Theorem fprodcom2 16183

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 5674	. . . . . . . . 9 ⊢ Rel ({𝑗} × 𝐵)
2	1	rgenw 3084	. . . . . . . 8 ⊢ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐵)
3		reliun 5800	. . . . . . . 8 ⊢ (Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐵))
4	2, 3	mpbir 234	. . . . . . 7 ⊢ Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
5		relcnv 6113	. . . . . . 7 ⊢ Rel ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷)
6		ancom 466	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑗 ∧ 𝑦 = 𝑘) ↔ (𝑦 = 𝑘 ∧ 𝑥 = 𝑗))
7		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
8		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
9	7, 8	opth 5450	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ (𝑥 = 𝑗 ∧ 𝑦 = 𝑘))
10	8, 7	opth 5450	. . . . . . . . . . . 12 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ↔ (𝑦 = 𝑘 ∧ 𝑥 = 𝑗))
11	6, 9, 10	3bitr4i 306	. . . . . . . . . . 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 644	. . . . . . . . 9 ⊢ (𝜑 → ((⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) ↔ (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷))))
15	14	2exbidv 1959	. . . . . . . 8 ⊢ (𝜑 → (∃𝑗∃𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) ↔ ∃𝑗∃𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷))))
16		eliunxp 5821	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗∃𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)))
17	7, 8	opelcnv 5867	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ⟨𝑦, 𝑥⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
18		eliunxp 5821	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑥⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ∃𝑘∃𝑗(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))
19		excom 2202	. . . . . . . . 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 300	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ∃𝑗∃𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))
21	15, 16, 20	3bitr4g 317	. . . . . . 7 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷)))
22	4, 5, 21	eqrelrdv 5774	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
23		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑥({𝑗} × 𝐵)
24		nfcv 2926	. . . . . . . 8 ⊢ Ⅎ𝑗{𝑥}
25		nfcsb1v 3875	. . . . . . . 8 ⊢ Ⅎ𝑗⦋𝑥 / 𝑗⦌𝐵
26	24, 25	nfxp 5689	. . . . . . 7 ⊢ Ⅎ𝑗({𝑥} × ⦋𝑥 / 𝑗⦌𝐵)
27		sneq 4594	. . . . . . . 8 ⊢ (𝑗 = 𝑥 → {𝑗} = {𝑥})
28		csbeq1a 3865	. . . . . . . 8 ⊢ (𝑗 = 𝑥 → 𝐵 = ⦋𝑥 / 𝑗⦌𝐵)
29	27, 28	xpeq12d 5687	. . . . . . 7 ⊢ (𝑗 = 𝑥 → ({𝑗} × 𝐵) = ({𝑥} × ⦋𝑥 / 𝑗⦌𝐵))
30	23, 26, 29	cbviun 4996	. . . . . 6 ⊢ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × ⦋𝑥 / 𝑗⦌𝐵)
31		nfcv 2926	. . . . . . . 8 ⊢ Ⅎ𝑦({𝑘} × 𝐷)
32		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑘{𝑦}
33		nfcsb1v 3875	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐷
34	32, 33	nfxp 5689	. . . . . . . 8 ⊢ Ⅎ𝑘({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
35		sneq 4594	. . . . . . . . 9 ⊢ (𝑘 = 𝑦 → {𝑘} = {𝑦})
36		csbeq1a 3865	. . . . . . . . 9 ⊢ (𝑘 = 𝑦 → 𝐷 = ⦋𝑦 / 𝑘⦌𝐷)
37	35, 36	xpeq12d 5687	. . . . . . . 8 ⊢ (𝑘 = 𝑦 → ({𝑘} × 𝐷) = ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
38	31, 34, 37	cbviun 4996	. . . . . . 7 ⊢ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) = ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
39	38	cnveqi 5860	. . . . . 6 ⊢ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) = ◡∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
40	22, 30, 39	3eqtr3g 2821	. . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 ({𝑥} × ⦋𝑥 / 𝑗⦌𝐵) = ◡∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
41	40	prodeq1d 16120	. . . 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 8038	. . . . . . 7 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → (1st ‘𝑤) = 𝑦)
43	42	csbeq1d 3855	. . . . . 6 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
44	8, 7	op2ndd 8039	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → (2nd ‘𝑤) = 𝑥)
45	44	csbeq1d 3855	. . . . . . 7 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → ⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑥 / 𝑗⦌𝐸)
46	45	csbeq2dv 3858	. . . . . 6 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → ⦋𝑦 / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
47	43, 46	eqtrd 2798	. . . . 5 ⊢ (𝑤 = ⟨𝑦, 𝑥⟩ → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
48	7, 8	op2ndd 8039	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
49	48	csbeq1d 3855	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸)
50	7, 8	op1std 8038	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
