Theorem fprodcom2fi 11516

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 4694	. . . . . . . . 9 ⊢ Rel ({𝑗} × 𝐵)
2	1	rgenw 2512	. . . . . . . 8 ⊢ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐵)
3		reliun 4706	. . . . . . . 8 ⊢ (Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐵))
4	2, 3	mpbir 145	. . . . . . 7 ⊢ Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
5		relcnv 4963	. . . . . . 7 ⊢ Rel ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷)
6		ancom 264	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑗 ∧ 𝑦 = 𝑘) ↔ (𝑦 = 𝑘 ∧ 𝑥 = 𝑗))
7		vex 2715	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
8		vex 2715	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
9	7, 8	opth 4197	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ (𝑥 = 𝑗 ∧ 𝑦 = 𝑘))
10	8, 7	opth 4197	. . . . . . . . . . . 12 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ↔ (𝑦 = 𝑘 ∧ 𝑥 = 𝑗))
11	6, 9, 10	3bitr4i 211	. . . . . . . . . . 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 465	. . . . . . . . 9 ⊢ (𝜑 → ((⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) ↔ (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷))))
15	14	2exbidv 1848	. . . . . . . 8 ⊢ (𝜑 → (∃𝑗∃𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) ↔ ∃𝑗∃𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷))))
16		eliunxp 4724	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗∃𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)))
17	7, 8	opelcnv 4767	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ⟨𝑦, 𝑥⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
18		eliunxp 4724	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑥⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ∃𝑘∃𝑗(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))
19		excom 1644	. . . . . . . . 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 205	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ∃𝑗∃𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))
21	15, 16, 20	3bitr4g 222	. . . . . . 7 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷)))
22	4, 5, 21	eqrelrdv 4681	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
23		nfcv 2299	. . . . . . 7 ⊢ Ⅎ𝑥({𝑗} × 𝐵)
24		nfcv 2299	. . . . . . . 8 ⊢ Ⅎ𝑗{𝑥}
25		nfcsb1v 3064	. . . . . . . 8 ⊢ Ⅎ𝑗⦋𝑥 / 𝑗⦌𝐵
26	24, 25	nfxp 4612	. . . . . . 7 ⊢ Ⅎ𝑗({𝑥} × ⦋𝑥 / 𝑗⦌𝐵)
27		sneq 3571	. . . . . . . 8 ⊢ (𝑗 = 𝑥 → {𝑗} = {𝑥})
28		csbeq1a 3040	. . . . . . . 8 ⊢ (𝑗 = 𝑥 → 𝐵 = ⦋𝑥 / 𝑗⦌𝐵)
29	27, 28	xpeq12d 4610	. . . . . . 7 ⊢ (𝑗 = 𝑥 → ({𝑗} × 𝐵) = ({𝑥} × ⦋𝑥 / 𝑗⦌𝐵))
30	23, 26, 29	cbviun 3886	. . . . . 6 ⊢ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × ⦋𝑥 / 𝑗⦌𝐵)
31		nfcv 2299	. . . . . . . 8 ⊢ Ⅎ𝑦({𝑘} × 𝐷)
32		nfcv 2299	. . . . . . . . 9 ⊢ Ⅎ𝑘{𝑦}
33		nfcsb1v 3064	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐷
34	32, 33	nfxp 4612	. . . . . . . 8 ⊢ Ⅎ𝑘({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
35		sneq 3571	. . . . . . . . 9 ⊢ (𝑘 = 𝑦 → {𝑘} = {𝑦})
36		csbeq1a 3040	. . . . . . . . 9 ⊢ (𝑘 = 𝑦 → 𝐷 = ⦋𝑦 / 𝑘⦌𝐷)
37	35, 36	xpeq12d 4610	. . . . . . . 8 ⊢ (𝑘 = 𝑦 → ({𝑘} × 𝐷) = ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
38	31, 34, 37	cbviun 3886	. . . . . . 7 ⊢ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) = ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
39	38	cnveqi 4760	. . . . . 6 ⊢ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) = ◡∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
40	22, 30, 39	3eqtr3g 2213	. . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 ({𝑥} × ⦋𝑥 / 𝑗⦌𝐵) = ◡∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
41	40	prodeq1d 11454	. . . 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 6093	. . . . . . 7 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → (1st ‘𝑤) = 𝑦)
43	42	csbeq1d 3038	. . . . . 6 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
44	8, 7	op2ndd 6094	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → (2nd ‘𝑤) = 𝑥)
45	44	csbeq1d 3038	. . . . . . 7 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → ⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑥 / 𝑗⦌𝐸)
46	45	csbeq2dv 3057	. . . . . 6 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → ⦋𝑦 / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
47	43, 46	eqtrd 2190	. . . . 5 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
48	7, 8	op2ndd 6094	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
49	48	csbeq1d 3038	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸)
