Theorem txhmeo 13904

Description: Lift a pair of homeomorphisms on the factors to a homeomorphism of product topologies. (Contributed by Mario Carneiro, 2-Sep-2015.)

Hypotheses

Ref	Expression
txhmeo.1	⊢ 𝑋 = ∪ 𝐽
txhmeo.2	⊢ 𝑌 = ∪ 𝐾
txhmeo.3	⊢ (𝜑 → 𝐹 ∈ (𝐽Homeo𝐿))
txhmeo.4	⊢ (𝜑 → 𝐺 ∈ (𝐾Homeo𝑀))

Assertion

Ref	Expression
txhmeo	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐽 ×t 𝐾)Homeo(𝐿 ×t 𝑀)))

Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐽,𝑦   𝑥,𝐾,𝑦   𝜑,𝑥,𝑦   𝑥,𝐺,𝑦   𝑥,𝐿,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥,𝑀,𝑦

Proof of Theorem txhmeo

Dummy variables 𝑣 𝑢 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		txhmeo.3	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐽Homeo𝐿))
2		hmeocn 13890	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐿) → 𝐹 ∈ (𝐽 Cn 𝐿))
3	1, 2	syl 14	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐿))
4		cntop1 13786	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐿) → 𝐽 ∈ Top)
5	3, 4	syl 14	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Top)
6		txhmeo.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
7	6	toptopon 13603	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
8	5, 7	sylib 122	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
9		txhmeo.4	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (𝐾Homeo𝑀))
10		hmeocn 13890	. . . . . 6 ⊢ (𝐺 ∈ (𝐾Homeo𝑀) → 𝐺 ∈ (𝐾 Cn 𝑀))
11	9, 10	syl 14	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝑀))
12		cntop1 13786	. . . . 5 ⊢ (𝐺 ∈ (𝐾 Cn 𝑀) → 𝐾 ∈ Top)
13	11, 12	syl 14	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Top)
14		txhmeo.2	. . . . 5 ⊢ 𝑌 = ∪ 𝐾
15	14	toptopon 13603	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
16	13, 15	sylib 122	. . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
17	8, 16	cnmpt1st 13873	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
18	8, 16, 17, 3	cnmpt21f 13877	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐹‘𝑥)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
19	8, 16	cnmpt2nd 13874	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
20	8, 16, 19, 11	cnmpt21f 13877	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐺‘𝑦)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
21	8, 16, 18, 20	cnmpt2t 13878	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
22		vex 2742	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
23		vex 2742	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
24	22, 23	op1std 6151	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑢) = 𝑥)
25	24	fveq2d 5521	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st ‘𝑢)) = (𝐹‘𝑥))
26	22, 23	op2ndd 6152	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑢) = 𝑦)
27	26	fveq2d 5521	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝐺‘(2nd ‘𝑢)) = (𝐺‘𝑦))
28	25, 27	opeq12d 3788	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → ⟨(𝐹‘(1st ‘𝑢)), (𝐺‘(2nd ‘𝑢))⟩ = ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
29	28	mpompt 5969	. . . . . . 7 ⊢ (𝑢 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐹‘(1st ‘𝑢)), (𝐺‘(2nd ‘𝑢))⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
