	2, 3	oveq12i 6064	. . . . 5 ⊢ (∏𝑘 ∈ ∅ 𝐵 · ∏𝑘 ∈ ∅ 𝐶) = (1 · 1)
5		prod0 12279	. . . . 5 ⊢ ∏𝑘 ∈ ∅ (𝐵 · 𝐶) = 1
6	1, 4, 5	3eqtr4ri 2266	. . . 4 ⊢ ∏𝑘 ∈ ∅ (𝐵 · 𝐶) = (∏𝑘 ∈ ∅ 𝐵 · ∏𝑘 ∈ ∅ 𝐶)
7		prodeq1 12247	. . . 4 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = ∏𝑘 ∈ ∅ (𝐵 · 𝐶))
8		prodeq1 12247	. . . . 5 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
9		prodeq1 12247	. . . . 5 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
10	8, 9	oveq12d 6070	. . . 4 ⊢ (𝐴 = ∅ → (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶) = (∏𝑘 ∈ ∅ 𝐵 · ∏𝑘 ∈ ∅ 𝐶))
11	6, 7, 10	3eqtr4a 2293	. . 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 531	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ)
14		nnuz 9896	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
15	13, 14	eleqtrdi 2327	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ (ℤ_≥‘1))
16		elnnuz 9897	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℕ ↔ 𝑝 ∈ (ℤ_≥‘1))
17	16	biimpri 133	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (ℤ_≥‘1) → 𝑝 ∈ ℕ)
18	17	adantl 277	. . . . . . . . . 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 5834	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
21	20	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
22		f1of 5616	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))⟶𝐴)
23	22	ad2antll 491	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
24		fco 5529	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
25	21, 23, 24	syl2anc 411	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
26	25	ad2antrr 488	. . . . . . . . . . . 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 110	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → 𝑝 ≤ (♯‘𝐴))
28		simplr 529	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → 𝑝 ∈ (ℤ_≥‘1))
29	13	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℕ)
30	29	nnzd 9705	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℤ)
31		elfz5 10357	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ (ℤ_≥‘1) ∧ (♯‘𝐴) ∈ ℤ) → (𝑝 ∈ (1...(♯‘𝐴)) ↔ 𝑝 ≤ (♯‘𝐴)))
32	28, 30, 31	syl2anc 411	. . . . . . . . . . . . 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 167	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → 𝑝 ∈ (1...(♯‘𝐴)))
34	26, 33	ffvelcdmd 5815	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝) ∈ ℂ)
35		1cnd 8295	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑝 ≤ (♯‘𝐴)) → 1 ∈ ℂ)
36	18	nnzd 9705	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) → 𝑝 ∈ ℤ)
37	13	adantr 276	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) → (♯‘𝐴) ∈ ℕ)
38	37	nnzd 9705	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) → (♯‘𝐴) ∈ ℤ)
39		zdcle 9659	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑝 ≤ (♯‘𝐴))
40	36, 38, 39	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) → DECID 𝑝 ≤ (♯‘𝐴))
41	34, 35, 40	ifcldadc 3654	. . . . . . . . . 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 4114	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑝 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑝 ≤ (♯‘𝐴)))
43		fveq2 5672	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑝 → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝))
44	42, 43	ifbieq1d 3647	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑝 → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 1) = if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝), 1))
45		eqid 2234	. . . . . . . . . . 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 5755	. . . . . . . . . 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 411	. . . . . . . . 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 2311	. . . . . . . 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 5834	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℂ)
51	50	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℂ)
52		fco 5529	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
53	51, 23, 52	syl2anc 411	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
54	53	ad2antrr 488	. . . . . . . . . . . 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	ffvelcdmd 5815	. . . . . . . . . . 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 3654	. . . . . . . . . 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 5672	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑝 → (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛) = (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝))
58	42, 57	ifbieq1d 3647	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑝 → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 1) = if(𝑝 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝), 1))
59		eqid 2234	. . . . . . . . . . 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 5755	. . . . . . . . . 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 411	. . . . . . . . 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 2311	. . . . . . . 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 488	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
64	63, 33	ffvelcdmd 5815	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (𝑓‘𝑝) ∈ 𝐴)
65		csbov12g 6092	. . . . . . . . . . . . . 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 8299	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 · 𝐶) ∈ ℂ)
68	67	ralrimiva 2617	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝐵 · 𝐶) ∈ ℂ)
69	68	ad3antrrr 492	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ∀𝑘 ∈ 𝐴 (𝐵 · 𝐶) ∈ ℂ)
