Theorem fprod2dlem 15184

Description: Lemma for fprod2d 15185- 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		simpr 477	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
2		fprod2d.7	. . . 4 ⊢ (𝜓 ↔ ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷)
3	1, 2	sylib 210	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷)
4		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑚∏𝑘 ∈ 𝐵 𝐶
5		nfcsb1v 3800	. . . . . . 7 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐵
6		nfcsb1v 3800	. . . . . . 7 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐶
7	5, 6	nfcprod 15115	. . . . . 6 ⊢ Ⅎ𝑗∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶
8		csbeq1a 3791	. . . . . . 7 ⊢ (𝑗 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑗⦌𝐵)
9		csbeq1a 3791	. . . . . . . 8 ⊢ (𝑗 = 𝑚 → 𝐶 = ⦋𝑚 / 𝑗⦌𝐶)
10	9	adantr 473	. . . . . . 7 ⊢ ((𝑗 = 𝑚 ∧ 𝑘 ∈ 𝐵) → 𝐶 = ⦋𝑚 / 𝑗⦌𝐶)
11	8, 10	prodeq12dv 15130	. . . . . 6 ⊢ (𝑗 = 𝑚 → ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶)
12	4, 7, 11	cbvprodi 15121	. . . . 5 ⊢ ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶 = ∏𝑚 ∈ {𝑦}∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶
13		fprod2d.6	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
14	13	unssbd 4048	. . . . . . . 8 ⊢ (𝜑 → {𝑦} ⊆ 𝐴)
15		vex 3412	. . . . . . . . 9 ⊢ 𝑦 ∈ V
16	15	snss 4586	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 ↔ {𝑦} ⊆ 𝐴)
17	14, 16	sylibr 226	. . . . . . 7 ⊢ (𝜑 → 𝑦 ∈ 𝐴)
18		fprod2d.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)
19	18	ralrimiva 3126	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ Fin)
20		nfcsb1v 3800	. . . . . . . . . . 11 ⊢ Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐵
21	20	nfel1 2940	. . . . . . . . . 10 ⊢ Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐵 ∈ Fin
22		csbeq1a 3791	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑗⦌𝐵)
23	22	eleq1d 2844	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (𝐵 ∈ Fin ↔ ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin))
24	21, 23	rspc 3523	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 𝐵 ∈ Fin → ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin))
25	17, 19, 24	sylc 65	. . . . . . . 8 ⊢ (𝜑 → ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin)
26		fprod2d.4	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)
27	26	ralrimivva 3135	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
28		nfcsb1v 3800	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐶
29	28	nfel1 2940	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ
30	20, 29	nfral 3168	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ
31		csbeq1a 3791	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑗⦌𝐶)
32	31	eleq1d 2844	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑦 → (𝐶 ∈ ℂ ↔ ⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ))
33	22, 32	raleqbidv 3335	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → (∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ ↔ ∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ))
34	30, 33	rspc 3523	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ → ∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ))
35	17, 27, 34	sylc 65	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)
36	35	r19.21bi 3152	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) → ⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)
37	25, 36	fprodcl 15156	. . . . . . 7 ⊢ (𝜑 → ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)
38		csbeq1 3785	. . . . . . . . 9 ⊢ (𝑚 = 𝑦 → ⦋𝑚 / 𝑗⦌𝐵 = ⦋𝑦 / 𝑗⦌𝐵)
39		csbeq1 3785	. . . . . . . . . 10 ⊢ (𝑚 = 𝑦 → ⦋𝑚 / 𝑗⦌𝐶 = ⦋𝑦 / 𝑗⦌𝐶)
40	39	adantr 473	. . . . . . . . 9 ⊢ ((𝑚 = 𝑦 ∧ 𝑘 ∈ ⦋𝑚 / 𝑗⦌𝐵) → ⦋𝑚 / 𝑗⦌𝐶 = ⦋𝑦 / 𝑗⦌𝐶)
41	38, 40	prodeq12dv 15130	. . . . . . . 8 ⊢ (𝑚 = 𝑦 → ∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶)
42	41	prodsn 15166	. . . . . . 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)
43	17, 37, 42	syl2anc 576	. . . . . 6 ⊢ (𝜑 → ∏𝑚 ∈ {𝑦}∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶)
44		nfcv 2926	. . . . . . . 8 ⊢ Ⅎ𝑚⦋𝑦 / 𝑗⦌𝐶
45		nfcsb1v 3800	. . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
