Theorem fprodss 14678

Description: Change the index set to a subset in a finite sum. (Contributed by Scott Fenton, 16-Dec-2017.)

Hypotheses

Ref	Expression
fprodss.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
fprodss.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
fprodss.3	⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 1)
fprodss.4	⊢ (𝜑 → 𝐵 ∈ Fin)

Assertion

Ref	Expression
fprodss	⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)

Distinct variable groups:   𝐴,𝑘   𝐵,𝑘   𝜑,𝑘

Allowed substitution hint:   𝐶(𝑘)

Proof of Theorem fprodss

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

Step	Hyp	Ref	Expression
1		fprodss.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		sseq2 3627	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ ∅))
3		ss0 3974	. . . . 5 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅)
4	2, 3	syl6bi 243	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 ⊆ 𝐵 → 𝐴 = ∅))
5		prodeq1 14639	. . . . . 6 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
6		prodeq1 14639	. . . . . . 7 ⊢ (𝐵 = ∅ → ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
7	6	eqcomd 2628	. . . . . 6 ⊢ (𝐵 = ∅ → ∏𝑘 ∈ ∅ 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)
8	5, 7	sylan9eq 2676	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)
9	8	expcom 451	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶))
10	4, 9	syld 47	. . 3 ⊢ (𝐵 = ∅ → (𝐴 ⊆ 𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶))
11	1, 10	syl5com 31	. 2 ⊢ (𝜑 → (𝐵 = ∅ → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶))
12		cnvimass 5485	. . . . . . . . 9 ⊢ (◡𝑓 “ 𝐴) ⊆ dom 𝑓
13		simprr 796	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)
14		f1of 6137	. . . . . . . . . . 11 ⊢ (𝑓:(1...(#‘𝐵))–1-1-onto→𝐵 → 𝑓:(1...(#‘𝐵))⟶𝐵)
15	13, 14	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(#‘𝐵))⟶𝐵)
16		fdm 6051	. . . . . . . . . 10 ⊢ (𝑓:(1...(#‘𝐵))⟶𝐵 → dom 𝑓 = (1...(#‘𝐵)))
17	15, 16	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → dom 𝑓 = (1...(#‘𝐵)))
18	12, 17	syl5sseq 3653	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (◡𝑓 “ 𝐴) ⊆ (1...(#‘𝐵)))
19		f1ofn 6138	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(#‘𝐵))–1-1-onto→𝐵 → 𝑓 Fn (1...(#‘𝐵)))
20	13, 19	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → 𝑓 Fn (1...(#‘𝐵)))
21		elpreima 6337	. . . . . . . . . . . 12 ⊢ (𝑓 Fn (1...(#‘𝐵)) → (𝑛 ∈ (◡𝑓 “ 𝐴) ↔ (𝑛 ∈ (1...(#‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴)))
22	20, 21	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (◡𝑓 “ 𝐴) ↔ (𝑛 ∈ (1...(#‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴)))
23	15	ffvelrnda 6359	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (1...(#‘𝐵))) → (𝑓‘𝑛) ∈ 𝐵)
24	23	ex 450	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (1...(#‘𝐵)) → (𝑓‘𝑛) ∈ 𝐵))
25	24	adantrd 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → ((𝑛 ∈ (1...(#‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴) → (𝑓‘𝑛) ∈ 𝐵))
26	22, 25	sylbid 230	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (◡𝑓 “ 𝐴) → (𝑓‘𝑛) ∈ 𝐵))
27	26	imp 445	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (◡𝑓 “ 𝐴)) → (𝑓‘𝑛) ∈ 𝐵)
28		fprodss.2	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
29	28	ex 450	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
30	29	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
31		eldif 3584	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (𝐵 ∖ 𝐴) ↔ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴))
32		fprodss.3	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 1)
33		ax-1cn 9994	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
34	32, 33	syl6eqel 2709	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ ℂ)
35	31, 34	sylan2br 493	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴)) → 𝐶 ∈ ℂ)
36	35	expr 643	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (¬ 𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
37	30, 36	pm2.61d 170	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
38	37	adantlr 751	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
39		eqid 2622	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐵 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ 𝐶)
40	38, 39	fmptd 6385	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (𝑘 ∈ 𝐵 ↦ 𝐶):𝐵⟶ℂ)
41	40	ffvelrnda 6359	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐵) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) ∈ ℂ)
