Theorem fprodss 15992

Description: Change the index set to a subset in a finite product. (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 3965	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ ∅))
3		ss0 4359	. . . . 5 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅)
4	2, 3	biimtrdi 256	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 ⊆ 𝐵 → 𝐴 = ∅))
5		prodeq1 15951	. . . . . 6 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
6		prodeq1 15951	. . . . . . 7 ⊢ (𝐵 = ∅ → ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
7	6	eqcomd 2771	. . . . . 6 ⊢ (𝐵 = ∅ → ∏𝑘 ∈ ∅ 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)
8	5, 7	sylan9eq 2820	. . . . 5 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)
9	8	expcom 418	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶))
10	4, 9	syld 48	. . 3 ⊢ (𝐵 = ∅ → (𝐴 ⊆ 𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶))
11	1, 10	syl5com 32	. 2 ⊢ (𝜑 → (𝐵 = ∅ → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶))
12		cnvimass 6075	. . . . . . . . 9 ⊢ (◡𝑓 “ 𝐴) ⊆ dom 𝑓
13		simprr 784	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)
14		f1of 6810	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → 𝑓:(1...(♯‘𝐵))⟶𝐵)
15	13, 14	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(♯‘𝐵))⟶𝐵)
16	12, 15	fssdm 6715	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (◡𝑓 “ 𝐴) ⊆ (1...(♯‘𝐵)))
17		f1ofn 6811	. . . . . . . . . . . 12 ⊢ (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → 𝑓 Fn (1...(♯‘𝐵)))
18		elpreima 7043	. . . . . . . . . . . 12 ⊢ (𝑓 Fn (1...(♯‘𝐵)) → (𝑛 ∈ (◡𝑓 “ 𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴)))
19	13, 17, 18	3syl 19	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (◡𝑓 “ 𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴)))
20	15	ffvelcdmda 7069	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (1...(♯‘𝐵))) → (𝑓‘𝑛) ∈ 𝐵)
21	20	ex 417	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (1...(♯‘𝐵)) → (𝑓‘𝑛) ∈ 𝐵))
22	21	adantrd 496	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ((𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴) → (𝑓‘𝑛) ∈ 𝐵))
23	19, 22	sylbid 243	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (◡𝑓 “ 𝐴) → (𝑓‘𝑛) ∈ 𝐵))
24	23	imp 411	. . . . . . . . 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 417	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
27	26	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
28		eldif 3917	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (𝐵 ∖ 𝐴) ↔ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴))
29		fprodss.3	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 1)
30		ax-1cn 11146	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
31	29, 30	eqeltrdi 2873	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ ℂ)
32	28, 31	sylan2br 606	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴)) → 𝐶 ∈ ℂ)
33	32	expr 461	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (¬ 𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
34	27, 33	pm2.61d 181	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
35	34	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
36	35	fmpttd 7100	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑘 ∈ 𝐵 ↦ 𝐶):𝐵⟶ℂ)
37	36	ffvelcdmda 7069	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐵) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) ∈ ℂ)
38	24, 37	syldan 602	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (◡𝑓 “ 𝐴)) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) ∈ ℂ)
39		eqid 2765	. . . . . . . . 9 ⊢ (ℤ≥‘1) = (ℤ≥‘1)
40		simprl 782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (♯‘𝐵) ∈ ℕ)
41		nnuz 12892	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
42	40, 41	eleqtrdi 2875	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (♯‘𝐵) ∈ (ℤ≥‘1))
43		ssidd 3962	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (1...(♯‘𝐵)) ⊆ (1...(♯‘𝐵)))
44	39, 42, 43	fprodntriv 15986	. . . . . . . 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 4087	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴)) → 𝑛 ∈ (1...(♯‘𝐵)))
46	45, 20	sylan2 604	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑓‘𝑛) ∈ 𝐵)
47		eldifn 4088	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴)) → ¬ 𝑛 ∈ (◡𝑓 “ 𝐴))
48	47	adantl 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ¬ 𝑛 ∈ (◡𝑓 “ 𝐴))
49	45	adantl 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → 𝑛 ∈ (1...(♯‘𝐵)))
50	19	adantr 485	. . . . . . . . . . . . 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 719	. . . . . . . . . . . 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 327	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ¬ (𝑓‘𝑛) ∈ 𝐴)
53	46, 52	eldifd 3918	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴))
54		difss 4092	. . . . . . . . . . . . 13 ⊢ (𝐵 ∖ 𝐴) ⊆ 𝐵
55		resmpt 6030	. . . . . . . . . . . . 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 6872	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴))‘(𝑓‘𝑛)) = ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛))
58		fvres 6890	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴) → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴))‘(𝑓‘𝑛)) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
59	57, 58	eqtr3id 2814	. . . . . . . . . 10 ⊢ ((𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴) → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
