Theorem fprodss 15304

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 3995	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ ∅))
3		ss0 4354	. . . . 5 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅)
4	2, 3	syl6bi 255	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 ⊆ 𝐵 → 𝐴 = ∅))
5		prodeq1 15265	. . . . . 6 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
6		prodeq1 15265	. . . . . . 7 ⊢ (𝐵 = ∅ → ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
7	6	eqcomd 2829	. . . . . 6 ⊢ (𝐵 = ∅ → ∏𝑘 ∈ ∅ 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)
8	5, 7	sylan9eq 2878	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)
9	8	expcom 416	. . . 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 5951	. . . . . . . . 9 ⊢ (◡𝑓 “ 𝐴) ⊆ dom 𝑓
13		simprr 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)
14		f1of 6617	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → 𝑓:(1...(♯‘𝐵))⟶𝐵)
15	13, 14	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(♯‘𝐵))⟶𝐵)
16	12, 15	fssdm 6532	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (◡𝑓 “ 𝐴) ⊆ (1...(♯‘𝐵)))
17		f1ofn 6618	. . . . . . . . . . . 12 ⊢ (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → 𝑓 Fn (1...(♯‘𝐵)))
18		elpreima 6830	. . . . . . . . . . . 12 ⊢ (𝑓 Fn (1...(♯‘𝐵)) → (𝑛 ∈ (◡𝑓 “ 𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴)))
19	13, 17, 18	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (◡𝑓 “ 𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴)))
20	15	ffvelrnda 6853	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (1...(♯‘𝐵))) → (𝑓‘𝑛) ∈ 𝐵)
21	20	ex 415	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (1...(♯‘𝐵)) → (𝑓‘𝑛) ∈ 𝐵))
22	21	adantrd 494	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ((𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴) → (𝑓‘𝑛) ∈ 𝐵))
23	19, 22	sylbid 242	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (◡𝑓 “ 𝐴) → (𝑓‘𝑛) ∈ 𝐵))
24	23	imp 409	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (◡𝑓 “ 𝐴)) → (𝑓‘𝑛) ∈ 𝐵)
25		fprodss.2	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
26	25	ex 415	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
27	26	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
28		eldif 3948	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (𝐵 ∖ 𝐴) ↔ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴))
29		fprodss.3	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 1)
30		ax-1cn 10597	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
31	29, 30	eqeltrdi 2923	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ ℂ)
32	28, 31	sylan2br 596	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴)) → 𝐶 ∈ ℂ)
33	32	expr 459	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (¬ 𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
34	27, 33	pm2.61d 181	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
35	34	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
36	35	fmpttd 6881	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑘 ∈ 𝐵 ↦ 𝐶):𝐵⟶ℂ)
37	36	ffvelrnda 6853	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐵) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) ∈ ℂ)
38	24, 37	syldan 593	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (◡𝑓 “ 𝐴)) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) ∈ ℂ)
39		eqid 2823	. . . . . . . . 9 ⊢ (ℤ≥‘1) = (ℤ≥‘1)
40		simprl 769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (♯‘𝐵) ∈ ℕ)
41		nnuz 12284	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
42	40, 41	eleqtrdi 2925	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (♯‘𝐵) ∈ (ℤ≥‘1))
43		ssidd 3992	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (1...(♯‘𝐵)) ⊆ (1...(♯‘𝐵)))
44	39, 42, 43	fprodntriv 15298	. . . . . . . 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))
45		eldifi 4105	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴)) → 𝑛 ∈ (1...(♯‘𝐵)))
46	45, 20	sylan2 594	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑓‘𝑛) ∈ 𝐵)
47		eldifn 4106	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴)) → ¬ 𝑛 ∈ (◡𝑓 “ 𝐴))
48	47	adantl 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ¬ 𝑛 ∈ (◡𝑓 “ 𝐴))
49	45	adantl 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → 𝑛 ∈ (1...(♯‘𝐵)))
50	19	adantr 483	. . . . . . . . . . . . 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)))
51	49, 50	mpbirand 705	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑛 ∈ (◡𝑓 “ 𝐴) ↔ (𝑓‘𝑛) ∈ 𝐴))
52	48, 51	mtbid 326	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ¬ (𝑓‘𝑛) ∈ 𝐴)
53	46, 52	eldifd 3949	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴))
54		difss 4110	. . . . . . . . . . . . 13 ⊢ (𝐵 ∖ 𝐴) ⊆ 𝐵
55		resmpt 5907	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∖ 𝐴) ⊆ 𝐵 → ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴)) = (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶))
56	54, 55	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴)) = (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)
57	56	fveq1i 6673	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴))‘(𝑓‘𝑛)) = ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛))
58		fvres 6691	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴) → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴))‘(𝑓‘𝑛)) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
59	57, 58	syl5eqr 2872	. . . . . . . . . 10 ⊢ ((𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴) → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
60	53, 59	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)))
61		1ex 10639	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