50	7, 8	op1std 6093	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
51	50	csbeq1d 3038	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑥 / 𝑗⦌𝐸)
52	51	csbeq2dv 3057	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋𝑦 / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
53	49, 52	eqtrd 2190	. . . . 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 6756	. . . . . . . . 9 ⊢ (𝑦 ∈ V → {𝑦} ∈ Fin)
56	55	elv 2716	. . . . . . . 8 ⊢ {𝑦} ∈ Fin
57		fprodcom2fi.d	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐷 ∈ Fin)
58	57	ralrimiva 2530	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ 𝐶 𝐷 ∈ Fin)
59	33	nfel1 2310	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐷 ∈ Fin
60	36	eleq1d 2226	. . . . . . . . . 10 ⊢ (𝑘 = 𝑦 → (𝐷 ∈ Fin ↔ ⦋𝑦 / 𝑘⦌𝐷 ∈ Fin))
61	59, 60	rspc 2810	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐶 → (∀𝑘 ∈ 𝐶 𝐷 ∈ Fin → ⦋𝑦 / 𝑘⦌𝐷 ∈ Fin))
62	58, 61	mpan9 279	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ⦋𝑦 / 𝑘⦌𝐷 ∈ Fin)
63		xpfi 6871	. . . . . . . 8 ⊢ (({𝑦} ∈ Fin ∧ ⦋𝑦 / 𝑘⦌𝐷 ∈ Fin) → ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin)
64	56, 62, 63	sylancr 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin)
65	64	ralrimiva 2530	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin)
66		disjsnxp 6181	. . . . . . 7 ⊢ Disj 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
67	66	a1i 9	. . . . . 6 ⊢ (𝜑 → Disj 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
68		iunfidisj 6887	. . . . . 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 1220	. . . . 5 ⊢ (𝜑 → ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin)
70		reliun 4706	. . . . . . 7 ⊢ (Rel ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ↔ ∀𝑦 ∈ 𝐶 Rel ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
71		relxp 4694	. . . . . . . 8 ⊢ Rel ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
72	71	a1i 9	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → Rel ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
73	70, 72	mprgbir 2515	. . . . . 6 ⊢ Rel ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
74	73	a1i 9	. . . . 5 ⊢ (𝜑 → Rel ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
75		csbeq1 3034	. . . . . . . 8 ⊢ (𝑥 = (2nd ‘𝑤) → ⦋𝑥 / 𝑗⦌𝐸 = ⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
76	75	csbeq2dv 3057	. . . . . . 7 ⊢ (𝑥 = (2nd ‘𝑤) → ⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 = ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
77	76	eleq1d 2226	. . . . . 6 ⊢ (𝑥 = (2nd ‘𝑤) → (⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 ∈ ℂ))
78		csbeq1 3034	. . . . . . . 8 ⊢ (𝑦 = (1st ‘𝑤) → ⦋𝑦 / 𝑘⦌𝐷 = ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
79		csbeq1 3034	. . . . . . . . 9 ⊢ (𝑦 = (1st ‘𝑤) → ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 = ⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
80	79	eleq1d 2226	. . . . . . . 8 ⊢ (𝑦 = (1st ‘𝑤) → (⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
81	78, 80	raleqbidv 2664	. . . . . . 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 108	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → 𝜑)
83	33, 36	opeliunxp2f 6182	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑦, 𝑥⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷))
84	17, 83	sylbbr 135	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷) → ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
85	84	adantl 275	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
86	22	adantr 274	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
87	85, 86	eleqtrrd 2237	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
88		eliun 3853	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝐴 ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵))
89	87, 88	sylib 121	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → ∃𝑗 ∈ 𝐴 ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵))
90		simpr 109	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵))
91		opelxp 4615	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵) ↔ (𝑥 ∈ {𝑗} ∧ 𝑦 ∈ 𝐵))
92	90, 91	sylib 121	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → (𝑥 ∈ {𝑗} ∧ 𝑦 ∈ 𝐵))
93	92	simpld 111	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → 𝑥 ∈ {𝑗})
94		elsni 3578	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {𝑗} → 𝑥 = 𝑗)
95	93, 94	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → 𝑥 = 𝑗)