51	50	csbeq1d 3855	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑥 / 𝑗⦌𝐸)
52	51	csbeq2dv 3858	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⦋𝑦 / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
53	49, 52	eqtrd 2798	. . . . 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 9102	. . . . . . . 8 ⊢ {𝑦} ∈ Fin
56		fprodcom2.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ Fin)
57	56	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ Fin)
58	33, 36	opeliunxp2f 8258	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑦, 𝑥⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷))
59	17, 58	sylbbr 239	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷) → ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
60	59	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
61	22	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
62	60, 61	eleqtrrd 2867	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
63		eliun 4958	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝐴 ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵))
64	62, 63	sylib 221	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → ∃𝑗 ∈ 𝐴 ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵))
65		opelxp 5692	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵) ↔ (𝑥 ∈ {𝑗} ∧ 𝑦 ∈ 𝐵))
66	65	bilani 510	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → (𝑥 ∈ {𝑗} ∧ 𝑦 ∈ 𝐵))
67	66	simpld 500	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → 𝑥 ∈ {𝑗})
68		elsni 4601	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ {𝑗} → 𝑥 = 𝑗)
69	67, 68	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → 𝑥 = 𝑗)
70		simpl 488	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → 𝑗 ∈ 𝐴)
71	69, 70	eqeltrd 2864	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵)) → 𝑥 ∈ 𝐴)
72	71	rexlimiva 3159	. . . . . . . . . . . 12 ⊢ (∃𝑗 ∈ 𝐴 ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵) → 𝑥 ∈ 𝐴)
73	64, 72	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → 𝑥 ∈ 𝐴)
74	73	expr 462	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷 → 𝑥 ∈ 𝐴))
75	74	ssrdv 3919	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ⦋𝑦 / 𝑘⦌𝐷 ⊆ 𝐴)
76	57, 75	ssfid 9291	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ⦋𝑦 / 𝑘⦌𝐷 ∈ Fin)
77		xpfi 9342	. . . . . . . 8 ⊢ (({𝑦} ∈ Fin ∧ ⦋𝑦 / 𝑘⦌𝐷 ∈ Fin) → ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin)
78	55, 76, 77	sylancr 599	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin)
79	78	ralrimiva 3158	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin)
80		iunfi 9363	. . . . . 6 ⊢ ((𝐶 ∈ Fin ∧ ∀𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin) → ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin)
81	54, 79, 80	syl2anc 596	. . . . 5 ⊢ (𝜑 → ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ∈ Fin)
82		reliun 5800	. . . . . . 7 ⊢ (Rel ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ↔ ∀𝑦 ∈ 𝐶 Rel ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
83		relxp 5674	. . . . . . . 8 ⊢ Rel ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
84	83	a1i 11	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → Rel ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
85	82, 84	mprgbir 3087	. . . . . 6 ⊢ Rel ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)
86	85	a1i 11	. . . . 5 ⊢ (𝜑 → Rel ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
87		csbeq1 3854	. . . . . . . 8 ⊢ (𝑥 = (2nd ‘𝑤) → ⦋𝑥 / 𝑗⦌𝐸 = ⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
88	87	csbeq2dv 3858	. . . . . . 7 ⊢ (𝑥 = (2nd ‘𝑤) → ⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 = ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
89	88	eleq1d 2849	. . . . . 6 ⊢ (𝑥 = (2nd ‘𝑤) → (⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 ∈ ℂ))
90		csbeq1 3854	. . . . . . . 8 ⊢ (𝑦 = (1st ‘𝑤) → ⦋𝑦 / 𝑘⦌𝐷 = ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
91		csbeq1 3854	. . . . . . . . 9 ⊢ (𝑦 = (1st ‘𝑤) → ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 = ⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
92	91	eleq1d 2849	. . . . . . . 8 ⊢ (𝑦 = (1st ‘𝑤) → (⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
93	90, 92	raleqbidv 3338	. . . . . . 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))
94		simpl 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → 𝜑)
95	25	nfcri 2918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵
96	68	equcomd 2054	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ {𝑗} → 𝑗 = 𝑥)
97	96, 28	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ {𝑗} → 𝐵 = ⦋𝑥 / 𝑗⦌𝐵)