46		csbeq1a 3791	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → ⦋𝑦 / 𝑗⦌𝐶 = ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
47	44, 45, 46	cbvprodi 15121	. . . . . . 7 ⊢ ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 = ∏𝑚 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
48		csbeq1 3785	. . . . . . . . 9 ⊢ (𝑚 = (2nd ‘𝑧) → ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
49		snfi 8383	. . . . . . . . . 10 ⊢ {𝑦} ∈ Fin
50		xpfi 8576	. . . . . . . . . 10 ⊢ (({𝑦} ∈ Fin ∧ ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin) → ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ∈ Fin)
51	49, 25, 50	sylancr 578	. . . . . . . . 9 ⊢ (𝜑 → ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ∈ Fin)
52		2ndconst 7597	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 → (2nd ↾ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)):({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)–1-1-onto→⦋𝑦 / 𝑗⦌𝐵)
53	17, 52	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (2nd ↾ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)):({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)–1-1-onto→⦋𝑦 / 𝑗⦌𝐵)
54		fvres 6512	. . . . . . . . . 10 ⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) → ((2nd ↾ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))‘𝑧) = (2nd ‘𝑧))
55	54	adantl 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → ((2nd ↾ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))‘𝑧) = (2nd ‘𝑧))
56	45	nfel1 2940	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ
57	46	eleq1d 2844	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ ↔ ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ))
58	56, 57	rspc 3523	. . . . . . . . . 10 ⊢ (𝑚 ∈ ⦋𝑦 / 𝑗⦌𝐵 → (∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ → ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ))
59	35, 58	mpan9 499	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ⦋𝑦 / 𝑗⦌𝐵) → ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)
60	48, 51, 53, 55, 59	fprodf1o 15150	. . . . . . . 8 ⊢ (𝜑 → ∏𝑚 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
61		elxp 5423	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ↔ ∃𝑚∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)))
62		nfv 1873	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗 𝑧 = ⟨𝑚, 𝑘⟩
63		nfv 1873	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗 𝑚 ∈ {𝑦}
64	20	nfcri 2920	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵
65	63, 64	nfan 1862	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗(𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)
66	62, 65	nfan 1862	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵))
67	66	nfex 2262	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵))
68		nfv 1873	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵))
69		opeq1 4671	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑗 → ⟨𝑚, 𝑘⟩ = ⟨𝑗, 𝑘⟩)
70	69	eqeq2d 2782	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑗 → (𝑧 = ⟨𝑚, 𝑘⟩ ↔ 𝑧 = ⟨𝑗, 𝑘⟩))
71		eleq1w 2842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑗 → (𝑚 ∈ {𝑦} ↔ 𝑗 ∈ {𝑦}))
72		velsn 4451	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ {𝑦} ↔ 𝑗 = 𝑦)
73	71, 72	syl6bb 279	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑗 → (𝑚 ∈ {𝑦} ↔ 𝑗 = 𝑦))
74	73	anbi1d 620	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) ↔ (𝑗 = 𝑦 ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)))
75	22	eleq2d 2845	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑦 → (𝑘 ∈ 𝐵 ↔ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵))
76	75	pm5.32i 567	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵) ↔ (𝑗 = 𝑦 ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵))
77	74, 76	syl6bbr 281	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) ↔ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)))
78	70, 77	anbi12d 621	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑗 → ((𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) ↔ (𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵))))
