Theorem fprodcom2fi 12132

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)
fprodcom2fi.d	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐷 ∈ Fin)
fprodcom2.4	⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) ↔ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))
fprodcom2.5	⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐸 ∈ ℂ)

Assertion

Ref	Expression
fprodcom2fi	⊢ (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 fprodcom2fi

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

Step	Hyp	Ref	Expression
1		relxp 4827	. . . . . . . . 9 ⊢ Rel ({𝑗} × 𝐵)
2	1	rgenw 2585	. . . . . . . 8 ⊢ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐵)
3		reliun 4839	. . . . . . . 8 ⊢ (Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐵))
4	2, 3	mpbir 146	. . . . . . 7 ⊢ Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
5		relcnv 5105	. . . . . . 7 ⊢ Rel ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷)
6		ancom 266	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑗 ∧ 𝑦 = 𝑘) ↔ (𝑦 = 𝑘 ∧ 𝑥 = 𝑗))
7		vex 2802	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
8		vex 2802	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
9	7, 8	opth 4322	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ (𝑥 = 𝑗 ∧ 𝑦 = 𝑘))
10	8, 7	opth 4322	. . . . . . . . . . . 12 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ↔ (𝑦 = 𝑘 ∧ 𝑥 = 𝑗))
11	6, 9, 10	3bitr4i 212	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩)
12	11	a1i 9	. . . . . . . . . 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 473	. . . . . . . . 9 ⊢ (𝜑 → ((⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) ↔ (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷))))
15	14	2exbidv 1914	. . . . . . . 8 ⊢ (𝜑 → (∃𝑗∃𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) ↔ ∃𝑗∃𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷))))
16		eliunxp 4860	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗∃𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)))
17	7, 8	opelcnv 4903	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ⟨𝑦, 𝑥⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
18		eliunxp 4860	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑥⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ∃𝑘∃𝑗(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))
19		excom 1710	. . . . . . . . 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 206	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ∃𝑗∃𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))
21	15, 16, 20	3bitr4g 223	. . . . . . 7 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷)))
22	4, 5, 21	eqrelrdv 4814	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
23		nfcv 2372	. . . . . . 7 ⊢ Ⅎ𝑥({𝑗} × 𝐵)
24		nfcv 2372	. . . . . . . 8 ⊢ Ⅎ𝑗{𝑥}
25		nfcsb1v 3157	. . . . . . . 8 ⊢ Ⅎ𝑗⦋𝑥 / 𝑗⦌𝐵
26	24, 25	nfxp 4745	. . . . . . 7 ⊢ Ⅎ𝑗({𝑥} × ⦋𝑥 / 𝑗⦌𝐵)
27		sneq 3677	. . . . . . . 8 ⊢ (𝑗 = 𝑥 → {𝑗} = {𝑥})
28		csbeq1a 3133	. . . . . . . 8 ⊢ (𝑗 = 𝑥 → 𝐵 = ⦋𝑥 / 𝑗⦌𝐵)
29	27, 28	xpeq12d 4743	. . . . . . 7 ⊢ (𝑗 = 𝑥 → ({𝑗} × 𝐵) = ({𝑥} × ⦋𝑥 / 𝑗⦌𝐵))
30	23, 26, 29	cbviun 4001	. . . . . 6 ⊢ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × ⦋𝑥 / 𝑗⦌𝐵)
31		nfcv 2372	. . . . . . . 8 ⊢ Ⅎ𝑦({𝑘} × 𝐷)
32		nfcv 2372	. . . . . . . . 9 ⊢ Ⅎ𝑘{𝑦}
33		nfcsb1v 3157	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐷
34	32, 33	nfxp 4745	. . . . . . . 8 ⊢ Ⅎ𝑘({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
35		sneq 3677	. . . . . . . . 9 ⊢ (𝑘 = 𝑦 → {𝑘} = {𝑦})
36		csbeq1a 3133	. . . . . . . . 9 ⊢ (𝑘 = 𝑦 → 𝐷 = ⦋𝑦 / 𝑘⦌𝐷)
37	35, 36	xpeq12d 4743	. . . . . . . 8 ⊢ (𝑘 = 𝑦 → ({𝑘} × 𝐷) = ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
38	31, 34, 37	cbviun 4001	. . . . . . 7 ⊢ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) = ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
39	38	cnveqi 4896	. . . . . 6 ⊢ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) = ◡∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
40	22, 30, 39	3eqtr3g 2285	. . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 ({𝑥} × ⦋𝑥 / 𝑗⦌𝐵) = ◡∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
41	40	prodeq1d 12070	. . . 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 6292	. . . . . . 7 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → (1st ‘𝑤) = 𝑦)
43	42	csbeq1d 3131	. . . . . 6 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
44	8, 7	op2ndd 6293	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → (2nd ‘𝑤) = 𝑥)
45	44	csbeq1d 3131	. . . . . . 7 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → ⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑥 / 𝑗⦌𝐸)
46	45	csbeq2dv 3150	. . . . . 6 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → ⦋𝑦 / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
47	43, 46	eqtrd 2262	. . . . 5 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
