Theorem fprod2dlem 16057

Description: Lemma for fprod2d 16058- 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 510	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷)
3		nfcv 2927	. . . . . 6 ⊢ Ⅎ𝑚∏𝑘 ∈ 𝐵 𝐶
4		nfcsb1v 3878	. . . . . . 7 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐵
5		nfcsb1v 3878	. . . . . . 7 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐶
6	4, 5	nfcprod 15986	. . . . . 6 ⊢ Ⅎ𝑗∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶
7		csbeq1a 3868	. . . . . . 7 ⊢ (𝑗 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑗⦌𝐵)
8		csbeq1a 3868	. . . . . . . 8 ⊢ (𝑗 = 𝑚 → 𝐶 = ⦋𝑚 / 𝑗⦌𝐶)
9	8	adantr 486	. . . . . . 7 ⊢ ((𝑗 = 𝑚 ∧ 𝑘 ∈ 𝐵) → 𝐶 = ⦋𝑚 / 𝑗⦌𝐶)
10	7, 9	prodeq12dv 16003	. . . . . 6 ⊢ (𝑗 = 𝑚 → ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶)
11	3, 6, 10	cbvprodi 15992	. . . . 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 4147	. . . . . . . 8 ⊢ (𝜑 → {𝑦} ⊆ 𝐴)
14		vex 3461	. . . . . . . . 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 3159	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ Fin)
19		nfcsb1v 3878	. . . . . . . . . . 11 ⊢ Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐵
20	19	nfel1 2943	. . . . . . . . . 10 ⊢ Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐵 ∈ Fin
21		csbeq1a 3868	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑗⦌𝐵)
22	21	eleq1d 2850	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (𝐵 ∈ Fin ↔ ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin))
23	20, 22	rspc 3571	. . . . . . . . 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 3210	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
27		nfcsb1v 3878	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐶
28	27	nfel1 2943	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ
29	19, 28	nfralw 3314	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ
30		csbeq1a 3868	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑗⦌𝐶)
31	30	eleq1d 2850	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑦 → (𝐶 ∈ ℂ ↔ ⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ))
32	21, 31	raleqbidv 3340	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → (∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ ↔ ∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ))
33	29, 32	rspc 3571	. . . . . . . . . 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 3259	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) → ⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)
36	24, 35	fprodcl 16029	. . . . . . 7 ⊢ (𝜑 → ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)
37		csbeq1 3857	. . . . . . . . 9 ⊢ (𝑚 = 𝑦 → ⦋𝑚 / 𝑗⦌𝐵 = ⦋𝑦 / 𝑗⦌𝐵)
38		csbeq1 3857	. . . . . . . . . 10 ⊢ (𝑚 = 𝑦 → ⦋𝑚 / 𝑗⦌𝐶 = ⦋𝑦 / 𝑗⦌𝐶)
39	38	adantr 486	. . . . . . . . 9 ⊢ ((𝑚 = 𝑦 ∧ 𝑘 ∈ ⦋𝑚 / 𝑗⦌𝐵) → ⦋𝑚 / 𝑗⦌𝐶 = ⦋𝑦 / 𝑗⦌𝐶)
40	37, 39	prodeq12dv 16003	. . . . . . . 8 ⊢ (𝑚 = 𝑦 → ∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶)
41	40	prodsn 16039	. . . . . . 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 596	. . . . . 6 ⊢ (𝜑 → ∏𝑚 ∈ {𝑦}∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶)
43		nfcv 2927	. . . . . . . 8 ⊢ Ⅎ𝑚⦋𝑦 / 𝑗⦌𝐶
44		nfcsb1v 3878	. . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
45		csbeq1a 3868	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → ⦋𝑦 / 𝑗⦌𝐶 = ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
46	43, 44, 45	cbvprodi 15992	. . . . . . 7 ⊢ ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 = ∏𝑚 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
47		csbeq1 3857	. . . . . . . . 9 ⊢ (𝑚 = (2nd ‘𝑧) → ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
48		snfi 9047	. . . . . . . . . 10 ⊢ {𝑦} ∈ Fin
49		xpfi 9286	. . . . . . . . . 10 ⊢ (({𝑦} ∈ Fin ∧ ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin) → ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ∈ Fin)
50	48, 24, 49	sylancr 599	. . . . . . . . 9 ⊢ (𝜑 → ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ∈ Fin)
51		2ndconst 8102	. . . . . . . . . 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 6904	. . . . . . . . . 10 ⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) → ((2nd ↾ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))‘𝑧) = (2nd ‘𝑧))
54	53	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → ((2nd ↾ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))‘𝑧) = (2nd ‘𝑧))
55	44	nfel1 2943	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ
56	45	eleq1d 2850	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ ↔ ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ))
57	55, 56	rspc 3571	. . . . . . . . . 10 ⊢ (𝑚 ∈ ⦋𝑦 / 𝑗⦌𝐵 → (∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ → ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ))
58	34, 57	mpan9 516	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ⦋𝑦 / 𝑗⦌𝐵) → ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)
59	47, 50, 52, 54, 58	fprodf1o 16023	. . . . . . . 8 ⊢ (𝜑 → ∏𝑚 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
60		elxp 5686	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ↔ ∃𝑚∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)))
61		nfv 1947	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗 𝑧 = ⟨𝑚, 𝑘⟩
62		nfv 1947	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗 𝑚 ∈ {𝑦}
63	19	nfcri 2919	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵
64	62, 63	nfan 1932	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗(𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)
65	61, 64	nfan 1932	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵))
66	65	nfex 2359	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵))
67		nfv 1947	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵))
68		opeq1 4840	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑗 → ⟨𝑚, 𝑘⟩ = ⟨𝑗, 𝑘⟩)
69	68	eqeq2d 2776	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑗 → (𝑧 = ⟨𝑚, 𝑘⟩ ↔ 𝑧 = ⟨𝑗, 𝑘⟩))
70		eleq1w 2848	. . . . . . . . . . . . . . . . . 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 643	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) ↔ (𝑗 = 𝑦 ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)))
74	21	eleq2d 2851	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑦 → (𝑘 ∈ 𝐵 ↔ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵))
75	74	pm5.32i 585	. . . . . . . . . . . . . . . 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 644	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑗 → ((𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) ↔ (𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵))))
78	77	exbidv 1954	. . . . . . . . . . . . 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 2376	. . . . . . . . . . . 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 1947	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝜑
82		nfcv 2927	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(2nd ‘𝑧)
83	82, 27	nfcsbw 3880	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
84	83	nfeq2 2944	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
85		nfv 1947	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝜑
86		nfcsb1v 3878	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
87	86	nfeq2 2944	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
88		fprod2d.1	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)
89	88	ad2antlr 740	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = 𝐶)
90	30	ad2antrl 741	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐶 = ⦋𝑦 / 𝑗⦌𝐶)
91		fveq2 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd ‘𝑧) = (2nd ‘⟨𝑗, 𝑘⟩))
92		vex 3461	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑗 ∈ V
93		vex 3461	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑘 ∈ V
94	92, 93	op2nd 8001	. . . . . . . . . . . . . . . . . 18 ⊢ (2nd ‘⟨𝑗, 𝑘⟩) = 𝑘
95	91, 94	eqtr2di 2817	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝑘 = (2nd ‘𝑧))
96	95	ad2antlr 740	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝑘 = (2nd ‘𝑧))
97		csbeq1a 3868	. . . . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
100	99	expl 463	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶))
101	85, 87, 100	exlimd 2257	. . . . . . . . . . . 12 ⊢ (𝜑 → (∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶))
102	81, 84, 101	exlimd 2257	. . . . . . . . . . 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 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
105	104	prodeq2dv 15999	. . . . . . . 8 ⊢ (𝜑 → ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
106	59, 105	eqtr4d 2803	. . . . . . 7 ⊢ (𝜑 → ∏𝑚 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)
107	46, 106	eqtrid 2812	. . . . . 6 ⊢ (𝜑 → ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)
108	42, 107	eqtrd 2800	. . . . 5 ⊢ (𝜑 → ∏𝑚 ∈ {𝑦}∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)
109	11, 108	eqtrid 2812	. . . 4 ⊢ (𝜑 → ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)
110	109	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)
111	2, 110	oveq12d 7437	. 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 2766	. . . 4 ⊢ (𝜑 → (𝑥 ∪ {𝑦}) = (𝑥 ∪ {𝑦}))
116		fprod2d.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin)
117	116, 12	ssfid 9236	. . . 4 ⊢ (𝜑 → (𝑥 ∪ {𝑦}) ∈ Fin)
118	12	sselda 3938	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝑗 ∈ 𝐴)
119	25	anassrs 473	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
120	17, 119	fprodcl 16029	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∏𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
121	118, 120	syldan 603	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → ∏𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
122	114, 115, 117, 121	fprodsplit 16043	. . 3 ⊢ (𝜑 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = (∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 · ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶))
123	122	adantr 486	. 2 ⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = (∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 · ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶))
124		eliun 4962	. . . . . . . . . 10 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝑥 𝑧 ∈ ({𝑗} × 𝐵))
125		xp1st 8024	. . . . . . . . . . . . . 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 2850	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → ((1st ‘𝑧) ∈ 𝑥 ↔ 𝑗 ∈ 𝑥))
129	128	biimparc 485	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ 𝑥 ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → (1st ‘𝑧) ∈ 𝑥)
130	129	rexlimiva 3160	. . . . . . . . . 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 8024	. . . . . . . . 9 ⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) → (1st ‘𝑧) ∈ {𝑦})
