Theorem fprod2dlem 16030

Description: Lemma for fprod2d 16031- induction step. (Contributed by Scott Fenton, 30-Jan-2018.)

Hypotheses

Ref	Expression
fprod2d.1	⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)
fprod2d.2	⊢ (𝜑 → 𝐴 ∈ Fin)
fprod2d.3	⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)
fprod2d.4	⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)
fprod2d.5	⊢ (𝜑 → ¬ 𝑦 ∈ 𝑥)
fprod2d.6	⊢ (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
fprod2d.7	⊢ (𝜓 ↔ ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷)

Assertion

Ref	Expression
fprod2dlem	⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)

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

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦,𝑧,𝑗,𝑘)   𝐴(𝑥,𝑦,𝑧)   𝐵(𝑥,𝑦,𝑗)   𝐶(𝑥,𝑦,𝑗,𝑘)   𝐷(𝑥,𝑦,𝑧)

Proof of Theorem fprod2dlem

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fprod2d.7	. . . 4 ⊢ (𝜓 ↔ ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷)
2	1	bilani 509	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷)
3		nfcv 2931	. . . . . 6 ⊢ Ⅎ𝑚∏𝑘 ∈ 𝐵 𝐶
4		nfcsb1v 3885	. . . . . . 7 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐵
5		nfcsb1v 3885	. . . . . . 7 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐶
6	4, 5	nfcprod 15959	. . . . . 6 ⊢ Ⅎ𝑗∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶
7		csbeq1a 3875	. . . . . . 7 ⊢ (𝑗 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑗⦌𝐵)
8		csbeq1a 3875	. . . . . . . 8 ⊢ (𝑗 = 𝑚 → 𝐶 = ⦋𝑚 / 𝑗⦌𝐶)
9	8	adantr 485	. . . . . . 7 ⊢ ((𝑗 = 𝑚 ∧ 𝑘 ∈ 𝐵) → 𝐶 = ⦋𝑚 / 𝑗⦌𝐶)
10	7, 9	prodeq12dv 15976	. . . . . 6 ⊢ (𝑗 = 𝑚 → ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶)
11	3, 6, 10	cbvprodi 15965	. . . . 5 ⊢ ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶 = ∏𝑚 ∈ {𝑦}∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶
12		fprod2d.6	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
13	12	unssbd 4155	. . . . . . . 8 ⊢ (𝜑 → {𝑦} ⊆ 𝐴)
14		vex 3467	. . . . . . . . 9 ⊢ 𝑦 ∈ V
15	14	snss 4752	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 ↔ {𝑦} ⊆ 𝐴)
16	13, 15	sylibr 237	. . . . . . 7 ⊢ (𝜑 → 𝑦 ∈ 𝐴)
17		fprod2d.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)
18	17	ralrimiva 3163	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ Fin)
19		nfcsb1v 3885	. . . . . . . . . . 11 ⊢ Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐵
20	19	nfel1 2947	. . . . . . . . . 10 ⊢ Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐵 ∈ Fin
21		csbeq1a 3875	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑗⦌𝐵)
22	21	eleq1d 2854	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (𝐵 ∈ Fin ↔ ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin))
23	20, 22	rspc 3578	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 𝐵 ∈ Fin → ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin))
24	16, 18, 23	sylc 66	. . . . . . . 8 ⊢ (𝜑 → ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin)
25		fprod2d.4	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)
26	25	ralrimivva 3214	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
27		nfcsb1v 3885	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐶
28	27	nfel1 2947	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ
29	19, 28	nfralw 3318	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ
30		csbeq1a 3875	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑗⦌𝐶)
31	30	eleq1d 2854	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑦 → (𝐶 ∈ ℂ ↔ ⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ))
32	21, 31	raleqbidv 3345	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → (∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ ↔ ∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ))
33	29, 32	rspc 3578	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ → ∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ))
34	16, 26, 33	sylc 66	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)
35	34	r19.21bi 3263	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) → ⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)
36	24, 35	fprodcl 16002	. . . . . . 7 ⊢ (𝜑 → ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)
37		csbeq1 3864	. . . . . . . . 9 ⊢ (𝑚 = 𝑦 → ⦋𝑚 / 𝑗⦌𝐵 = ⦋𝑦 / 𝑗⦌𝐵)