48	7, 8	op2ndd 6293	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
49	48	csbeq1d 3131	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸)
50	7, 8	op1std 6292	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
51	50	csbeq1d 3131	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑥 / 𝑗⦌𝐸)
52	51	csbeq2dv 3150	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋𝑦 / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
53	49, 52	eqtrd 2262	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
54		fprodcom2.2	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Fin)
55		snfig 6965	. . . . . . . . 9 ⊢ (𝑦 ∈ V → {𝑦} ∈ Fin)
56	55	elv 2803	. . . . . . . 8 ⊢ {𝑦} ∈ Fin
57		fprodcom2fi.d	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐷 ∈ Fin)
58	57	ralrimiva 2603	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ 𝐶 𝐷 ∈ Fin)
59	33	nfel1 2383	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐷 ∈ Fin
60	36	eleq1d 2298	. . . . . . . . . 10 ⊢ (𝑘 = 𝑦 → (𝐷 ∈ Fin ↔ ⦋𝑦 / 𝑘⦌𝐷 ∈ Fin))
61	59, 60	rspc 2901	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐶 → (∀𝑘 ∈ 𝐶 𝐷 ∈ Fin → ⦋𝑦 / 𝑘⦌𝐷 ∈ Fin))
62	58, 61	mpan9 281	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ⦋𝑦 / 𝑘⦌𝐷 ∈ Fin)
63		xpfi 7090	. . . . . . . 8 ⊢ (({𝑦} ∈ Fin ∧ ⦋𝑦 / 𝑘⦌𝐷 ∈ Fin) → ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin)
64	56, 62, 63	sylancr 414	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin)
65	64	ralrimiva 2603	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin)
66		disjsnxp 6381	. . . . . . 7 ⊢ Disj 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
67	66	a1i 9	. . . . . 6 ⊢ (𝜑 → Disj 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
68		iunfidisj 7109	. . . . . 6 ⊢ ((𝐶 ∈ Fin ∧ ∀𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin ∧ Disj 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin)
69	54, 65, 67, 68	syl3anc 1271	. . . . 5 ⊢ (𝜑 → ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin)
70		reliun 4839	. . . . . . 7 ⊢ (Rel ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ↔ ∀𝑦 ∈ 𝐶 Rel ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
71		relxp 4827	. . . . . . . 8 ⊢ Rel ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
72	71	a1i 9	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → Rel ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
73	70, 72	mprgbir 2588	. . . . . 6 ⊢ Rel ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
74	73	a1i 9	. . . . 5 ⊢ (𝜑 → Rel ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
75		csbeq1 3127	. . . . . . . 8 ⊢ (𝑥 = (2nd ‘𝑤) → ⦋𝑥 / 𝑗⦌𝐸 = ⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
76	75	csbeq2dv 3150	. . . . . . 7 ⊢ (𝑥 = (2nd ‘𝑤) → ⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 = ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
77	76	eleq1d 2298	. . . . . 6 ⊢ (𝑥 = (2nd ‘𝑤) → (⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 ∈ ℂ))
78		csbeq1 3127	. . . . . . . 8 ⊢ (𝑦 = (1st ‘𝑤) → ⦋𝑦 / 𝑘⦌𝐷 = ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
79		csbeq1 3127	. . . . . . . . 9 ⊢ (𝑦 = (1st ‘𝑤) → ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 = ⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
80	79	eleq1d 2298	. . . . . . . 8 ⊢ (𝑦 = (1st ‘𝑤) → (⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
81	78, 80	raleqbidv 2744	. . . . . . 7 ⊢ (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))
82		simpl 109	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → 𝜑)
83	33, 36	opeliunxp2f 6382	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑦, 𝑥⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷))
84	17, 83	sylbbr 136	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷) → ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
85	84	adantl 277	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
86	22	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
87	85, 86	eleqtrrd 2309	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
88		eliun 3968	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝐴 ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵))
89	87, 88	sylib 122	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → ∃𝑗 ∈ 𝐴 ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵))
90		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵))
91		opelxp 4748	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵) ↔ (𝑥 ∈ {𝑗} ∧ 𝑦 ∈ 𝐵))
92	90, 91	sylib 122	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → (𝑥 ∈ {𝑗} ∧ 𝑦 ∈ 𝐵))
93	92	simpld 112	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → 𝑥 ∈ {𝑗})
94		elsni 3684	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {𝑗} → 𝑥 = 𝑗)
95	93, 94	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → 𝑥 = 𝑗)
96		simpl 109	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → 𝑗 ∈ 𝐴)