62	61	elsn2 4606	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ {1} ↔ 𝐶 = 1)
63	29, 62	sylibr 236	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ {1})
64	63	fmpttd 6881	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶):(𝐵 ∖ 𝐴)⟶{1})
65	64	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶):(𝐵 ∖ 𝐴)⟶{1})
66	65, 53	ffvelrnd 6854	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) ∈ {1})
67		elsni 4586	. . . . . . . . . 10 ⊢ (((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) ∈ {1} → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) = 1)
68	66, 67	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)
69	60, 68	eqtr3d 2860	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) = 1)
70		fzssuz 12951	. . . . . . . . 9 ⊢ (1...(♯‘𝐵)) ⊆ (ℤ≥‘1)
71	70	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (1...(♯‘𝐵)) ⊆ (ℤ≥‘1))
72	16, 38, 44, 69, 71	prodss 15303	. . . . . . 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)))
73	1	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝐴 ⊆ 𝐵)
74	73	resmptd 5910	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) = (𝑘 ∈ 𝐴 ↦ 𝐶))
75	74	fveq1d 6674	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚))
76		fvres 6691	. . . . . . . . . 10 ⊢ (𝑚 ∈ 𝐴 → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
77	75, 76	sylan9req 2879	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
78	77	prodeq2dv 15279	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
79		fveq2 6672	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
80		fzfid 13344	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (1...(♯‘𝐵)) ∈ Fin)
81	80, 15	fisuppfi 8843	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (◡𝑓 “ 𝐴) ∈ Fin)
82		f1of1 6616	. . . . . . . . . . . 12 ⊢ (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → 𝑓:(1...(♯‘𝐵))–1-1→𝐵)
83	13, 82	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(♯‘𝐵))–1-1→𝐵)
84		f1ores 6631	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘𝐵))–1-1→𝐵 ∧ (◡𝑓 “ 𝐴) ⊆ (1...(♯‘𝐵))) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→(𝑓 “ (◡𝑓 “ 𝐴)))
85	83, 16, 84	syl2anc 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→(𝑓 “ (◡𝑓 “ 𝐴)))
86		f1ofo 6624	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → 𝑓:(1...(♯‘𝐵))–onto→𝐵)
87	13, 86	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(♯‘𝐵))–onto→𝐵)
88		foimacnv 6634	. . . . . . . . . . . 12 ⊢ ((𝑓:(1...(♯‘𝐵))–onto→𝐵 ∧ 𝐴 ⊆ 𝐵) → (𝑓 “ (◡𝑓 “ 𝐴)) = 𝐴)
89	87, 73, 88	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑓 “ (◡𝑓 “ 𝐴)) = 𝐴)
90	89	f1oeq3d 6614	. . . . . . . . . 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))
91	85, 90	mpbid 234	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→𝐴)
92		fvres 6691	. . . . . . . . . 10 ⊢ (𝑛 ∈ (◡𝑓 “ 𝐴) → ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑛) = (𝑓‘𝑛))
93	92	adantl 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (◡𝑓 “ 𝐴)) → ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑛) = (𝑓‘𝑛))
94	73	sselda 3969	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → 𝑚 ∈ 𝐵)
95	36	ffvelrnda 6853	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐵) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) ∈ ℂ)
96	94, 95	syldan 593	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) ∈ ℂ)
97	79, 81, 91, 93, 96	fprodf1o 15302	. . . . . . . 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)))
98	78, 97	eqtrd 2858	. . . . . . 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)))
99		eqidd 2824	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (1...(♯‘𝐵))) → (𝑓‘𝑛) = (𝑓‘𝑛))
100	79, 80, 13, 99, 95	fprodf1o 15302	. . . . . . 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)))
101	72, 98, 100	3eqtr4d 2868	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ∏𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
102		prodfc 15301	. . . . . 6 ⊢ ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐶
103		prodfc 15301	. . . . . 6 ⊢ ∏𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ∏𝑘 ∈ 𝐵 𝐶
104	101, 102, 103	3eqtr3g 2881	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)
105	104	expr 459	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐵) ∈ ℕ) → (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶))
106	105	exlimdv 1934	. . 3 ⊢ ((𝜑 ∧ (♯‘𝐵) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶))
107	106	expimpd 456	. 2 ⊢ (𝜑 → (((♯‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵) → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶))
108		fprodss.4	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
109		fz1f1o 15069	. . 3 ⊢ (𝐵 ∈ Fin → (𝐵 = ∅ ∨ ((♯‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)))
110	108, 109	syl 17	. 2 ⊢ (𝜑 → (𝐵 = ∅ ∨ ((♯‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)))
111	11, 107, 110	mpjaod 856	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ∖ cdif 3935   ⊆ wss 3938  ∅c0 4293  {csn 4569   ↦ cmpt 5148  ^◡ccnv 5556   ↾ cres 5559   “ cima 5560   Fn wfn 6352  ⟶wf 6353  –1-1→wf1 6354  –onto→wfo 6355  –1-1-onto→wf1o 6356  ‘cfv 6357  (class class class)co 7158  Fincfn 8511  ℂcc 10537  1c1 10540  ℕcn 11640  ℤ_≥cuz 12246  ...cfz 12895  ♯chash 13693  ∏cprod 15261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-sup 8908  df-oi 8976  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-n0 11901  df-z 11985  df-uz 12247  df-rp 12393  df-fz 12896  df-fzo 13037  df-seq 13373  df-exp 13433  df-hash 13694  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-clim 14847  df-prod 15262

This theorem is referenced by:  fprodsplit  15322