79	78	exbidv 1880	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑗 → (∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) ↔ ∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵))))
80	67, 68, 79	cbvexv1 2276	. . . . . . . . . . . 12 ⊢ (∃𝑚∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) ↔ ∃𝑗∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)))
81	61, 80	bitri 267	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ↔ ∃𝑗∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)))
82		nfv 1873	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝜑
83		nfcv 2926	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(2nd ‘𝑧)
84	83, 28	nfcsb 3802	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
85	84	nfeq2 2941	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
86		nfv 1873	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝜑
87		nfcsb1v 3800	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
88	87	nfeq2 2941	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
89		fprod2d.1	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)
90	89	ad2antlr 714	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = 𝐶)
91	31	ad2antrl 715	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐶 = ⦋𝑦 / 𝑗⦌𝐶)
92		fveq2 6493	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd ‘𝑧) = (2nd ‘⟨𝑗, 𝑘⟩))
93		vex 3412	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑗 ∈ V
94		vex 3412	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑘 ∈ V
95	93, 94	op2nd 7503	. . . . . . . . . . . . . . . . . 18 ⊢ (2nd ‘⟨𝑗, 𝑘⟩) = 𝑘
96	92, 95	syl6req 2825	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝑘 = (2nd ‘𝑧))
97	96	ad2antlr 714	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝑘 = (2nd ‘𝑧))
98		csbeq1a 3791	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (2nd ‘𝑧) → ⦋𝑦 / 𝑗⦌𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
99	97, 98	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → ⦋𝑦 / 𝑗⦌𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
100	90, 91, 99	3eqtrd 2812	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
101	100	expl 450	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶))
102	86, 88, 101	exlimd 2146	. . . . . . . . . . . 12 ⊢ (𝜑 → (∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶))
103	82, 85, 102	exlimd 2146	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑗∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶))
104	81, 103	syl5bi 234	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶))
105	104	imp 398	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
106	105	prodeq2dv 15127	. . . . . . . 8 ⊢ (𝜑 → ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
107	60, 106	eqtr4d 2811	. . . . . . 7 ⊢ (𝜑 → ∏𝑚 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)
108	47, 107	syl5eq 2820	. . . . . 6 ⊢ (𝜑 → ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)
109	43, 108	eqtrd 2808	. . . . 5 ⊢ (𝜑 → ∏𝑚 ∈ {𝑦}∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)
110	12, 109	syl5eq 2820	. . . 4 ⊢ (𝜑 → ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)
111	110	adantr 473	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)
112	3, 111	oveq12d 6988	. 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))
113		fprod2d.5	. . . . 5 ⊢ (𝜑 → ¬ 𝑦 ∈ 𝑥)
114		disjsn 4515	. . . . 5 ⊢ ((𝑥 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝑥)
115	113, 114	sylibr 226	. . . 4 ⊢ (𝜑 → (𝑥 ∩ {𝑦}) = ∅)
116		eqidd 2773	. . . 4 ⊢ (𝜑 → (𝑥 ∪ {𝑦}) = (𝑥 ∪ {𝑦}))
117		fprod2d.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin)
118	117, 13	ssfid 8528	. . . 4 ⊢ (𝜑 → (𝑥 ∪ {𝑦}) ∈ Fin)
119	13	sselda 3854	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝑗 ∈ 𝐴)
120	26	anassrs 460	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
121	18, 120	fprodcl 15156	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∏𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
122	119, 121	syldan 582	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → ∏𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