97	95, 96	eqeltrd 2306	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → 𝑥 ∈ 𝐴)
98	97	rexlimiva 2643	. . . . . . . . . . 11 ⊢ (∃𝑗 ∈ 𝐴 ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵) → 𝑥 ∈ 𝐴)
99	89, 98	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → 𝑥 ∈ 𝐴)
100	25	nfcri 2366	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵
101	94	equcomd 1753	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ {𝑗} → 𝑗 = 𝑥)
102	101, 28	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ {𝑗} → 𝐵 = ⦋𝑥 / 𝑗⦌𝐵)
103	102	eleq2d 2299	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ {𝑗} → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵))
104	103	biimpa 296	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ {𝑗} ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵)
105	91, 104	sylbi 121	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵) → 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵)
106	105	a1i 9	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵) → 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵))
107	100, 106	rexlimi 2641	. . . . . . . . . . 11 ⊢ (∃𝑗 ∈ 𝐴 ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵) → 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵)
108	89, 107	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵)
109		fprodcom2.5	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐸 ∈ ℂ)
110	109	ralrimivva 2612	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ)
111		nfcsb1v 3157	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗⦋𝑥 / 𝑗⦌𝐸
112	111	nfel1 2383	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ
113	25, 112	nfralw 2567	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗∀𝑘 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ
114		csbeq1a 3133	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑥 → 𝐸 = ⦋𝑥 / 𝑗⦌𝐸)
115	114	eleq1d 2298	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑥 → (𝐸 ∈ ℂ ↔ ⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
116	28, 115	raleqbidv 2744	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑥 → (∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ ↔ ∀𝑘 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
117	113, 116	rspc 2901	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ → ∀𝑘 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
118	110, 117	mpan9 281	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
119		nfcsb1v 3157	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
120	119	nfel1 2383	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ
121		csbeq1a 3133	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑦 → ⦋𝑥 / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
122	121	eleq1d 2298	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑦 → (⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
123	120, 122	rspc 2901	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵 → (∀𝑘 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ → ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
124	118, 123	syl5com 29	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵 → ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
125	124	impr 379	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵)) → ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
126	82, 99, 108, 125	syl12anc 1269	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
127	126	ralrimivva 2612	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ∀𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
128	127	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → ∀𝑦 ∈ 𝐶 ∀𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
129		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
130		eliun 3968	. . . . . . . . 9 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ↔ ∃𝑦 ∈ 𝐶 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
131	129, 130	sylib 122	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → ∃𝑦 ∈ 𝐶 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
132		xp1st 6309	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) → (1st ‘𝑤) ∈ {𝑦})
133	132	adantl 277	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ {𝑦})
134		elsni 3684	. . . . . . . . . . 11 ⊢ ((1st ‘𝑤) ∈ {𝑦} → (1st ‘𝑤) = 𝑦)
135	133, 134	syl 14	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (1st ‘𝑤) = 𝑦)
136		simpl 109	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → 𝑦 ∈ 𝐶)
137	135, 136	eqeltrd 2306	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ 𝐶)
138	137	rexlimiva 2643	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐶 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) → (1st ‘𝑤) ∈ 𝐶)
139	131, 138	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ 𝐶)
140	81, 128, 139	rspcdva 2912	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → ∀𝑥 ∈ ⦋ (1st ‘𝑤) / 𝑘⦌𝐷⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