38		csbeq1 3864	. . . . . . . . . 10 ⊢ (𝑚 = 𝑦 → ⦋𝑚 / 𝑗⦌𝐶 = ⦋𝑦 / 𝑗⦌𝐶)
39	38	adantr 485	. . . . . . . . 9 ⊢ ((𝑚 = 𝑦 ∧ 𝑘 ∈ ⦋𝑚 / 𝑗⦌𝐵) → ⦋𝑚 / 𝑗⦌𝐶 = ⦋𝑦 / 𝑗⦌𝐶)
40	37, 39	prodeq12dv 15976	. . . . . . . 8 ⊢ (𝑚 = 𝑦 → ∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶)
41	40	prodsn 16012	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ) → ∏𝑚 ∈ {𝑦}∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶)
42	16, 36, 41	syl2anc 595	. . . . . 6 ⊢ (𝜑 → ∏𝑚 ∈ {𝑦}∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶)
43		nfcv 2931	. . . . . . . 8 ⊢ Ⅎ𝑚⦋𝑦 / 𝑗⦌𝐶
44		nfcsb1v 3885	. . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
45		csbeq1a 3875	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → ⦋𝑦 / 𝑗⦌𝐶 = ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
46	43, 44, 45	cbvprodi 15965	. . . . . . 7 ⊢ ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 = ∏𝑚 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
47		csbeq1 3864	. . . . . . . . 9 ⊢ (𝑚 = (2nd ‘𝑧) → ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
48		snfi 9036	. . . . . . . . . 10 ⊢ {𝑦} ∈ Fin
49		xpfi 9275	. . . . . . . . . 10 ⊢ (({𝑦} ∈ Fin ∧ ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin) → ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ∈ Fin)
50	48, 24, 49	sylancr 598	. . . . . . . . 9 ⊢ (𝜑 → ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ∈ Fin)
51		2ndconst 8092	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 → (2nd ↾ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)):({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)–1-1-onto→⦋𝑦 / 𝑗⦌𝐵)
52	16, 51	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (2nd ↾ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)):({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)–1-1-onto→⦋𝑦 / 𝑗⦌𝐵)
53		fvres 6898	. . . . . . . . . 10 ⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) → ((2nd ↾ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))‘𝑧) = (2nd ‘𝑧))
54	53	adantl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → ((2nd ↾ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))‘𝑧) = (2nd ‘𝑧))
55	44	nfel1 2947	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ
56	45	eleq1d 2854	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ ↔ ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ))
57	55, 56	rspc 3578	. . . . . . . . . 10 ⊢ (𝑚 ∈ ⦋𝑦 / 𝑗⦌𝐵 → (∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ → ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ))
58	34, 57	mpan9 515	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ⦋𝑦 / 𝑗⦌𝐵) → ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)
59	47, 50, 52, 54, 58	fprodf1o 15996	. . . . . . . 8 ⊢ (𝜑 → ∏𝑚 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
60		elxp 5682	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ↔ ∃𝑚∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)))
61		nfv 1941	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗 𝑧 = ⟨𝑚, 𝑘⟩
62		nfv 1941	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗 𝑚 ∈ {𝑦}
63	19	nfcri 2923	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵
64	62, 63	nfan 1926	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗(𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)
65	61, 64	nfan 1926	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵))
66	65	nfex 2363	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵))
67		nfv 1941	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵))
68		opeq1 4839	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑗 → ⟨𝑚, 𝑘⟩ = ⟨𝑗, 𝑘⟩)
69	68	eqeq2d 2780	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑗 → (𝑧 = ⟨𝑚, 𝑘⟩ ↔ 𝑧 = ⟨𝑗, 𝑘⟩))
70		eleq1w 2852	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑗 → (𝑚 ∈ {𝑦} ↔ 𝑗 ∈ {𝑦}))
71		velsn 4607	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ {𝑦} ↔ 𝑗 = 𝑦)
72	70, 71	bitrdi 290	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑗 → (𝑚 ∈ {𝑦} ↔ 𝑗 = 𝑦))
73	72	anbi1d 642	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) ↔ (𝑗 = 𝑦 ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)))
74	21	eleq2d 2855	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑦 → (𝑘 ∈ 𝐵 ↔ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵))