96		simpl 108	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → 𝑗 ∈ 𝐴)
97	95, 96	eqeltrd 2234	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → 𝑥 ∈ 𝐴)
98	97	rexlimiva 2569	. . . . . . . . . . 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 2293	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵
101	94	equcomd 1687	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ {𝑗} → 𝑗 = 𝑥)
102	101, 28	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ {𝑗} → 𝐵 = ⦋𝑥 / 𝑗⦌𝐵)
103	102	eleq2d 2227	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ {𝑗} → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵))
104	103	biimpa 294	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ {𝑗} ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵)
105	91, 104	sylbi 120	. . . . . . . . . . . . 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 2567	. . . . . . . . . . 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 2539	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ)
111		nfcsb1v 3064	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗⦋𝑥 / 𝑗⦌𝐸
112	111	nfel1 2310	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ
113	25, 112	nfralw 2494	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗∀𝑘 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ
114		csbeq1a 3040	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑥 → 𝐸 = ⦋𝑥 / 𝑗⦌𝐸)
115	114	eleq1d 2226	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑥 → (𝐸 ∈ ℂ ↔ ⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
116	28, 115	raleqbidv 2664	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑥 → (∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ ↔ ∀𝑘 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
117	113, 116	rspc 2810	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ → ∀𝑘 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
118	110, 117	mpan9 279	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
119		nfcsb1v 3064	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
120	119	nfel1 2310	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ
121		csbeq1a 3040	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑦 → ⦋𝑥 / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
122	121	eleq1d 2226	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑦 → (⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
123	120, 122	rspc 2810	. . . . . . . . . . . 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 377	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵)) → ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
126	82, 99, 108, 125	syl12anc 1218	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
127	126	ralrimivva 2539	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ∀𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
128	127	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → ∀𝑦 ∈ 𝐶 ∀𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
129		simpr 109	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
130		eliun 3853	. . . . . . . . 9 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ↔ ∃𝑦 ∈ 𝐶 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
131	129, 130	sylib 121	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → ∃𝑦 ∈ 𝐶 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
132		xp1st 6110	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) → (1st ‘𝑤) ∈ {𝑦})
133	132	adantl 275	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ {𝑦})
134		elsni 3578	. . . . . . . . . . 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 108	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → 𝑦 ∈ 𝐶)
137	135, 136	eqeltrd 2234	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ 𝐶)
138	137	rexlimiva 2569	. . . . . . . 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 2821	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → ∀𝑥 ∈ ⦋ (1st ‘𝑤) / 𝑘⦌𝐷⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
141		xp2nd 6111	. . . . . . . . . 10 ⊢ (𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) → (2nd ‘𝑤) ∈ ⦋𝑦 / 𝑘⦌𝐷)
142	141	adantl 275	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈ ⦋𝑦 / 𝑘⦌𝐷)
143	135	csbeq1d 3038	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → ⦋(1st ‘𝑤) / 𝑘⦌𝐷 = ⦋𝑦 / 𝑘⦌𝐷)
144	142, 143	eleqtrrd 2237	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈ ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
145	144	rexlimiva 2569	. . . . . . 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 2821	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 ∈ ℂ)