141		xp2nd 6310	. . . . . . . . . 10 ⊢ (𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) → (2nd ‘𝑤) ∈ ⦋𝑦 / 𝑘⦌𝐷)
142	141	adantl 277	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈ ⦋𝑦 / 𝑘⦌𝐷)
143	135	csbeq1d 3131	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → ⦋(1st ‘𝑤) / 𝑘⦌𝐷 = ⦋𝑦 / 𝑘⦌𝐷)
144	142, 143	eleqtrrd 2309	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈ ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
145	144	rexlimiva 2643	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐶 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) → (2nd ‘𝑤) ∈ ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
146	131, 145	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈ ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
147	77, 140, 146	rspcdva 2912	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 ∈ ℂ)
148	47, 53, 69, 74, 147	fprodcnv 12131	. . . 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)
149	41, 148	eqtr4d 2265	. . 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)
150		fprodcom2.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin)
151		fprodcom2.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)
152	151	ralrimiva 2603	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ Fin)
153	25	nfel1 2383	. . . . . 6 ⊢ Ⅎ𝑗⦋𝑥 / 𝑗⦌𝐵 ∈ Fin
154	28	eleq1d 2298	. . . . . 6 ⊢ (𝑗 = 𝑥 → (𝐵 ∈ Fin ↔ ⦋𝑥 / 𝑗⦌𝐵 ∈ Fin))
155	153, 154	rspc 2901	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 𝐵 ∈ Fin → ⦋𝑥 / 𝑗⦌𝐵 ∈ Fin))
156	152, 155	mpan9 281	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ⦋𝑥 / 𝑗⦌𝐵 ∈ Fin)
157	53, 150, 156, 125	fprod2d 12129	. . 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)
158	47, 54, 62, 126	fprod2d 12129	. . 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)
159	149, 157, 158	3eqtr4d 2272	. 2 ⊢ (𝜑 → ∏𝑥 ∈ 𝐴 ∏𝑦 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 = ∏𝑦 ∈ 𝐶 ∏𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
160		nfcv 2372	. . 3 ⊢ Ⅎ𝑥∏𝑘 ∈ 𝐵 𝐸
161		nfcv 2372	. . . . 5 ⊢ Ⅎ𝑗𝑦
162	161, 111	nfcsbw 3161	. . . 4 ⊢ Ⅎ𝑗⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
163	25, 162	nfcprod 12061	. . 3 ⊢ Ⅎ𝑗∏𝑦 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
164		nfcv 2372	. . . . 5 ⊢ Ⅎ𝑦𝐸
165		nfcsb1v 3157	. . . . 5 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐸
166		csbeq1a 3133	. . . . 5 ⊢ (𝑘 = 𝑦 → 𝐸 = ⦋𝑦 / 𝑘⦌𝐸)
167	164, 165, 166	cbvprodi 12066	. . . 4 ⊢ ∏𝑘 ∈ 𝐵 𝐸 = ∏𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐸
168	114	csbeq2dv 3150	. . . . . 6 ⊢ (𝑗 = 𝑥 → ⦋𝑦 / 𝑘⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
169	168	adantr 276	. . . . 5 ⊢ ((𝑗 = 𝑥 ∧ 𝑦 ∈ 𝐵) → ⦋𝑦 / 𝑘⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
170	28, 169	prodeq12dv 12075	. . . 4 ⊢ (𝑗 = 𝑥 → ∏𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐸 = ∏𝑦 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
171	167, 170	eqtrid 2274	. . 3 ⊢ (𝑗 = 𝑥 → ∏𝑘 ∈ 𝐵 𝐸 = ∏𝑦 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
172	160, 163, 171	cbvprodi 12066	. 2 ⊢ ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐸 = ∏𝑥 ∈ 𝐴 ∏𝑦 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
173		nfcv 2372	. . 3 ⊢ Ⅎ𝑦∏𝑗 ∈ 𝐷 𝐸
174	33, 119	nfcprod 12061	. . 3 ⊢ Ⅎ𝑘∏𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
175		nfcv 2372	. . . . 5 ⊢ Ⅎ𝑥𝐸
176	175, 111, 114	cbvprodi 12066	. . . 4 ⊢ ∏𝑗 ∈ 𝐷 𝐸 = ∏𝑥 ∈ 𝐷 ⦋𝑥 / 𝑗⦌𝐸
177	121	adantr 276	. . . . 5 ⊢ ((𝑘 = 𝑦 ∧ 𝑥 ∈ 𝐷) → ⦋𝑥 / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
178	36, 177	prodeq12dv 12075	. . . 4 ⊢ (𝑘 = 𝑦 → ∏𝑥 ∈ 𝐷 ⦋𝑥 / 𝑗⦌𝐸 = ∏𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
179	176, 178	eqtrid 2274	. . 3 ⊢ (𝑘 = 𝑦 → ∏𝑗 ∈ 𝐷 𝐸 = ∏𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
180	173, 174, 179	cbvprodi 12066	. 2 ⊢ ∏𝑘 ∈ 𝐶 ∏𝑗 ∈ 𝐷 𝐸 = ∏𝑦 ∈ 𝐶 ∏𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
181	159, 172, 180	3eqtr4g 2287	1 ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐸 = ∏𝑘 ∈ 𝐶 ∏𝑗 ∈ 𝐷 𝐸)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  Vcvv 2799  ⦋csb 3124  {csn 3666  ⟨cop 3669  ∪ ciun 3964  Disj wdisj 4058   × cxp 4716  ^◡ccnv 4717  Rel wrel 4723  ‘cfv 5317  1^st c1st 6282  2^nd c2nd 6283  Fincfn 6885  ℂcc 7993  ∏cprod 12056

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-disj 4059  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-isom 5326  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-frec 6535  df-1o 6560  df-oadd 6564  df-er 6678  df-en 6886  df-dom 6887  df-fin 6888  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-n0 9366  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-fz 10201  df-fzo 10335  df-seqfrec 10665  df-exp 10756  df-ihash 10993  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505  df-clim 11785  df-proddc 12057

This theorem is referenced by:  fprodcom  12133  fprod0diagfz  12134