75	74	pm5.32i 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵) ↔ (𝑗 = 𝑦 ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵))
76	73, 75	bitr4di 292	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) ↔ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)))
77	69, 76	anbi12d 643	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑗 → ((𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) ↔ (𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵))))
78	77	exbidv 1948	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑗 → (∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) ↔ ∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵))))
79	66, 67, 78	cbvexv1 2380	. . . . . . . . . . . 12 ⊢ (∃𝑚∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) ↔ ∃𝑗∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)))
80	60, 79	bitri 278	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ↔ ∃𝑗∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)))
81		nfv 1941	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝜑
82		nfcv 2931	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(2nd ‘𝑧)
83	82, 27	nfcsbw 3887	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
84	83	nfeq2 2948	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
85		nfv 1941	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝜑
86		nfcsb1v 3885	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
87	86	nfeq2 2948	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
88		fprod2d.1	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)
89	88	ad2antlr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = 𝐶)
90	30	ad2antrl 740	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐶 = ⦋𝑦 / 𝑗⦌𝐶)
91		fveq2 6879	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd ‘𝑧) = (2nd ‘⟨𝑗, 𝑘⟩))
92		vex 3467	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑗 ∈ V
93		vex 3467	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑘 ∈ V
94	92, 93	op2nd 7991	. . . . . . . . . . . . . . . . . 18 ⊢ (2nd ‘⟨𝑗, 𝑘⟩) = 𝑘
95	91, 94	eqtr2di 2821	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝑘 = (2nd ‘𝑧))
96	95	ad2antlr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝑘 = (2nd ‘𝑧))
97		csbeq1a 3875	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (2nd ‘𝑧) → ⦋𝑦 / 𝑗⦌𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
98	96, 97	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → ⦋𝑦 / 𝑗⦌𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
99	89, 90, 98	3eqtrd 2808	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
100	99	expl 462	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶))
101	85, 87, 100	exlimd 2260	. . . . . . . . . . . 12 ⊢ (𝜑 → (∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶))
102	81, 84, 101	exlimd 2260	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑗∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶))
103	80, 102	biimtrid 245	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶))
104	103	imp 411	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
105	104	prodeq2dv 15972	. . . . . . . 8 ⊢ (𝜑 → ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
106	59, 105	eqtr4d 2807	. . . . . . 7 ⊢ (𝜑 → ∏𝑚 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)
107	46, 106	eqtrid 2816	. . . . . 6 ⊢ (𝜑 → ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)
108	42, 107	eqtrd 2804	. . . . 5 ⊢ (𝜑 → ∏𝑚 ∈ {𝑦}∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)
109	11, 108	eqtrid 2816	. . . 4 ⊢ (𝜑 → ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)
110	109	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)
111	2, 110	oveq12d 7426	. 2 ⊢ ((𝜑 ∧ 𝜓) → (∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 · ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶) = (∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷 · ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷))
112		fprod2d.5	. . . . 5 ⊢ (𝜑 → ¬ 𝑦 ∈ 𝑥)
113		disjsn 4679	. . . . 5 ⊢ ((𝑥 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝑥)
114	112, 113	sylibr 237	. . . 4 ⊢ (𝜑 → (𝑥 ∩ {𝑦}) = ∅)
115		eqidd 2770	. . . 4 ⊢ (𝜑 → (𝑥 ∪ {𝑦}) = (𝑥 ∪ {𝑦}))
116		fprod2d.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin)