98	97	eleq2d 2850	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ {𝑗} → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵))
99	98	biimpa 482	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ {𝑗} ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵)
100	65, 99	sylbi 220	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵) → 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵)
101	100	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵) → 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵))
102	95, 101	rexlimi 3266	. . . . . . . . . . 11 ⊢ (∃𝑗 ∈ 𝐴 ⟨𝑥, 𝑦⟩ ∈ ({𝑗} × 𝐵) → 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵)
103	64, 102	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵)
104		fprodcom2.5	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐸 ∈ ℂ)
105	104	ralrimivva 3209	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ)
106		nfcsb1v 3875	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗⦋𝑥 / 𝑗⦌𝐸
107	106	nfel1 2942	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ
108	25, 107	nfralw 3313	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗∀𝑘 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ
109		csbeq1a 3865	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑥 → 𝐸 = ⦋𝑥 / 𝑗⦌𝐸)
110	109	eleq1d 2849	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑥 → (𝐸 ∈ ℂ ↔ ⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
111	28, 110	raleqbidv 3338	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑥 → (∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ ↔ ∀𝑘 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
112	108, 111	rspc 3568	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ → ∀𝑘 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
113	105, 112	mpan9 516	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
114		nfcsb1v 3875	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
115	114	nfel1 2942	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ
116		csbeq1a 3865	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑦 → ⦋𝑥 / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
117	116	eleq1d 2849	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑦 → (⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
118	115, 117	rspc 3568	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵 → (∀𝑘 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ → ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
119	113, 118	syl5com 32	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵 → ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ))
120	119	impr 460	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ⦋𝑥 / 𝑗⦌𝐵)) → ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
121	94, 73, 103, 120	syl12anc 850	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 ∈ ⦋𝑦 / 𝑘⦌𝐷)) → ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
122	121	ralrimivva 3209	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ∀𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
123	122	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → ∀𝑦 ∈ 𝐶 ∀𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
124		eliun 4958	. . . . . . . . 9 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) ↔ ∃𝑦 ∈ 𝐶 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
125	124	bilani 510	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → ∃𝑦 ∈ 𝐶 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷))
126		xp1st 8060	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) → (1st ‘𝑤) ∈ {𝑦})
127	126	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ {𝑦})
128		elsni 4601	. . . . . . . . . . 11 ⊢ ((1st ‘𝑤) ∈ {𝑦} → (1st ‘𝑤) = 𝑦)
129	127, 128	syl 18	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (1st ‘𝑤) = 𝑦)
130		simpl 488	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → 𝑦 ∈ 𝐶)
131	129, 130	eqeltrd 2864	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ 𝐶)
132	131	rexlimiva 3159	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐶 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) → (1st ‘𝑤) ∈ 𝐶)
133	125, 132	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ 𝐶)
134	93, 123, 133	rspcdva 3581	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → ∀𝑥 ∈ ⦋ (1st ‘𝑤) / 𝑘⦌𝐷⦋(1st ‘𝑤) / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 ∈ ℂ)
135		xp2nd 8061	. . . . . . . . . 10 ⊢ (𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) → (2nd ‘𝑤) ∈ ⦋𝑦 / 𝑘⦌𝐷)
136	135	adantl 487	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈ ⦋𝑦 / 𝑘⦌𝐷)
137	129	csbeq1d 3855	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → ⦋(1st ‘𝑤) / 𝑘⦌𝐷 = ⦋𝑦 / 𝑘⦌𝐷)