70		nfcsb1v 3173	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋(𝑓‘𝑝) / 𝑘⦌(𝐵 · 𝐶)
71	70	nfel1 2397	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘⦋(𝑓‘𝑝) / 𝑘⦌(𝐵 · 𝐶) ∈ ℂ
72		csbeq1a 3149	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑓‘𝑝) → (𝐵 · 𝐶) = ⦋(𝑓‘𝑝) / 𝑘⦌(𝐵 · 𝐶))
73	72	eleq1d 2303	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑓‘𝑝) → ((𝐵 · 𝐶) ∈ ℂ ↔ ⦋(𝑓‘𝑝) / 𝑘⦌(𝐵 · 𝐶) ∈ ℂ))
74	71, 73	rspc 2917	. . . . . . . . . . . . . . 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 2234	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) = (𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))
77	76	fvmpts 5757	. . . . . . . . . . . . . 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 411	. . . . . . . . . . . . 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 2617	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
80	79	ad3antrrr 492	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
81		nfcsb1v 3173	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘⦋(𝑓‘𝑝) / 𝑘⦌𝐵
82	81	nfel1 2397	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋(𝑓‘𝑝) / 𝑘⦌𝐵 ∈ ℂ
83		csbeq1a 3149	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑓‘𝑝) → 𝐵 = ⦋(𝑓‘𝑝) / 𝑘⦌𝐵)
84	83	eleq1d 2303	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑓‘𝑝) → (𝐵 ∈ ℂ ↔ ⦋(𝑓‘𝑝) / 𝑘⦌𝐵 ∈ ℂ))
85	82, 84	rspc 2917	. . . . . . . . . . . . . . . 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 2234	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
88	87	fvmpts 5757	. . . . . . . . . . . . . . 15 ⊢ (((𝑓‘𝑝) ∈ 𝐴 ∧ ⦋(𝑓‘𝑝) / 𝑘⦌𝐵 ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑝)) = ⦋(𝑓‘𝑝) / 𝑘⦌𝐵)
89	64, 86, 88	syl2anc 411	. . . . . . . . . . . . . 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 2617	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ)
91	90	ad3antrrr 492	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ)
92		nfcsb1v 3173	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘⦋(𝑓‘𝑝) / 𝑘⦌𝐶
93	92	nfel1 2397	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋(𝑓‘𝑝) / 𝑘⦌𝐶 ∈ ℂ
94		csbeq1a 3149	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑓‘𝑝) → 𝐶 = ⦋(𝑓‘𝑝) / 𝑘⦌𝐶)
95	94	eleq1d 2303	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑓‘𝑝) → (𝐶 ∈ ℂ ↔ ⦋(𝑓‘𝑝) / 𝑘⦌𝐶 ∈ ℂ))
96	93, 95	rspc 2917	. . . . . . . . . . . . . . . 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 2234	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ 𝐶)
99	98	fvmpts 5757	. . . . . . . . . . . . . . 15 ⊢ (((𝑓‘𝑝) ∈ 𝐴 ∧ ⦋(𝑓‘𝑝) / 𝑘⦌𝐶 ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑝)) = ⦋(𝑓‘𝑝) / 𝑘⦌𝐶)
100	64, 97, 99	syl2anc 411	. . . . . . . . . . . . . 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 6070	. . . . . . . . . . . . 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 2277	. . . . . . . . . . . 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 5750	. . . . . . . . . . . . 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 411	. . . . . . . . . . . 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 5750	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑝 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑝) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑝)))
106	63, 33, 105	syl2anc 411	. . . . . . . . . . . . 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 5750	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑝 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑝) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑝)))
108	63, 33, 107	syl2anc 411	. . . . . . . . . . . . 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 6070	. . . . . . . . . . . 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 2277	. . . . . . . . . . 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 3631	. . . . . . . . . . 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 3631	. . . . . . . . . . . 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 3631	. . . . . . . . . . . 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 6070	. . . . . . . . . . 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 2277	. . . . . . . . . 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 2238	. . . . . . . . . . 11 ⊢ 1 = (1 · 1)
117		simpr 110	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑝 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑝 ≤ (♯‘𝐴)) → ¬ 𝑝 ≤ (♯‘𝐴))
118	117	iffalsed 3634	. . . . . . . . . . 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 3634	. . . . . . . . . . . 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 3634	. . . . . . . . . . . 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 6070	. . . . . . . . . . 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 2293	. . . . . . . . . 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 844	. . . . . . . . . . 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 806	. . . . . . . . 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 2311	. . . . . . . . . . . 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 2311	. . . . . . . . . . 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 3654	. . . . . . . . . 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 5672	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑝 → (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛) = (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝))