60	53, 59	syl 18	. . . . . . . . 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 11191	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
62	61	elsn2 4627	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ {1} ↔ 𝐶 = 1)
63	29, 62	sylibr 237	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ {1})
64	63	fmpttd 7100	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶):(𝐵 ∖ 𝐴)⟶{1})
65	64	ad2antrr 738	. . . . . . . . . . 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	ffvelcdmd 7070	. . . . . . . . . 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 4602	. . . . . . . . . 10 ⊢ (((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) ∈ {1} → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) = 1)
68	66, 67	syl 18	. . . . . . . . 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 2802	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) = 1)
70		fzssuz 13584	. . . . . . . . 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 15991	. . . . . . 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 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝐴 ⊆ 𝐵)
74	73	resmptd 6033	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) = (𝑘 ∈ 𝐴 ↦ 𝐶))
75	74	fveq1d 6873	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚))
76		fvres 6890	. . . . . . . . . 10 ⊢ (𝑚 ∈ 𝐴 → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
77	75, 76	sylan9req 2821	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
78	77	prodeq2dv 15966	. . . . . . . 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 6871	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
80		fzfid 14000	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (1...(♯‘𝐵)) ∈ Fin)
81	80, 15	fisuppfi 9319	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (◡𝑓 “ 𝐴) ∈ Fin)
82		f1of1 6809	. . . . . . . . . . . 12 ⊢ (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → 𝑓:(1...(♯‘𝐵))–1-1→𝐵)
83	13, 82	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(♯‘𝐵))–1-1→𝐵)
84		f1ores 6825	. . . . . . . . . . 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 595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→(𝑓 “ (◡𝑓 “ 𝐴)))
86		f1ofo 6818	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → 𝑓:(1...(♯‘𝐵))–onto→𝐵)
87	13, 86	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(♯‘𝐵))–onto→𝐵)
88		foimacnv 6828	. . . . . . . . . . . 12 ⊢ ((𝑓:(1...(♯‘𝐵))–onto→𝐵 ∧ 𝐴 ⊆ 𝐵) → (𝑓 “ (◡𝑓 “ 𝐴)) = 𝐴)
89	87, 73, 88	syl2anc 595	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑓 “ (◡𝑓 “ 𝐴)) = 𝐴)
90	89	f1oeq3d 6807	. . . . . . . . . 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 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→𝐴)
92		fvres 6890	. . . . . . . . . 10 ⊢ (𝑛 ∈ (◡𝑓 “ 𝐴) → ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑛) = (𝑓‘𝑛))
93	92	adantl 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (◡𝑓 “ 𝐴)) → ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑛) = (𝑓‘𝑛))
94	73	sselda 3939	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → 𝑚 ∈ 𝐵)
95	36	ffvelcdmda 7069	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐵) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) ∈ ℂ)
96	94, 95	syldan 602	. . . . . . . . 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 15990	. . . . . . . 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 2800	. . . . . . 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 2766	. . . . . . . 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 15990	. . . . . . 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 2810	. . . . . 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 15989	. . . . . 6 ⊢ ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐶
103		prodfc 15989	. . . . . 6 ⊢ ∏𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ∏𝑘 ∈ 𝐵 𝐶
104	101, 102, 103	3eqtr3g 2823	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)
105	104	expr 461	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐵) ∈ ℕ) → (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶))
106	105	exlimdv 1956	. . 3 ⊢ ((𝜑 ∧ (♯‘𝐵) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶))
107	106	expimpd 458	. 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 15751	. . 3 ⊢ (𝐵 ∈ Fin → (𝐵 = ∅ ∨ ((♯‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)))
110	108, 109	syl 18	. 2 ⊢ (𝜑 → (𝐵 = ∅ ∨ ((♯‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)))
111	11, 107, 110	mpjaod 873	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1563  ∃wex 1802   ∈ wcel 2145   ∖ cdif 3904   ⊆ wss 3907  ∅c0 4288  {csn 4585   ↦ cmpt 5186  ^◡ccnv 5651   ↾ cres 5654   “ cima 5655   Fn wfn 6520  ⟶wf 6521  –1-1→wf1 6522  –onto→wfo 6523  –1-1-onto→wf1o 6524  ‘cfv 6525  (class class class)co 7400  Fincfn 8931  ℂcc 11086  1c1 11089  ℕcn 12224  ℤ_≥cuz 12853  ...cfz 13526  ♯chash 14357  ∏cprod 15947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-inf2 9598  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-pre-sup 11166

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-sup 9390  df-oi 9460  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12225  df-2 12294  df-3 12295  df-n0 12496  df-z 12583  df-uz 12854  df-rp 13008  df-fz 13527  df-fzo 13674  df-seq 14029  df-exp 14089  df-hash 14358  df-cj 15140  df-re 15141  df-im 15142  df-sqrt 15276  df-abs 15277  df-clim 15529  df-prod 15948

This theorem is referenced by:  fprodsplit  16010