Theorem fprodcom2 14714

Description: Interchange order of multiplication. Note that 𝐵(𝑗) and 𝐷(𝑘) are not necessarily constant expressions. (Contributed by Scott Fenton, 1-Feb-2018.) (Proof shortened by JJ, 2-Aug-2021.)

Hypotheses

Ref	Expression
fprodcom2.1	⊢ (𝜑 → 𝐴 ∈ Fin)
fprodcom2.2	⊢ (𝜑 → 𝐶 ∈ Fin)
fprodcom2.3	⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)
fprodcom2.4	⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) ↔ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))
fprodcom2.5	⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐸 ∈ ℂ)

Assertion

Ref	Expression
fprodcom2	⊢ (𝜑 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐸 = ∏𝑘 ∈ 𝐶 ∏𝑗 ∈ 𝐷 𝐸)

Distinct variable groups:   𝐴,𝑗,𝑘   𝐵,𝑘   𝐶,𝑗,𝑘   𝐷,𝑗   𝜑,𝑗,𝑘

Allowed substitution hints:   𝐵(𝑗)   𝐷(𝑘)   𝐸(𝑗,𝑘)

Proof of Theorem fprodcom2

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relxp 5227	. . . . . . . . 9 ⊢ Rel ({𝑗} × 𝐵)
2	1	rgenw 2924	. . . . . . . 8 ⊢ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐵)
3		reliun 5239	. . . . . . . 8 ⊢ (Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐵))
4	2, 3	mpbir 221	. . . . . . 7 ⊢ Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
5		relcnv 5503	. . . . . . 7 ⊢ Rel ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷)
6		ancom 466	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑗 ∧ 𝑦 = 𝑘) ↔ (𝑦 = 𝑘 ∧ 𝑥 = 𝑗))
7		vex 3203	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
8		vex 3203	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
9	7, 8	opth 4945	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ (𝑥 = 𝑗 ∧ 𝑦 = 𝑘))
10	8, 7	opth 4945	. . . . . . . . . . . 12 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ↔ (𝑦 = 𝑘 ∧ 𝑥 = 𝑗))
11	6, 9, 10	3bitr4i 292	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩)
12	11	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩))
13		fprodcom2.4	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) ↔ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))
14	12, 13	anbi12d 747	. . . . . . . . 9 ⊢ (𝜑 → ((⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) ↔ (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷))))
15	14	2exbidv 1852	. . . . . . . 8 ⊢ (𝜑 → (∃𝑗∃𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) ↔ ∃𝑗∃𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷))))
16		eliunxp 5259	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗∃𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)))
17	7, 8	opelcnv 5304	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ⟨𝑦, 𝑥⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
18		eliunxp 5259	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑥⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ∃𝑘∃𝑗(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))
19		excom 2042	. . . . . . . . 9 ⊢ (∃𝑘∃𝑗(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)) ↔ ∃𝑗∃𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))
20	17, 18, 19	3bitri 286	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ∃𝑗∃𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))
21	15, 16, 20	3bitr4g 303	. . . . . . 7 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷)))
22	4, 5, 21	eqrelrdv 5216	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
23		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑥({𝑗} × 𝐵)
24		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑗{𝑥}
25		nfcsb1v 3549	. . . . . . . 8 ⊢ Ⅎ𝑗⦋𝑥 / 𝑗⦌𝐵
26	24, 25	nfxp 5142	. . . . . . 7 ⊢ Ⅎ𝑗({𝑥} × ⦋𝑥 / 𝑗⦌𝐵)
27		sneq 4187	. . . . . . . 8 ⊢ (𝑗 = 𝑥 → {𝑗} = {𝑥})
28		csbeq1a 3542	. . . . . . . 8 ⊢ (𝑗 = 𝑥 → 𝐵 = ⦋𝑥 / 𝑗⦌𝐵)
29	27, 28	xpeq12d 5140	. . . . . . 7 ⊢ (𝑗 = 𝑥 → ({𝑗} × 𝐵) = ({𝑥} × ⦋𝑥 / 𝑗⦌𝐵))
30	23, 26, 29	cbviun 4557	. . . . . 6 ⊢ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × ⦋𝑥 / 𝑗⦌𝐵)
31		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑦({𝑘} × 𝐷)
32		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑘{𝑦}
33		nfcsb1v 3549	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐷
34	32, 33	nfxp 5142	. . . . . . . 8 ⊢ Ⅎ𝑘({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
35		sneq 4187	. . . . . . . . 9 ⊢ (𝑘 = 𝑦 → {𝑘} = {𝑦})
36		csbeq1a 3542	. . . . . . . . 9 ⊢ (𝑘 = 𝑦 → 𝐷 = ⦋𝑦 / 𝑘⦌𝐷)
37	35, 36	xpeq12d 5140	. . . . . . . 8 ⊢ (𝑘 = 𝑦 → ({𝑘} × 𝐷) = ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
38	31, 34, 37	cbviun 4557	. . . . . . 7 ⊢ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) = ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
39	38	cnveqi 5297	. . . . . 6 ⊢ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) = ◡∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
40	22, 30, 39	3eqtr3g 2679	. . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 ({𝑥} × ⦋𝑥 / 𝑗⦌𝐵) = ◡∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
41	40	prodeq1d 14651	. . . 4 ⊢ (𝜑 → ∏𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × ⦋𝑥 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ∏𝑧 ∈ ◡ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸)
42	8, 7	op1std 7178	. . . . . . 7 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → (1st ‘𝑤) = 𝑦)
43	42	csbeq1d 3540	. . . . . 6 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
44	8, 7	op2ndd 7179	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → (2nd ‘𝑤) = 𝑥)
45	44	csbeq1d 3540	. . . . . . 7 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → ⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑥 / 𝑗⦌𝐸)
46	45	csbeq2dv 3992	. . . . . 6 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → ⦋𝑦 / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
47	43, 46	eqtrd 2656	. . . . 5 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
48	7, 8	op2ndd 7179	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
49	48	csbeq1d 3540	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸)
50	7, 8	op1std 7178	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
51	50	csbeq1d 3540	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑥 / 𝑗⦌𝐸)
52	51	csbeq2dv 3992	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋𝑦 / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
53	49, 52	eqtrd 2656	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
54		fprodcom2.2	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Fin)
55		snfi 8038	. . . . . . . 8 ⊢ {𝑦} ∈ Fin
56		fprodcom2.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ Fin)
57	56	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ Fin)
58	33, 36	opeliunxp2f 7336	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑦, 𝑥⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷))
59	17, 58	sylbbr 226	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷) → ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
60	59	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
61	22	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
62	60, 61	eleqtrrd 2704	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
63		eliun 4524	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝐴 ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵))
64	62, 63	sylib 208	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → ∃𝑗 ∈ 𝐴 ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵))
65		simpr 477	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵))
66		opelxp 5146	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵) ↔ (𝑥 ∈ {𝑗} ∧ 𝑦 ∈ 𝐵))
67	65, 66	sylib 208	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → (𝑥 ∈ {𝑗} ∧ 𝑦 ∈ 𝐵))
68	67	simpld 475	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → 𝑥 ∈ {𝑗})
69		elsni 4194	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ {𝑗} → 𝑥 = 𝑗)
70	68, 69	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → 𝑥 = 𝑗)
71		simpl 473	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → 𝑗 ∈ 𝐴)
72	70, 71	eqeltrd 2701	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → 𝑥 ∈ 𝐴)
73	72	rexlimiva 3028	. . . . . . . . . . . 12 ⊢ (∃𝑗 ∈ 𝐴 ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵) → 𝑥 ∈ 𝐴)