130	42, 129	ifbieq1d 3647	. . . . . . . . . . 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 2234	. . . . . . . . . . 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 5755	. . . . . . . . . 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 411	. . . . . . . . 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 6070	. . . . . . . . 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 2277	. . . . . . . 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 12235	. . . . . . 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 5672	. . . . . . . 8 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑛)))
138		simprr 533	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
139	67	fmpttd 5834	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)):𝐴⟶ℂ)
140	139	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)):𝐴⟶ℂ)
141	140	ffvelcdmda 5814	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑚) ∈ ℂ)
142		fvco3 5750	. . . . . . . . 9 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘(𝑓‘𝑛)))
143	23, 142	sylan 283	. . . . . . . 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 12277	. . . . . . 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 5672	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
146	21	ffvelcdmda 5814	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) ∈ ℂ)
147		fvco3 5750	. . . . . . . . . 10 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
148	23, 147	sylan 283	. . . . . . . . 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 12277	. . . . . . . 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 5672	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
151	51	ffvelcdmda 5814	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) ∈ ℂ)
152		fvco3 5750	. . . . . . . . . 10 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
153	23, 152	sylan 283	. . . . . . . . 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 12277	. . . . . . . 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 6070	. . . . . . 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 2277	. . . . . 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 12281	. . . . . . . 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 276	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶))
160		prodfct 12281	. . . . . . . . 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 12281	. . . . . . . . 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 6070	. . . . . . 7 ⊢ (𝜑 → (∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) · ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚)) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶))
165	164	adantr 276	. . . . . 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 2275	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶))
167	166	expr 375	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → ∏𝑘 ∈ 𝐴 (𝐵 · 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 · ∏𝑘 ∈ 𝐴 𝐶)))
168	167	exlimdv 1868	. . 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 363	. 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 12068	. . 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 726	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 104 ↔ wb 105 ∨ wo 716 DECID wdc 842 = wceq 1398 ∃wex 1541 ∈ wcel 2205 ∀wral 2522 ⦋csb 3140 ∅c0 3510 ifcif 3622 class class class wbr 4111 ↦ cmpt 4173 ∘ ccom 4755 ⟶wf 5350 –1-1-onto→wf1o 5353 ‘cfv 5354 (class class class)co 6052 Fincfn 6977 ℂcc 8130 1c1 8133 · cmul 8137 ≤ cle 8314 ℕcn 9242 ℤcz 9582 ℤ_≥cuz 9859 ...cfz 10348 seqcseq 10816 ♯chash 11146 ∏cprod 12244

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4227 ax-sep 4230 ax-nul 4238 ax-pow 4289 ax-pr 4324 ax-un 4556 ax-setind 4661 ax-iinf 4712 ax-cnex 8223 ax-resscn 8224 ax-1cn 8225 ax-1re 8226 ax-icn 8227 ax-addcl 8228 ax-addrcl 8229 ax-mulcl 8230 ax-mulrcl 8231 ax-addcom 8232 ax-mulcom 8233 ax-addass 8234 ax-mulass 8235 ax-distr 8236 ax-i2m1 8237 ax-0lt1 8238 ax-1rid 8239 ax-0id 8240 ax-rnegex 8241 ax-precex 8242 ax-cnre 8243 ax-pre-ltirr 8244 ax-pre-ltwlin 8245 ax-pre-lttrn 8246 ax-pre-apti 8247 ax-pre-ltadd 8248 ax-pre-mulgt0 8249 ax-pre-mulext 8250 ax-arch 8251 ax-caucvg 8252

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3045 df-csb 3141 df-dif 3215 df-un 3217 df-in 3219 df-ss 3226 df-nul 3511 df-if 3623 df-pw 3673 df-sn 3697 df-pr 3698 df-op 3700 df-uni 3917 df-int 3952 df-iun 3995 df-br 4112 df-opab 4174 df-mpt 4175 df-tr 4211 df-id 4416 df-po 4419 df-iso 4420 df-iord 4489 df-on 4491 df-ilim 4492 df-suc 4494 df-iom 4715 df-xp 4757 df-rel 4758 df-cnv 4759 df-co 4760 df-dm 4761 df-rn 4762 df-res 4763 df-ima 4764 df-iota 5314 df-fun 5356 df-fn 5357 df-f 5358 df-f1 5359 df-fo 5360 df-f1o 5361 df-fv 5362 df-isom 5363 df-riota 6005 df-ov 6055 df-oprab 6056 df-mpo 6057 df-1st 6336 df-2nd 6337 df-recs 6538 df-irdg 6603 df-frec 6624 df-1o 6649 df-oadd 6653 df-er 6769 df-en 6978 df-dom 6979 df-fin 6980 df-pnf 8315 df-mnf 8316 df-xr 8317 df-ltxr 8318 df-le 8319 df-sub 8451 df-neg 8452 df-reap 8854 df-ap 8861 df-div 8952 df-inn 9243 df-2 9301 df-3 9302 df-4 9303 df-n0 9502 df-z 9583 df-uz 9860 df-q 9958 df-rp 9993 df-fz 10349 df-fzo 10484 df-seqfrec 10817 df-exp 10908 df-ihash 11147 df-cj 11535 df-re 11536 df-im 11537 df-rsqrt 11691 df-abs 11692 df-clim 11972 df-proddc 12245

This theorem is referenced by: fprodsplitdc 12290 fproddivap 12324 gausslemma2dlem5 15988 gausslemma2dlem6 15989

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