Theorem fprodf1o 12282

Description: Re-index a finite product using a bijection. (Contributed by Scott Fenton, 7-Dec-2017.)

Hypotheses

Ref	Expression
fprodf1o.1	⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)
fprodf1o.2	⊢ (𝜑 → 𝐶 ∈ Fin)
fprodf1o.3	⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
fprodf1o.4	⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)
fprodf1o.5	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)

Assertion

Ref	Expression
fprodf1o	⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷)

Distinct variable groups:   𝐴,𝑘,𝑛   𝐵,𝑛   𝐶,𝑛   𝐷,𝑘   𝑛,𝐹   𝑘,𝐺   𝜑,𝑘,𝑛

Allowed substitution hints:   𝐵(𝑘)   𝐶(𝑘)   𝐷(𝑛)   𝐹(𝑘)   𝐺(𝑛)

Proof of Theorem fprodf1o

Dummy variables 𝑓 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prod0 12279	. . . 4 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1
2		fprodf1o.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
3	2	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐹:𝐶–1-1-onto→𝐴)
4		f1oeq2 5605	. . . . . . . . 9 ⊢ (𝐶 = ∅ → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:∅–1-1-onto→𝐴))
5	4	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = ∅) → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:∅–1-1-onto→𝐴))
6	3, 5	mpbid 147	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐹:∅–1-1-onto→𝐴)
7		f1ofo 5623	. . . . . . 7 ⊢ (𝐹:∅–1-1-onto→𝐴 → 𝐹:∅–onto→𝐴)
8	6, 7	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐹:∅–onto→𝐴)
9		fo00 5654	. . . . . . 7 ⊢ (𝐹:∅–onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
10	9	simprbi 275	. . . . . 6 ⊢ (𝐹:∅–onto→𝐴 → 𝐴 = ∅)
11	8, 10	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐴 = ∅)
12	11	prodeq1d 12258	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = ∅) → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
13		prodeq1 12247	. . . . . 6 ⊢ (𝐶 = ∅ → ∏𝑛 ∈ 𝐶 𝐷 = ∏𝑛 ∈ ∅ 𝐷)
14		prod0 12279	. . . . . 6 ⊢ ∏𝑛 ∈ ∅ 𝐷 = 1
15	13, 14	eqtrdi 2283	. . . . 5 ⊢ (𝐶 = ∅ → ∏𝑛 ∈ 𝐶 𝐷 = 1)
16	15	adantl 277	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = ∅) → ∏𝑛 ∈ 𝐶 𝐷 = 1)
17	1, 12, 16	3eqtr4a 2293	. . 3 ⊢ ((𝜑 ∧ 𝐶 = ∅) → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷)
18	17	ex 115	. 2 ⊢ (𝜑 → (𝐶 = ∅ → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷))
19		2fveq3 5677	. . . . . . . 8 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘(𝑓‘𝑛))))
20		simprl 531	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → (♯‘𝐶) ∈ ℕ)
21		simprr 533	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)
22		f1of 5616	. . . . . . . . . . . 12 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶⟶𝐴)
23	2, 22	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴)
24	23	ffvelcdmda 5814	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐶) → (𝐹‘𝑚) ∈ 𝐴)
25		fprodf1o.5	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
26	25	fmpttd 5834	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
27	26	ffvelcdmda 5814	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐹‘𝑚) ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) ∈ ℂ)
28	24, 27	syldan 282	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐶) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) ∈ ℂ)
29	28	adantlr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑚 ∈ 𝐶) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) ∈ ℂ)
30		simpr 110	. . . . . . . . . . . 12 ⊢ (((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶) → 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)
31		f1oco 5639	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐶–1-1-onto→𝐴 ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶) → (𝐹 ∘ 𝑓):(1...(♯‘𝐶))–1-1-onto→𝐴)
32	2, 30, 31	syl2an 289	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → (𝐹 ∘ 𝑓):(1...(♯‘𝐶))–1-1-onto→𝐴)
33		f1of 5616	. . . . . . . . . . 11 ⊢ ((𝐹 ∘ 𝑓):(1...(♯‘𝐶))–1-1-onto→𝐴 → (𝐹 ∘ 𝑓):(1...(♯‘𝐶))⟶𝐴)
34	32, 33	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → (𝐹 ∘ 𝑓):(1...(♯‘𝐶))⟶𝐴)
35		fvco3 5750	. . . . . . . . . 10 ⊢ (((𝐹 ∘ 𝑓):(1...(♯‘𝐶))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐶))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ (𝐹 ∘ 𝑓))‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘((𝐹 ∘ 𝑓)‘𝑛)))
36	34, 35	sylan 283	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑛 ∈ (1...(♯‘𝐶))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ (𝐹 ∘ 𝑓))‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘((𝐹 ∘ 𝑓)‘𝑛)))
37		f1of 5616	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶 → 𝑓:(1...(♯‘𝐶))⟶𝐶)
38	37	adantl 277	. . . . . . . . . . . 12 ⊢ (((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶) → 𝑓:(1...(♯‘𝐶))⟶𝐶)
39	38	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → 𝑓:(1...(♯‘𝐶))⟶𝐶)
40		fvco3 5750	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘𝐶))⟶𝐶 ∧ 𝑛 ∈ (1...(♯‘𝐶))) → ((𝐹 ∘ 𝑓)‘𝑛) = (𝐹‘(𝑓‘𝑛)))
41	39, 40	sylan 283	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑛 ∈ (1...(♯‘𝐶))) → ((𝐹 ∘ 𝑓)‘𝑛) = (𝐹‘(𝑓‘𝑛)))
42	41	fveq2d 5676	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑛 ∈ (1...(♯‘𝐶))) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘((𝐹 ∘ 𝑓)‘𝑛)) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘(𝑓‘𝑛))))
43	36, 42	eqtrd 2267	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑛 ∈ (1...(♯‘𝐶))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ (𝐹 ∘ 𝑓))‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘(𝑓‘𝑛))))