74	64, 73	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → 𝑥 ∈ 𝐴)
75	74	expr 643	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷 → 𝑥 ∈ 𝐴))
76	75	ssrdv 3609	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ⦋𝑦 / 𝑘⦌𝐷 ⊆ 𝐴)
77	57, 76	ssfid 8183	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ⦋𝑦 / 𝑘⦌𝐷 ∈ Fin)
78		xpfi 8231	. . . . . . . 8 ⊢ (({𝑦} ∈ Fin ∧ ⦋𝑦 / 𝑘⦌𝐷 ∈ Fin) → ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin)
79	55, 77, 78	sylancr 695	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin)
80	79	ralrimiva 2966	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin)
81		iunfi 8254	. . . . . 6 ⊢ ((𝐶 ∈ Fin ∧ ∀𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin) → ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin)
82	54, 80, 81	syl2anc 693	. . . . 5 ⊢ (𝜑 → ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin)
83		reliun 5239	. . . . . . 7 ⊢ (Rel ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ↔ ∀𝑦 ∈ 𝐶 Rel ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
84		relxp 5227	. . . . . . . 8 ⊢ Rel ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
85	84	a1i 11	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → Rel ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
86	83, 85	mprgbir 2927	. . . . . 6 ⊢ Rel ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
87	86	a1i 11	. . . . 5 ⊢ (𝜑 → Rel ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
88		simpr 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
89		eliun 4524	. . . . . . . 8 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ↔ ∃𝑦 ∈ 𝐶 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
90	88, 89	sylib 208	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → ∃𝑦 ∈ 𝐶 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
91		xp2nd 7199	. . . . . . . . . 10 ⊢ (𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) → (2nd ‘𝑤) ∈ ⦋𝑦 / 𝑘⦌𝐷)
92	91	adantl 482	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈ ⦋𝑦 / 𝑘⦌𝐷)
93		xp1st 7198	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) → (1st ‘𝑤) ∈ {𝑦})
94	93	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ {𝑦})
95		elsni 4194	. . . . . . . . . . 11 ⊢ ((1st ‘𝑤) ∈ {𝑦} → (1st ‘𝑤) = 𝑦)
96	94, 95	syl 17	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (1st ‘𝑤) = 𝑦)
97	96	csbeq1d 3540	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → ⦋(1st ‘𝑤) / 𝑘⦌𝐷 = ⦋𝑦 / 𝑘⦌𝐷)
98	92, 97	eleqtrrd 2704	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈ ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
99	98	rexlimiva 3028	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐶 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) → (2nd ‘𝑤) ∈ ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
100	90, 99	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈ ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
101		simpl 473	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → 𝑦 ∈ 𝐶)
102	96, 101	eqeltrd 2701	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ 𝐶)
103	102	rexlimiva 3028	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐶 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) → (1st ‘𝑤) ∈ 𝐶)
104	90, 103	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ 𝐶)
105		simpl 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → 𝜑)
106	25	nfcri 2758	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵
107	69	equcomd 1946	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ {𝑗} → 𝑗 = 𝑥)
108	107, 28	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ {𝑗} → 𝐵 = ⦋𝑥 / 𝑗⦌𝐵)
109	108	eleq2d 2687	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ {𝑗} → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵))
110	109	biimpa 501	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ {𝑗} ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵)
111	66, 110	sylbi 207	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵) → 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵)
112	111	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵) → 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵))