133	131, 132	anim12i 625	. . . . . . . 8 ⊢ ((𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → ((1st ‘𝑧) ∈ 𝑥 ∧ (1st ‘𝑧) ∈ {𝑦}))
134		elin 3922	. . . . . . . 8 ⊢ (𝑧 ∈ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) ↔ (𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)))
135		elin 3922	. . . . . . . 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 2851	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ (1st ‘𝑧) ∈ ∅))
138		noel 4291	. . . . . . . . 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 3944	. . . . 5 ⊢ (𝜑 → (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) ⊆ ∅)
143		ss0 4359	. . . . 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 5062	. . . . . 6 ⊢ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵))
146		nfcv 2927	. . . . . . . . 9 ⊢ Ⅎ𝑚({𝑗} × 𝐵)
147		nfcv 2927	. . . . . . . . . 10 ⊢ Ⅎ𝑗{𝑚}
148	147, 4	nfxp 5696	. . . . . . . . 9 ⊢ Ⅎ𝑗({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)
149		sneq 4601	. . . . . . . . . 10 ⊢ (𝑗 = 𝑚 → {𝑗} = {𝑚})
150	149, 7	xpeq12d 5694	. . . . . . . . 9 ⊢ (𝑗 = 𝑚 → ({𝑗} × 𝐵) = ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵))
151	146, 148, 150	cbviun 5001	. . . . . . . 8 ⊢ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = ∪ 𝑚 ∈ {𝑦} ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)
152		sneq 4601	. . . . . . . . . 10 ⊢ (𝑚 = 𝑦 → {𝑚} = {𝑦})
153	152, 37	xpeq12d 5694	. . . . . . . . 9 ⊢ (𝑚 = 𝑦 → ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) = ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))
154	14, 153	iunxsn 5059	. . . . . . . 8 ⊢ ∪ 𝑚 ∈ {𝑦} ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) = ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)
155	151, 154	eqtri 2788	. . . . . . 7 ⊢ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)
156	155	uneq2i 4119	. . . . . 6 ⊢ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵)) = (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))
157	145, 156	eqtri 2788	. . . . 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 9047	. . . . . . 7 ⊢ {𝑗} ∈ Fin
160	118, 17	syldan 603	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝐵 ∈ Fin)
161		xpfi 9286	. . . . . . 7 ⊢ (({𝑗} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑗} × 𝐵) ∈ Fin)
162	159, 160, 161	sylancr 599	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → ({𝑗} × 𝐵) ∈ Fin)
163	162	ralrimiva 3159	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
164		iunfi 9307	. . . . 5 ⊢ (((𝑥 ∪ {𝑦}) ∈ Fin ∧ ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin) → ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
165	117, 163, 164	syl2anc 596	. . . 4 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
166		eliun 4962	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ↔ ∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵))
167		elxp 5686	. . . . . . . 8 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) ↔ ∃𝑚∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵)))
168		simprl 783	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑧 = ⟨𝑚, 𝑘⟩)
169		simprrl 793	. . . . . . . . . . . . . . 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 4846	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → ⟨𝑚, 𝑘⟩ = ⟨𝑗, 𝑘⟩)
173	168, 172	eqtrd 2800	. . . . . . . . . . . 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 779	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝜑)
176	118	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑗 ∈ 𝐴)
177		simprrr 794	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑘 ∈ 𝐵)
178	175, 176, 177, 25	syl12anc 850	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝐶 ∈ ℂ)
179	174, 178	eqeltrd 2865	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝐷 ∈ ℂ)
180	179	ex 418	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → ((𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵)) → 𝐷 ∈ ℂ))
181	180	exlimdvv 1967	. . . . . . . 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 3168	. . . . . 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 412	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)) → 𝐷 ∈ ℂ)
186	144, 158, 165, 185	fprodsplit 16043	. . 3 ⊢ (𝜑 → ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷 · ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷))
187	186	adantr 486	. 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 2810	1 ⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  ⦋csb 3854   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  {csn 4591  ⟨cop 4597  ∪ ciun 4958   × cxp 5661   ↾ cres 5665  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7419  1^st c1st 7990  2^nd c2nd 7991  Fincfn 8949  ℂcc 11113   · cmul 11120  ∏cprod 15980

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-inf2 9617  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192  ax-pre-sup 11193

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-sup 9409  df-oi 9479  df-card 9941  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-div 11887  df-nn 12249  df-2 12318  df-3 12319  df-n0 12520  df-z 12607  df-uz 12879  df-rp 13033  df-fz 13552  df-fzo 13700  df-seq 14056  df-exp 14116  df-hash 14385  df-cj 15174  df-re 15175  df-im 15176  df-sqrt 15310  df-abs 15311  df-clim 15563  df-prod 15981

This theorem is used by:  fprod2d  16058