44	19, 20, 21, 29, 43	fprodseq 12277	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → ∏𝑚 ∈ 𝐶 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐶), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ (𝐹 ∘ 𝑓))‘𝑛), 1)))‘(♯‘𝐶)))
45		eqid 2234	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
46		fprodf1o.1	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)
47		fprodf1o.4	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)
48	23	ffvelcdmda 5814	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) ∈ 𝐴)
49	47, 48	eqeltrrd 2312	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐺 ∈ 𝐴)
50	46	eleq1d 2303	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐺 → (𝐵 ∈ ℂ ↔ 𝐷 ∈ ℂ))
51	25	ralrimiva 2617	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
52	51	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
53	50, 52, 49	rspcdva 2928	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐷 ∈ ℂ)
54	45, 46, 49, 53	fvmptd3 5773	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝐺) = 𝐷)
55	47	fveq2d 5676	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝐺))
56		simpr 110	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝑛 ∈ 𝐶)
57		eqid 2234	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝐶 ↦ 𝐷) = (𝑛 ∈ 𝐶 ↦ 𝐷)
58	57	fvmpt2 5763	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝐷 ∈ ℂ) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = 𝐷)
59	56, 53, 58	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = 𝐷)
60	54, 55, 59	3eqtr4rd 2278	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)))
61	60	ralrimiva 2617	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑛 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)))
62		nffvmpt1 5683	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚)
63	62	nfeq1 2396	. . . . . . . . . . 11 ⊢ Ⅎ𝑛((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚))
64		fveq2 5672	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚))
65		2fveq3 5677	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)))
66	64, 65	eqeq12d 2249	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)) ↔ ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚))))
67	63, 66	rspc 2917	. . . . . . . . . 10 ⊢ (𝑚 ∈ 𝐶 → (∀𝑛 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚))))
68	61, 67	mpan9 281	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐶) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)))
69	68	adantlr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑚 ∈ 𝐶) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)))
70	69	prodeq2dv 12260	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → ∏𝑚 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ∏𝑚 ∈ 𝐶 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)))
71		fveq2 5672	. . . . . . . 8 ⊢ (𝑚 = ((𝐹 ∘ 𝑓)‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘((𝐹 ∘ 𝑓)‘𝑛)))
72	26	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
73	72	ffvelcdmda 5814	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) ∈ ℂ)
74	71, 20, 32, 73, 36	fprodseq 12277	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐶), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ (𝐹 ∘ 𝑓))‘𝑛), 1)))‘(♯‘𝐶)))
75	44, 70, 74	3eqtr4rd 2278	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ∏𝑚 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚))
76	51	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
77		prodfct 12281	. . . . . . 7 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐵)
78	76, 77	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐵)
79	53	ralrimiva 2617	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ 𝐶 𝐷 ∈ ℂ)
80	79	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → ∀𝑛 ∈ 𝐶 𝐷 ∈ ℂ)
81		prodfct 12281	. . . . . . 7 ⊢ (∀𝑛 ∈ 𝐶 𝐷 ∈ ℂ → ∏𝑚 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ∏𝑛 ∈ 𝐶 𝐷)
82	80, 81	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → ∏𝑚 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ∏𝑛 ∈ 𝐶 𝐷)
83	75, 78, 82	3eqtr3d 2275	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷)
84	83	expr 375	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐶) ∈ ℕ) → (𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷))
85	84	exlimdv 1868	. . 3 ⊢ ((𝜑 ∧ (♯‘𝐶) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷))
86	85	expimpd 363	. 2 ⊢ (𝜑 → (((♯‘𝐶) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶) → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷))
87		fprodf1o.2	. . 3 ⊢ (𝜑 → 𝐶 ∈ Fin)
88		fz1f1o 12068	. . 3 ⊢ (𝐶 ∈ Fin → (𝐶 = ∅ ∨ ((♯‘𝐶) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)))
89	87, 88	syl 14	. 2 ⊢ (𝜑 → (𝐶 = ∅ ∨ ((♯‘𝐶) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)))
90	18, 86, 89	mpjaod 726	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∀wral 2522  ∅c0 3510  ifcif 3622   class class class wbr 4111   ↦ cmpt 4173   ∘ ccom 4755  ⟶wf 5350  –onto→wfo 5352  –1-1-onto→wf1o 5353  ‘cfv 5354  (class class class)co 6052  Fincfn 6977  ℂcc 8130  1c1 8133   · cmul 8137   ≤ cle 8314  ℕcn 9242  ...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:  fprodssdc  12284  fprodshft  12312  fprodrev  12313  fprod2dlemstep  12316  fprodcnv  12319  eulerthlemth  12937  gausslemma2dlem1  15983