117	116, 12	ssfid 9225	. . . 4 ⊢ (𝜑 → (𝑥 ∪ {𝑦}) ∈ Fin)
118	12	sselda 3945	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝑗 ∈ 𝐴)
119	25	anassrs 472	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
120	17, 119	fprodcl 16002	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∏𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
121	118, 120	syldan 602	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → ∏𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
122	114, 115, 117, 121	fprodsplit 16016	. . 3 ⊢ (𝜑 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = (∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 · ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶))
123	122	adantr 485	. 2 ⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = (∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 · ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶))
124		eliun 4961	. . . . . . . . . 10 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝑥 𝑧 ∈ ({𝑗} × 𝐵))
125		xp1st 8014	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ {𝑗})
126		elsni 4608	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑧) ∈ {𝑗} → (1st ‘𝑧) = 𝑗)
127	125, 126	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) = 𝑗)
128	127	eleq1d 2854	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → ((1st ‘𝑧) ∈ 𝑥 ↔ 𝑗 ∈ 𝑥))
129	128	biimparc 484	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ 𝑥 ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → (1st ‘𝑧) ∈ 𝑥)
130	129	rexlimiva 3164	. . . . . . . . . 10 ⊢ (∃𝑗 ∈ 𝑥 𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ 𝑥)
131	124, 130	sylbi 220	. . . . . . . . 9 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ 𝑥)
132		xp1st 8014	. . . . . . . . 9 ⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) → (1st ‘𝑧) ∈ {𝑦})
133	131, 132	anim12i 624	. . . . . . . 8 ⊢ ((𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → ((1st ‘𝑧) ∈ 𝑥 ∧ (1st ‘𝑧) ∈ {𝑦}))
134		elin 3929	. . . . . . . 8 ⊢ (𝑧 ∈ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) ↔ (𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)))
135		elin 3929	. . . . . . . 8 ⊢ ((1st ‘𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ ((1st ‘𝑧) ∈ 𝑥 ∧ (1st ‘𝑧) ∈ {𝑦}))
136	133, 134, 135	3imtr4i 295	. . . . . . 7 ⊢ (𝑧 ∈ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → (1st ‘𝑧) ∈ (𝑥 ∩ {𝑦}))
137	114	eleq2d 2855	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ (1st ‘𝑧) ∈ ∅))
138		noel 4299	. . . . . . . . 9 ⊢ ¬ (1st ‘𝑧) ∈ ∅
139	138	pm2.21i 120	. . . . . . . 8 ⊢ ((1st ‘𝑧) ∈ ∅ → 𝑧 ∈ ∅)
140	137, 139	biimtrdi 256	. . . . . . 7 ⊢ (𝜑 → ((1st ‘𝑧) ∈ (𝑥 ∩ {𝑦}) → 𝑧 ∈ ∅))
141	136, 140	syl5 35	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → 𝑧 ∈ ∅))
142	141	ssrdv 3951	. . . . 5 ⊢ (𝜑 → (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) ⊆ ∅)
143		ss0 4365	. . . . 5 ⊢ ((∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) ⊆ ∅ → (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) = ∅)
144	142, 143	syl 18	. . . 4 ⊢ (𝜑 → (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) = ∅)
145		iunxun 5061	. . . . . 6 ⊢ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵))
146		nfcv 2931	. . . . . . . . 9 ⊢ Ⅎ𝑚({𝑗} × 𝐵)
147		nfcv 2931	. . . . . . . . . 10 ⊢ Ⅎ𝑗{𝑚}
148	147, 4	nfxp 5692	. . . . . . . . 9 ⊢ Ⅎ𝑗({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)
149		sneq 4601	. . . . . . . . . 10 ⊢ (𝑗 = 𝑚 → {𝑗} = {𝑚})
150	149, 7	xpeq12d 5690	. . . . . . . . 9 ⊢ (𝑗 = 𝑚 → ({𝑗} × 𝐵) = ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵))
151	146, 148, 150	cbviun 5000	. . . . . . . 8 ⊢ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = ∪ 𝑚 ∈ {𝑦} ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)
152		sneq 4601	. . . . . . . . . 10 ⊢ (𝑚 = 𝑦 → {𝑚} = {𝑦})
153	152, 37	xpeq12d 5690	. . . . . . . . 9 ⊢ (𝑚 = 𝑦 → ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) = ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))
154	14, 153	iunxsn 5058	. . . . . . . 8 ⊢ ∪ 𝑚 ∈ {𝑦} ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) = ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)
155	151, 154	eqtri 2792	. . . . . . 7 ⊢ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)