138	136, 137	eleqtrrd 2867	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈ ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
139	138	rexlimiva 3159	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐶 𝑤 ∈ ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷) → (2nd ‘𝑤) ∈ ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
140	125, 139	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈ ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
141	89, 134, 140	rspcdva 3581	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑦 ∈ 𝐶 ({𝑦} × ⦋𝑦 / 𝑘⦌𝐷)) → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 ∈ ℂ)
142	47, 53, 81, 86, 141	fprodcnv 16182	. . . 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)
143	41, 142	eqtr4d 2801	. . 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)
144		fprodcom2.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)
145	144	ralrimiva 3158	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ Fin)
146	25	nfel1 2942	. . . . . 6 ⊢ Ⅎ𝑗⦋𝑥 / 𝑗⦌𝐵 ∈ Fin
147	28	eleq1d 2849	. . . . . 6 ⊢ (𝑗 = 𝑥 → (𝐵 ∈ Fin ↔ ⦋𝑥 / 𝑗⦌𝐵 ∈ Fin))
148	146, 147	rspc 3568	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 𝐵 ∈ Fin → ⦋𝑥 / 𝑗⦌𝐵 ∈ Fin))
149	145, 148	mpan9 516	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ⦋𝑥 / 𝑗⦌𝐵 ∈ Fin)
150	53, 56, 149, 120	fprod2d 16180	. . 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)
151	47, 54, 76, 121	fprod2d 16180	. . 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)
152	143, 150, 151	3eqtr4d 2808	. 2 ⊢ (𝜑 → ∏𝑥 ∈ 𝐴 ∏𝑦 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸 = ∏𝑦 ∈ 𝐶 ∏𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
153		nfcv 2926	. . 3 ⊢ Ⅎ𝑥∏𝑘 ∈ 𝐵 𝐸
154		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑗𝑦
155	154, 106	nfcsbw 3877	. . . 4 ⊢ Ⅎ𝑗⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
156	25, 155	nfcprod 16110	. . 3 ⊢ Ⅎ𝑗∏𝑦 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
157		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑦𝐸
158		nfcsb1v 3875	. . . . 5 ⊢ Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐸
159		csbeq1a 3865	. . . . 5 ⊢ (𝑘 = 𝑦 → 𝐸 = ⦋𝑦 / 𝑘⦌𝐸)
160	157, 158, 159	cbvprodi 16116	. . . 4 ⊢ ∏𝑘 ∈ 𝐵 𝐸 = ∏𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐸
161	109	csbeq2dv 3858	. . . . . 6 ⊢ (𝑗 = 𝑥 → ⦋𝑦 / 𝑘⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
162	161	adantr 486	. . . . 5 ⊢ ((𝑗 = 𝑥 ∧ 𝑦 ∈ 𝐵) → ⦋𝑦 / 𝑘⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
163	28, 162	prodeq12dv 16125	. . . 4 ⊢ (𝑗 = 𝑥 → ∏𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐸 = ∏𝑦 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
164	160, 163	eqtrid 2810	. . 3 ⊢ (𝑗 = 𝑥 → ∏𝑘 ∈ 𝐵 𝐸 = ∏𝑦 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
165	153, 156, 164	cbvprodi 16116	. 2 ⊢ ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐸 = ∏𝑥 ∈ 𝐴 ∏𝑦 ∈ ⦋ 𝑥 / 𝑗⦌𝐵⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
166		nfcv 2926	. . 3 ⊢ Ⅎ𝑦∏𝑗 ∈ 𝐷 𝐸
167	33, 114	nfcprod 16110	. . 3 ⊢ Ⅎ𝑘∏𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
168		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑥𝐸
169	168, 106, 109	cbvprodi 16116	. . . 4 ⊢ ∏𝑗 ∈ 𝐷 𝐸 = ∏𝑥 ∈ 𝐷 ⦋𝑥 / 𝑗⦌𝐸
170	116	adantr 486	. . . . 5 ⊢ ((𝑘 = 𝑦 ∧ 𝑥 ∈ 𝐷) → ⦋𝑥 / 𝑗⦌𝐸 = ⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
171	36, 170	prodeq12dv 16125	. . . 4 ⊢ (𝑘 = 𝑦 → ∏𝑥 ∈ 𝐷 ⦋𝑥 / 𝑗⦌𝐸 = ∏𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
172	169, 171	eqtrid 2810	. . 3 ⊢ (𝑘 = 𝑦 → ∏𝑗 ∈ 𝐷 𝐸 = ∏𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸)
173	166, 167, 172	cbvprodi 16116	. 2 ⊢ ∏𝑘 ∈ 𝐶 ∏𝑗 ∈ 𝐷 𝐸 = ∏𝑦 ∈ 𝐶 ∏𝑥 ∈ ⦋ 𝑦 / 𝑘⦌𝐷⦋𝑦 / 𝑘⦌⦋𝑥 / 𝑗⦌𝐸
174	152, 165, 173	3eqtr4g 2823	1 ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐸 = ∏𝑘 ∈ 𝐶 ∏𝑗 ∈ 𝐷 𝐸)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  ⦋csb 3851  {csn 4588  ⟨cop 4838  ∪ ciun 4954   × cxp 5654  ^◡ccnv 5655  Rel wrel 5661  ‘cfv 6561  1^st c1st 8026  2^nd c2nd 8027  Fincfn 9004  ℂcc 11228  ∏cprod 16104

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-inf2 9672  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307  ax-pre-sup 11308

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-er 8748  df-en 9005  df-dom 9006  df-sdom 9007  df-fin 9008  df-sup 9464  df-oi 9534  df-card 10050  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-div 12004  df-nn 12366  df-2 12435  df-3 12436  df-n0 12637  df-z 12724  df-uz 12996  df-rp 13151  df-fz 13670  df-fzo 13819  df-seq 14175  df-exp 14235  df-hash 14505  df-cj 15298  df-re 15299  df-im 15300  df-sqrt 15434  df-abs 15435  df-clim 15687  df-prod 16105

This theorem is used by:  fprodcom  16184  fprod0diag  16185