123	115, 116, 118, 122	fprodsplit 15170	. . 3 ⊢ (𝜑 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = (∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 · ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶))
124	123	adantr 473	. 2 ⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = (∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 · ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶))
125		eliun 4790	. . . . . . . . . 10 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝑥 𝑧 ∈ ({𝑗} × 𝐵))
126		xp1st 7526	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ {𝑗})
127		elsni 4452	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑧) ∈ {𝑗} → (1st ‘𝑧) = 𝑗)
128	126, 127	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) = 𝑗)
129	128	eleq1d 2844	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → ((1st ‘𝑧) ∈ 𝑥 ↔ 𝑗 ∈ 𝑥))
130	129	biimparc 472	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ 𝑥 ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → (1st ‘𝑧) ∈ 𝑥)
131	130	rexlimiva 3220	. . . . . . . . . 10 ⊢ (∃𝑗 ∈ 𝑥 𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ 𝑥)
132	125, 131	sylbi 209	. . . . . . . . 9 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ 𝑥)
133		xp1st 7526	. . . . . . . . 9 ⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) → (1st ‘𝑧) ∈ {𝑦})
134	132, 133	anim12i 603	. . . . . . . 8 ⊢ ((𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → ((1st ‘𝑧) ∈ 𝑥 ∧ (1st ‘𝑧) ∈ {𝑦}))
135		elin 4053	. . . . . . . 8 ⊢ (𝑧 ∈ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) ↔ (𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)))
136		elin 4053	. . . . . . . 8 ⊢ ((1st ‘𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ ((1st ‘𝑧) ∈ 𝑥 ∧ (1st ‘𝑧) ∈ {𝑦}))
137	134, 135, 136	3imtr4i 284	. . . . . . 7 ⊢ (𝑧 ∈ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → (1st ‘𝑧) ∈ (𝑥 ∩ {𝑦}))
138	115	eleq2d 2845	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ (1st ‘𝑧) ∈ ∅))
139		noel 4178	. . . . . . . . 9 ⊢ ¬ (1st ‘𝑧) ∈ ∅
140	139	pm2.21i 117	. . . . . . . 8 ⊢ ((1st ‘𝑧) ∈ ∅ → 𝑧 ∈ ∅)
141	138, 140	syl6bi 245	. . . . . . 7 ⊢ (𝜑 → ((1st ‘𝑧) ∈ (𝑥 ∩ {𝑦}) → 𝑧 ∈ ∅))
142	137, 141	syl5 34	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → 𝑧 ∈ ∅))
143	142	ssrdv 3860	. . . . 5 ⊢ (𝜑 → (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) ⊆ ∅)
144		ss0 4232	. . . . 5 ⊢ ((∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) ⊆ ∅ → (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) = ∅)
145	143, 144	syl 17	. . . 4 ⊢ (𝜑 → (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) = ∅)
146		iunxun 4876	. . . . . 6 ⊢ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵))
147		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑚({𝑗} × 𝐵)
148		nfcv 2926	. . . . . . . . . 10 ⊢ Ⅎ𝑗{𝑚}
149	148, 5	nfxp 5433	. . . . . . . . 9 ⊢ Ⅎ𝑗({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)
150		sneq 4445	. . . . . . . . . 10 ⊢ (𝑗 = 𝑚 → {𝑗} = {𝑚})
151	150, 8	xpeq12d 5431	. . . . . . . . 9 ⊢ (𝑗 = 𝑚 → ({𝑗} × 𝐵) = ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵))
152	147, 149, 151	cbviun 4825	. . . . . . . 8 ⊢ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = ∪ 𝑚 ∈ {𝑦} ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)
153		sneq 4445	. . . . . . . . . 10 ⊢ (𝑚 = 𝑦 → {𝑚} = {𝑦})
154	153, 38	xpeq12d 5431	. . . . . . . . 9 ⊢ (𝑚 = 𝑦 → ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) = ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))
155	15, 154	iunxsn 4873	. . . . . . . 8 ⊢ ∪ 𝑚 ∈ {𝑦} ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) = ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)
156	152, 155	eqtri 2796	. . . . . . 7 ⊢ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)
157	156	uneq2i 4021	. . . . . 6 ⊢ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵)) = (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))
158	146, 157	eqtri 2796	. . . . 5 ⊢ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))
159	158	a1i 11	. . . 4 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)))
160		snfi 8383	. . . . . . 7 ⊢ {𝑗} ∈ Fin
161	119, 18	syldan 582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝐵 ∈ Fin)
162		xpfi 8576	. . . . . . 7 ⊢ (({𝑗} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑗} × 𝐵) ∈ Fin)
163	160, 161, 162	sylancr 578	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → ({𝑗} × 𝐵) ∈ Fin)
164	163	ralrimiva 3126	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
165		iunfi 8599	. . . . 5 ⊢ (((𝑥 ∪ {𝑦}) ∈ Fin ∧ ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin) → ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
166	118, 164, 165	syl2anc 576	. . . 4 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
167		eliun 4790	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ↔ ∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵))
168		elxp 5423	. . . . . . . 8 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) ↔ ∃𝑚∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵)))
169		simprl 758	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑧 = ⟨𝑚, 𝑘⟩)
170		simprrl 768	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑚 ∈ {𝑗})
171		elsni 4452	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ {𝑗} → 𝑚 = 𝑗)
172	170, 171	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑚 = 𝑗)
173	172	opeq1d 4677	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → ⟨𝑚, 𝑘⟩ = ⟨𝑗, 𝑘⟩)
174	169, 173	eqtrd 2808	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑧 = ⟨𝑗, 𝑘⟩)
175	174, 89	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝐷 = 𝐶)
176		simpll 754	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝜑)
177	119	adantr 473	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑗 ∈ 𝐴)
178		simprrr 769	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑘 ∈ 𝐵)
179	176, 177, 178, 26	syl12anc 824	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝐶 ∈ ℂ)
180	175, 179	eqeltrd 2860	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝐷 ∈ ℂ)
181	180	ex 405	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → ((𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵)) → 𝐷 ∈ ℂ))
182	181	exlimdvv 1893	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → (∃𝑚∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵)) → 𝐷 ∈ ℂ))
183	168, 182	syl5bi 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → (𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
184	183	rexlimdva 3223	. . . . . 6 ⊢ (𝜑 → (∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
185	167, 184	syl5bi 234	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
186	185	imp 398	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)) → 𝐷 ∈ ℂ)
187	145, 159, 166, 186	fprodsplit 15170	. . 3 ⊢ (𝜑 → ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷 · ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷))
188	187	adantr 473	. 2 ⊢ ((𝜑 ∧ 𝜓) → ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷 · ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷))
189	112, 124, 188	3eqtr4d 2818	1 ⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507  ∃wex 1742   ∈ wcel 2048  ∀wral 3082  ∃wrex 3083  ⦋csb 3782   ∪ cun 3823   ∩ cin 3824   ⊆ wss 3825  ∅c0 4173  {csn 4435  ⟨cop 4441  ∪ ciun 4786   × cxp 5398   ↾ cres 5402  –1-1-onto→wf1o 6181  ‘cfv 6182  (class class class)co 6970  1^st c1st 7492  2^nd c2nd 7493  Fincfn 8298  ℂcc 10325   · cmul 10332  ∏cprod 15109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-inf2 8890  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404  ax-pre-sup 10405

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-se 5360  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-isom 6191  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7494  df-2nd 7495  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-1o 7897  df-oadd 7901  df-er 8081  df-en 8299  df-dom 8300  df-sdom 8301  df-fin 8302  df-sup 8693  df-oi 8761  df-card 9154  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-div 11091  df-nn 11432  df-2 11496  df-3 11497  df-n0 11701  df-z 11787  df-uz 12052  df-rp 12198  df-fz 12702  df-fzo 12843  df-seq 13178  df-exp 13238  df-hash 13499  df-cj 14309  df-re 14310  df-im 14311  df-sqrt 14445  df-abs 14446  df-clim 14696  df-prod 15110

This theorem is referenced by:  fprod2d  15185