156	155	uneq2i 4127	. . . . . 6 ⊢ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵)) = (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))
157	145, 156	eqtri 2792	. . . . 5 ⊢ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))
158	157	a1i 11	. . . 4 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)))
159		snfi 9036	. . . . . . 7 ⊢ {𝑗} ∈ Fin
160	118, 17	syldan 602	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝐵 ∈ Fin)
161		xpfi 9275	. . . . . . 7 ⊢ (({𝑗} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑗} × 𝐵) ∈ Fin)
162	159, 160, 161	sylancr 598	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → ({𝑗} × 𝐵) ∈ Fin)
163	162	ralrimiva 3163	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
164		iunfi 9296	. . . . 5 ⊢ (((𝑥 ∪ {𝑦}) ∈ Fin ∧ ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin) → ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
165	117, 163, 164	syl2anc 595	. . . 4 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
166		eliun 4961	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ↔ ∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵))
167		elxp 5682	. . . . . . . 8 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) ↔ ∃𝑚∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵)))
168		simprl 782	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑧 = ⟨𝑚, 𝑘⟩)
169		simprrl 792	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑚 ∈ {𝑗})
170		elsni 4608	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ {𝑗} → 𝑚 = 𝑗)
171	169, 170	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑚 = 𝑗)
172	171	opeq1d 4845	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → ⟨𝑚, 𝑘⟩ = ⟨𝑗, 𝑘⟩)
173	168, 172	eqtrd 2804	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑧 = ⟨𝑗, 𝑘⟩)
174	173, 88	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝐷 = 𝐶)
175		simpll 778	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝜑)
176	118	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑗 ∈ 𝐴)
177		simprrr 793	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑘 ∈ 𝐵)
178	175, 176, 177, 25	syl12anc 849	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝐶 ∈ ℂ)
179	174, 178	eqeltrd 2869	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝐷 ∈ ℂ)
180	179	ex 417	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → ((𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵)) → 𝐷 ∈ ℂ))
181	180	exlimdvv 1961	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → (∃𝑚∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵)) → 𝐷 ∈ ℂ))
182	167, 181	biimtrid 245	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → (𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
183	182	rexlimdva 3172	. . . . . 6 ⊢ (𝜑 → (∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
184	166, 183	biimtrid 245	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
185	184	imp 411	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)) → 𝐷 ∈ ℂ)
186	144, 158, 165, 185	fprodsplit 16016	. . 3 ⊢ (𝜑 → ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷 · ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷))
187	186	adantr 485	. 2 ⊢ ((𝜑 ∧ 𝜓) → ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷 · ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷))
188	111, 123, 187	3eqtr4d 2814	1 ⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  ⦋csb 3861   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  {csn 4591  ⟨cop 4597  ∪ ciun 4957   × cxp 5657   ↾ cres 5661  –1-1-onto→wf1o 6533  ‘cfv 6534  (class class class)co 7408  1^st c1st 7980  2^nd c2nd 7981  Fincfn 8939  ℂcc 11094   · cmul 11101  ∏cprod 15953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-inf2 9606  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173  ax-pre-sup 11174

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-isom 6543  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-oi 9468  df-card 9921  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-div 11868  df-nn 12230  df-2 12299  df-3 12300  df-n0 12501  df-z 12588  df-uz 12859  df-rp 13013  df-fz 13532  df-fzo 13679  df-seq 14034  df-exp 14094  df-hash 14363  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-clim 15535  df-prod 15954

This theorem is referenced by:  fprod2d  16031