113	106, 112	rexlimi 3024	. . . . . . . . . . 11 ⊢ (∃𝑗 ∈ 𝐴 ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵) → 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵)
114	64, 113	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵)
115		fprodcom2.5	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐸 ∈ ℂ)
116	115	ralrimivva 2971	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ)
117		nfcsb1v 3549	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗⦋𝑥 / 𝑗⦌𝐸
118	117	nfel1 2779	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ
119	25, 118	nfral 2945	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗∀𝑘 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ
120		csbeq1a 3542	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑥 → 𝐸 = ⦋𝑥 / 𝑗⦌𝐸)
121	120	eleq1d 2686	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑥 → (𝐸 ∈ ℂ ↔ ⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
122	28, 121	raleqbidv 3152	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑥 → (∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ ↔ ∀𝑘 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
123	119, 122	rspc 3303	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ → ∀𝑘 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
124	116, 123	mpan9 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
125		nfcsb1v 3549	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
126	125	nfel1 2779	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ
127		csbeq1a 3542	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑦 → ⦋𝑥 / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
128	127	eleq1d 2686	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑦 → (⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
129	126, 128	rspc 3303	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵 → (∀𝑘 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ → ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
130	124, 129	syl5com 31	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵 → ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
131	130	impr 649	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵)) → ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
132	105, 74, 114, 131	syl12anc 1324	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
133	132	ralrimivva 2971	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ∀𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
134	133	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → ∀𝑦 ∈ 𝐶 ∀𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
135		csbeq1 3536	. . . . . . . . 9 ⊢ (𝑦 = (1st ‘𝑤) → ⦋𝑦 / 𝑘⦌𝐷 = ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
136		csbeq1 3536	. . . . . . . . . 10 ⊢ (𝑦 = (1st ‘𝑤) → ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 = ⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
137	136	eleq1d 2686	. . . . . . . . 9 ⊢ (𝑦 = (1st ‘𝑤) → (⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
138	135, 137	raleqbidv 3152	. . . . . . . 8 ⊢ (𝑦 = (1st ‘𝑤) → (∀𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ ↔ ∀𝑥 ∈ ⦋ (1st ‘𝑤) / 𝑘⦌𝐷⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
139	138	rspcv 3305	. . . . . . 7 ⊢ ((1st ‘𝑤) ∈ 𝐶 → (∀𝑦 ∈ 𝐶 ∀𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ → ∀𝑥 ∈ ⦋ (1st ‘𝑤) / 𝑘⦌𝐷⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
140	104, 134, 139	sylc 65	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → ∀𝑥 ∈ ⦋ (1st ‘𝑤) / 𝑘⦌𝐷⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
141		csbeq1 3536	. . . . . . . . 9 ⊢ (𝑥 = (2nd ‘𝑤) → ⦋𝑥 / 𝑗⦌𝐸 = ⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
142	141	csbeq2dv 3992	. . . . . . . 8 ⊢ (𝑥 = (2nd ‘𝑤) → ⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 = ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
143	142	eleq1d 2686	. . . . . . 7 ⊢ (𝑥 = (2nd ‘𝑤) → (⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 ∈ ℂ))