42	27, 41	syldan 487	. . . . . . . 8 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (◡𝑓 “ 𝐴)) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) ∈ ℂ)
43		eqid 2622	. . . . . . . . 9 ⊢ (ℤ≥‘1) = (ℤ≥‘1)
44		simprl 794	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (#‘𝐵) ∈ ℕ)
45		nnuz 11723	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
46	44, 45	syl6eleq 2711	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (#‘𝐵) ∈ (ℤ≥‘1))
47		ssid 3624	. . . . . . . . . 10 ⊢ (1...(#‘𝐵)) ⊆ (1...(#‘𝐵))
48	47	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (1...(#‘𝐵)) ⊆ (1...(#‘𝐵)))
49	43, 46, 48	fprodntriv 14672	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → ∃𝑚 ∈ (ℤ≥‘1)∃𝑦(𝑦 ≠ 0 ∧ seq𝑚( · , (𝑛 ∈ (ℤ≥‘1) ↦ if(𝑛 ∈ (1...(#‘𝐵)), ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)), 1))) ⇝ 𝑦))
50		eldifi 3732	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴)) → 𝑛 ∈ (1...(#‘𝐵)))
51	50, 23	sylan2 491	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑓‘𝑛) ∈ 𝐵)
52		eldifn 3733	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴)) → ¬ 𝑛 ∈ (◡𝑓 “ 𝐴))
53	52	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ¬ 𝑛 ∈ (◡𝑓 “ 𝐴))
54	22	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑛 ∈ (◡𝑓 “ 𝐴) ↔ (𝑛 ∈ (1...(#‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴)))
55	50	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → 𝑛 ∈ (1...(#‘𝐵)))
56	55	biantrurd 529	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ((𝑓‘𝑛) ∈ 𝐴 ↔ (𝑛 ∈ (1...(#‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴)))
57	54, 56	bitr4d 271	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑛 ∈ (◡𝑓 “ 𝐴) ↔ (𝑓‘𝑛) ∈ 𝐴))
58	53, 57	mtbid 314	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ¬ (𝑓‘𝑛) ∈ 𝐴)
59	51, 58	eldifd 3585	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴))
60		difss 3737	. . . . . . . . . . . . 13 ⊢ (𝐵 ∖ 𝐴) ⊆ 𝐵
61		resmpt 5449	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∖ 𝐴) ⊆ 𝐵 → ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴)) = (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶))
62	60, 61	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴)) = (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)
63	62	fveq1i 6192	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴))‘(𝑓‘𝑛)) = ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛))
64		fvres 6207	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴) → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴))‘(𝑓‘𝑛)) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
65	63, 64	syl5eqr 2670	. . . . . . . . . 10 ⊢ ((𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴) → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
66	59, 65	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
67		1ex 10035	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
68	67	elsn2 4211	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ {1} ↔ 𝐶 = 1)
69	32, 68	sylibr 224	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ {1})
70		eqid 2622	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶) = (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)
71	69, 70	fmptd 6385	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶):(𝐵 ∖ 𝐴)⟶{1})
72	71	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶):(𝐵 ∖ 𝐴)⟶{1})
73	72, 59	ffvelrnd 6360	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) ∈ {1})
74		elsni 4194	. . . . . . . . . 10 ⊢ (((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) ∈ {1} → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) = 1)
75	73, 74	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) = 1)
76	66, 75	eqtr3d 2658	. . . . . . . 8 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) = 1)
77		fzssuz 12382	. . . . . . . . 9 ⊢ (1...(#‘𝐵)) ⊆ (ℤ≥‘1)
78	77	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (1...(#‘𝐵)) ⊆ (ℤ≥‘1))
79	18, 42, 49, 76, 78	prodss 14677	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → ∏𝑛 ∈ (◡𝑓 “ 𝐴)((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) = ∏𝑛 ∈ (1...(#‘𝐵))((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
80	1	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → 𝐴 ⊆ 𝐵)
81	80	resmptd 5452	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) = (𝑘 ∈ 𝐴 ↦ 𝐶))
82	81	fveq1d 6193	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚))
83		fvres 6207	. . . . . . . . . 10 ⊢ (𝑚 ∈ 𝐴 → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