148	47, 53, 69, 74, 147	fprodcnv 11515	. . . 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 2193	. . 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 2530	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ Fin)
153	25	nfel1 2310	. . . . . 6 ⊢ Ⅎ𝑗⦋𝑥 / 𝑗⦌𝐵 ∈ Fin
154	28	eleq1d 2226	. . . . . 6 ⊢ (𝑗 = 𝑥 → (𝐵 ∈ Fin ↔ ⦋𝑥 / 𝑗⦌𝐵 ∈ Fin))
155	153, 154	rspc 2810	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 𝐵 ∈ Fin → ⦋𝑥 / 𝑗⦌𝐵 ∈ Fin))
156	152, 155	mpan9 279	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ⦋𝑥 / 𝑗⦌𝐵 ∈ Fin)
157	53, 150, 156, 125	fprod2d 11513	. . 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 11513	. . 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 2200	. 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 2299	. . 3 ⊢ Ⅎ𝑥∏𝑘 ∈ 𝐵 𝐸
161		nfcv 2299	. . . . 5 ⊢ Ⅎ𝑗𝑦
162	161, 111	nfcsbw 3067	. . . 4 ⊢ Ⅎ𝑗⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
163	25, 162	nfcprod 11445	. . 3 ⊢ Ⅎ𝑗∏𝑦 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
164		nfcv 2299	. . . . 5 ⊢ Ⅎ𝑦𝐸
165		nfcsb1v 3064	. . . . 5 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐸
166		csbeq1a 3040	. . . . 5 ⊢ (𝑘 = 𝑦 → 𝐸 = ⦋𝑦 / 𝑘⦌𝐸)
167	164, 165, 166	cbvprodi 11450	. . . 4 ⊢ ∏𝑘 ∈ 𝐵 𝐸 = ∏𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐸
168	114	csbeq2dv 3057	. . . . . 6 ⊢ (𝑗 = 𝑥 → ⦋𝑦 / 𝑘⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
169	168	adantr 274	. . . . 5 ⊢ ((𝑗 = 𝑥 ∧ 𝑦 ∈ 𝐵) → ⦋𝑦 / 𝑘⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
170	28, 169	prodeq12dv 11459	. . . 4 ⊢ (𝑗 = 𝑥 → ∏𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐸 = ∏𝑦 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
171	167, 170	syl5eq 2202	. . 3 ⊢ (𝑗 = 𝑥 → ∏𝑘 ∈ 𝐵 𝐸 = ∏𝑦 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
172	160, 163, 171	cbvprodi 11450	. 2 ⊢ ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐸 = ∏𝑥 ∈ 𝐴 ∏𝑦 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
173		nfcv 2299	. . 3 ⊢ Ⅎ𝑦∏𝑗 ∈ 𝐷 𝐸
174	33, 119	nfcprod 11445	. . 3 ⊢ Ⅎ𝑘∏𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
175		nfcv 2299	. . . . 5 ⊢ Ⅎ𝑥𝐸
176	175, 111, 114	cbvprodi 11450	. . . 4 ⊢ ∏𝑗 ∈ 𝐷 𝐸 = ∏𝑥 ∈ 𝐷 ⦋𝑥 / 𝑗⦌𝐸
177	121	adantr 274	. . . . 5 ⊢ ((𝑘 = 𝑦 ∧ 𝑥 ∈ 𝐷) → ⦋𝑥 / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
178	36, 177	prodeq12dv 11459	. . . 4 ⊢ (𝑘 = 𝑦 → ∏𝑥 ∈ 𝐷 ⦋𝑥 / 𝑗⦌𝐸 = ∏𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
179	176, 178	syl5eq 2202	. . 3 ⊢ (𝑘 = 𝑦 → ∏𝑗 ∈ 𝐷 𝐸 = ∏𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
180	173, 174, 179	cbvprodi 11450	. 2 ⊢ ∏𝑘 ∈ 𝐶 ∏𝑗 ∈ 𝐷 𝐸 = ∏𝑦 ∈ 𝐶 ∏𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
181	159, 172, 180	3eqtr4g 2215	1 ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐸 = ∏𝑘 ∈ 𝐶 ∏𝑗 ∈ 𝐷 𝐸)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1335  ∃wex 1472   ∈ wcel 2128  ∀wral 2435  ∃wrex 2436  Vcvv 2712  ⦋csb 3031  {csn 3560  ⟨cop 3563  ∪ ciun 3849  Disj wdisj 3942   × cxp 4583  ^◡ccnv 4584  Rel wrel 4590  ‘cfv 5169  1^st c1st 6083  2^nd c2nd 6084  Fincfn 6682  ℂcc 7724  ∏cprod 11440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546  ax-cnex 7817  ax-resscn 7818  ax-1cn 7819  ax-1re 7820  ax-icn 7821  ax-addcl 7822  ax-addrcl 7823  ax-mulcl 7824  ax-mulrcl 7825  ax-addcom 7826  ax-mulcom 7827  ax-addass 7828  ax-mulass 7829  ax-distr 7830  ax-i2m1 7831  ax-0lt1 7832  ax-1rid 7833  ax-0id 7834  ax-rnegex 7835  ax-precex 7836  ax-cnre 7837  ax-pre-ltirr 7838  ax-pre-ltwlin 7839  ax-pre-lttrn 7840  ax-pre-apti 7841  ax-pre-ltadd 7842  ax-pre-mulgt0 7843  ax-pre-mulext 7844  ax-arch 7845  ax-caucvg 7846

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-disj 3943  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-isom 5178  df-riota 5777  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-irdg 6314  df-frec 6335  df-1o 6360  df-oadd 6364  df-er 6477  df-en 6683  df-dom 6684  df-fin 6685  df-pnf 7908  df-mnf 7909  df-xr 7910  df-ltxr 7911  df-le 7912  df-sub 8042  df-neg 8043  df-reap 8444  df-ap 8451  df-div 8540  df-inn 8828  df-2 8886  df-3 8887  df-4 8888  df-n0 9085  df-z 9162  df-uz 9434  df-q 9522  df-rp 9554  df-fz 9906  df-fzo 10035  df-seqfrec 10338  df-exp 10412  df-ihash 10643  df-cj 10735  df-re 10736  df-im 10737  df-rsqrt 10891  df-abs 10892  df-clim 11169  df-proddc 11441

This theorem is referenced by:  fprodcom  11517  fprod0diagfz  11518