144	143	rspcv 3305	. . . . . 6 ⊢ ((2nd ‘𝑤) ∈ ⦋(1st ‘𝑤) / 𝑘⦌𝐷 → (∀𝑥 ∈ ⦋ (1st ‘𝑤) / 𝑘⦌𝐷⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 ∈ ℂ))
145	100, 140, 144	sylc 65	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 ∈ ℂ)
146	47, 53, 82, 87, 145	fprodcnv 14713	. . . 4 ⊢ (𝜑 → ∏𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ∏𝑧 ∈ ◡ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸)
147	41, 146	eqtr4d 2659	. . 3 ⊢ (𝜑 → ∏𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × ⦋𝑥 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ∏𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
148		fprodcom2.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)
149	148	ralrimiva 2966	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ Fin)
150	25	nfel1 2779	. . . . . 6 ⊢ Ⅎ𝑗⦋𝑥 / 𝑗⦌𝐵 ∈ Fin
151	28	eleq1d 2686	. . . . . 6 ⊢ (𝑗 = 𝑥 → (𝐵 ∈ Fin ↔ ⦋𝑥 / 𝑗⦌𝐵 ∈ Fin))
152	150, 151	rspc 3303	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 𝐵 ∈ Fin → ⦋𝑥 / 𝑗⦌𝐵 ∈ Fin))
153	149, 152	mpan9 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ⦋𝑥 / 𝑗⦌𝐵 ∈ Fin)
154	53, 56, 153, 131	fprod2d 14711	. . 3 ⊢ (𝜑 → ∏𝑥 ∈ 𝐴 ∏𝑦 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 = ∏𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × ⦋𝑥 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸)
155	47, 54, 77, 132	fprod2d 14711	. . 3 ⊢ (𝜑 → ∏𝑦 ∈ 𝐶 ∏𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 = ∏𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
156	147, 154, 155	3eqtr4d 2666	. 2 ⊢ (𝜑 → ∏𝑥 ∈ 𝐴 ∏𝑦 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 = ∏𝑦 ∈ 𝐶 ∏𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
157		nfcv 2764	. . 3 ⊢ Ⅎ𝑥∏𝑘 ∈ 𝐵 𝐸
158		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑗𝑦
159	158, 117	nfcsb 3551	. . . 4 ⊢ Ⅎ𝑗⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
160	25, 159	nfcprod 14641	. . 3 ⊢ Ⅎ𝑗∏𝑦 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
161		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑦𝐸
162		nfcsb1v 3549	. . . . 5 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐸
163		csbeq1a 3542	. . . . 5 ⊢ (𝑘 = 𝑦 → 𝐸 = ⦋𝑦 / 𝑘⦌𝐸)
164	161, 162, 163	cbvprodi 14647	. . . 4 ⊢ ∏𝑘 ∈ 𝐵 𝐸 = ∏𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐸
165	120	csbeq2dv 3992	. . . . . 6 ⊢ (𝑗 = 𝑥 → ⦋𝑦 / 𝑘⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
166	165	adantr 481	. . . . 5 ⊢ ((𝑗 = 𝑥 ∧ 𝑦 ∈ 𝐵) → ⦋𝑦 / 𝑘⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
167	28, 166	prodeq12dv 14656	. . . 4 ⊢ (𝑗 = 𝑥 → ∏𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐸 = ∏𝑦 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
168	164, 167	syl5eq 2668	. . 3 ⊢ (𝑗 = 𝑥 → ∏𝑘 ∈ 𝐵 𝐸 = ∏𝑦 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
169	157, 160, 168	cbvprodi 14647	. 2 ⊢ ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐸 = ∏𝑥 ∈ 𝐴 ∏𝑦 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
170		nfcv 2764	. . 3 ⊢ Ⅎ𝑦∏𝑗 ∈ 𝐷 𝐸
171	33, 125	nfcprod 14641	. . 3 ⊢ Ⅎ𝑘∏𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
172		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑥𝐸
173	172, 117, 120	cbvprodi 14647	. . . 4 ⊢ ∏𝑗 ∈ 𝐷 𝐸 = ∏𝑥 ∈ 𝐷 ⦋𝑥 / 𝑗⦌𝐸
174	127	adantr 481	. . . . 5 ⊢ ((𝑘 = 𝑦 ∧ 𝑥 ∈ 𝐷) → ⦋𝑥 / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
175	36, 174	prodeq12dv 14656	. . . 4 ⊢ (𝑘 = 𝑦 → ∏𝑥 ∈ 𝐷 ⦋𝑥 / 𝑗⦌𝐸 = ∏𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
176	173, 175	syl5eq 2668	. . 3 ⊢ (𝑘 = 𝑦 → ∏𝑗 ∈ 𝐷 𝐸 = ∏𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
177	170, 171, 176	cbvprodi 14647	. 2 ⊢ ∏𝑘 ∈ 𝐶 ∏𝑗 ∈ 𝐷 𝐸 = ∏𝑦 ∈ 𝐶 ∏𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
178	156, 169, 177	3eqtr4g 2681	1 ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐸 = ∏𝑘 ∈ 𝐶 ∏𝑗 ∈ 𝐷 𝐸)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  ⦋csb 3533  {csn 4177  ⟨cop 4183  ∪ ciun 4520   × cxp 5112  ^◡ccnv 5113  Rel wrel 5119  ‘cfv 5888  1^st c1st 7166  2^nd c2nd 7167  Fincfn 7955  ℂcc 9934  ∏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:  fprodcom  14716  fprod0diag  14717