84	82, 83	sylan9req 2677	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
85	84	prodeq2dv 14653	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
86		fveq2 6191	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
87		fzfid 12772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (1...(#‘𝐵)) ∈ Fin)
88	87, 15	fisuppfi 8283	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (◡𝑓 “ 𝐴) ∈ Fin)
89		f1of1 6136	. . . . . . . . . . . 12 ⊢ (𝑓:(1...(#‘𝐵))–1-1-onto→𝐵 → 𝑓:(1...(#‘𝐵))–1-1→𝐵)
90	13, 89	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(#‘𝐵))–1-1→𝐵)
91		f1ores 6151	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(#‘𝐵))–1-1→𝐵 ∧ (◡𝑓 “ 𝐴) ⊆ (1...(#‘𝐵))) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→(𝑓 “ (◡𝑓 “ 𝐴)))
92	90, 18, 91	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→(𝑓 “ (◡𝑓 “ 𝐴)))
93		f1ofo 6144	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(#‘𝐵))–1-1-onto→𝐵 → 𝑓:(1...(#‘𝐵))–onto→𝐵)
94	13, 93	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(#‘𝐵))–onto→𝐵)
95		foimacnv 6154	. . . . . . . . . . . 12 ⊢ ((𝑓:(1...(#‘𝐵))–onto→𝐵 ∧ 𝐴 ⊆ 𝐵) → (𝑓 “ (◡𝑓 “ 𝐴)) = 𝐴)
96	94, 80, 95	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (𝑓 “ (◡𝑓 “ 𝐴)) = 𝐴)
97		f1oeq3 6129	. . . . . . . . . . 11 ⊢ ((𝑓 “ (◡𝑓 “ 𝐴)) = 𝐴 → ((𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→(𝑓 “ (◡𝑓 “ 𝐴)) ↔ (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→𝐴))
98	96, 97	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → ((𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→(𝑓 “ (◡𝑓 “ 𝐴)) ↔ (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→𝐴))
99	92, 98	mpbid 222	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→𝐴)
100		fvres 6207	. . . . . . . . . 10 ⊢ (𝑛 ∈ (◡𝑓 “ 𝐴) → ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑛) = (𝑓‘𝑛))
101	100	adantl 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (◡𝑓 “ 𝐴)) → ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑛) = (𝑓‘𝑛))
102	80	sselda 3603	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → 𝑚 ∈ 𝐵)
103	40	ffvelrnda 6359	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐵) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) ∈ ℂ)
104	102, 103	syldan 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) ∈ ℂ)
105	86, 88, 99, 101, 104	fprodf1o 14676	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ∏𝑛 ∈ (◡𝑓 “ 𝐴)((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
106	85, 105	eqtrd 2656	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ∏𝑛 ∈ (◡𝑓 “ 𝐴)((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
107		eqidd 2623	. . . . . . . 8 ⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (1...(#‘𝐵))) → (𝑓‘𝑛) = (𝑓‘𝑛))
108	86, 87, 13, 107, 103	fprodf1o 14676	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → ∏𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ∏𝑛 ∈ (1...(#‘𝐵))((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
109	79, 106, 108	3eqtr4d 2666	. . . . . 6 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ∏𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
110		prodfc 14675	. . . . . 6 ⊢ ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐶
111		prodfc 14675	. . . . . 6 ⊢ ∏𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ∏𝑘 ∈ 𝐵 𝐶
112	109, 110, 111	3eqtr3g 2679	. . . . 5 ⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)
113	112	expr 643	. . . 4 ⊢ ((𝜑 ∧ (#‘𝐵) ∈ ℕ) → (𝑓:(1...(#‘𝐵))–1-1-onto→𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶))
114	113	exlimdv 1861	. . 3 ⊢ ((𝜑 ∧ (#‘𝐵) ∈ ℕ) → (∃𝑓 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶))
115	114	expimpd 629	. 2 ⊢ (𝜑 → (((#‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵) → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶))
116		fprodss.4	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
117		fz1f1o 14441	. . 3 ⊢ (𝐵 ∈ Fin → (𝐵 = ∅ ∨ ((#‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)))
118	116, 117	syl 17	. 2 ⊢ (𝜑 → (𝐵 = ∅ ∨ ((#‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)))
119	11, 115, 118	mpjaod 396	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  {csn 4177   ↦ cmpt 4729  ^◡ccnv 5113  dom cdm 5114   ↾ cres 5116   “ cima 5117   Fn wfn 5883  ⟶wf 5884  –1-1→wf1 5885  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  ℂcc 9934  1c1 9937  ℕcn 11020  ℤ_≥cuz 11687  ...cfz 12326  #chash 13117  ∏cprod 14635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-prod 14636

This theorem is referenced by:  fprodsplit  14696