30	29	eqcomi 2181	. . . . . 6 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = (𝑢 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐹‘(1st ‘𝑢)), (𝐺‘(2nd ‘𝑢))⟩)
31		eqid 2177	. . . . . . . . . 10 ⊢ ∪ 𝐿 = ∪ 𝐿
32	6, 31	cnf 13789	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽 Cn 𝐿) → 𝐹:𝑋⟶∪ 𝐿)
33	3, 32	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐿)
34		xp1st 6168	. . . . . . . 8 ⊢ (𝑢 ∈ (𝑋 × 𝑌) → (1st ‘𝑢) ∈ 𝑋)
35		ffvelcdm 5651	. . . . . . . 8 ⊢ ((𝐹:𝑋⟶∪ 𝐿 ∧ (1st ‘𝑢) ∈ 𝑋) → (𝐹‘(1st ‘𝑢)) ∈ ∪ 𝐿)
36	33, 34, 35	syl2an 289	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑋 × 𝑌)) → (𝐹‘(1st ‘𝑢)) ∈ ∪ 𝐿)
37		eqid 2177	. . . . . . . . . 10 ⊢ ∪ 𝑀 = ∪ 𝑀
38	14, 37	cnf 13789	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝐾 Cn 𝑀) → 𝐺:𝑌⟶∪ 𝑀)
39	11, 38	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑌⟶∪ 𝑀)
40		xp2nd 6169	. . . . . . . 8 ⊢ (𝑢 ∈ (𝑋 × 𝑌) → (2nd ‘𝑢) ∈ 𝑌)
41		ffvelcdm 5651	. . . . . . . 8 ⊢ ((𝐺:𝑌⟶∪ 𝑀 ∧ (2nd ‘𝑢) ∈ 𝑌) → (𝐺‘(2nd ‘𝑢)) ∈ ∪ 𝑀)
42	39, 40, 41	syl2an 289	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑋 × 𝑌)) → (𝐺‘(2nd ‘𝑢)) ∈ ∪ 𝑀)
43	36, 42	opelxpd 4661	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑋 × 𝑌)) → ⟨(𝐹‘(1st ‘𝑢)), (𝐺‘(2nd ‘𝑢))⟩ ∈ (∪ 𝐿 × ∪ 𝑀))
44	6, 31	hmeof1o 13894	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐽Homeo𝐿) → 𝐹:𝑋–1-1-onto→∪ 𝐿)
45	1, 44	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→∪ 𝐿)
46		f1ocnv 5476	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐿 → ◡𝐹:∪ 𝐿–1-1-onto→𝑋)
47		f1of 5463	. . . . . . . . 9 ⊢ (◡𝐹:∪ 𝐿–1-1-onto→𝑋 → ◡𝐹:∪ 𝐿⟶𝑋)
48	45, 46, 47	3syl 17	. . . . . . . 8 ⊢ (𝜑 → ◡𝐹:∪ 𝐿⟶𝑋)
49		xp1st 6168	. . . . . . . 8 ⊢ (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) → (1st ‘𝑣) ∈ ∪ 𝐿)
50		ffvelcdm 5651	. . . . . . . 8 ⊢ ((◡𝐹:∪ 𝐿⟶𝑋 ∧ (1st ‘𝑣) ∈ ∪ 𝐿) → (◡𝐹‘(1st ‘𝑣)) ∈ 𝑋)
51	48, 49, 50	syl2an 289	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀)) → (◡𝐹‘(1st ‘𝑣)) ∈ 𝑋)
52	14, 37	hmeof1o 13894	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝐾Homeo𝑀) → 𝐺:𝑌–1-1-onto→∪ 𝑀)
53	9, 52	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝑌–1-1-onto→∪ 𝑀)
54		f1ocnv 5476	. . . . . . . . 9 ⊢ (𝐺:𝑌–1-1-onto→∪ 𝑀 → ◡𝐺:∪ 𝑀–1-1-onto→𝑌)
55		f1of 5463	. . . . . . . . 9 ⊢ (◡𝐺:∪ 𝑀–1-1-onto→𝑌 → ◡𝐺:∪ 𝑀⟶𝑌)
56	53, 54, 55	3syl 17	. . . . . . . 8 ⊢ (𝜑 → ◡𝐺:∪ 𝑀⟶𝑌)
57		xp2nd 6169	. . . . . . . 8 ⊢ (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) → (2nd ‘𝑣) ∈ ∪ 𝑀)
58		ffvelcdm 5651	. . . . . . . 8 ⊢ ((◡𝐺:∪ 𝑀⟶𝑌 ∧ (2nd ‘𝑣) ∈ ∪ 𝑀) → (◡𝐺‘(2nd ‘𝑣)) ∈ 𝑌)
59	56, 57, 58	syl2an 289	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀)) → (◡𝐺‘(2nd ‘𝑣)) ∈ 𝑌)
60	51, 59	opelxpd 4661	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀)) → ⟨(◡𝐹‘(1st ‘𝑣)), (◡𝐺‘(2nd ‘𝑣))⟩ ∈ (𝑋 × 𝑌))
61	45	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → 𝐹:𝑋–1-1-onto→∪ 𝐿)
62	34	ad2antrl 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (1st ‘𝑢) ∈ 𝑋)
63	49	ad2antll 491	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (1st ‘𝑣) ∈ ∪ 𝐿)
64		f1ocnvfvb 5783	. . . . . . . . . 10 ⊢ ((𝐹:𝑋–1-1-onto→∪ 𝐿 ∧ (1st ‘𝑢) ∈ 𝑋 ∧ (1st ‘𝑣) ∈ ∪ 𝐿) → ((𝐹‘(1st ‘𝑢)) = (1st ‘𝑣) ↔ (◡𝐹‘(1st ‘𝑣)) = (1st ‘𝑢)))
65	61, 62, 63, 64	syl3anc 1238	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → ((𝐹‘(1st ‘𝑢)) = (1st ‘𝑣) ↔ (◡𝐹‘(1st ‘𝑣)) = (1st ‘𝑢)))
66		eqcom 2179	. . . . . . . . 9 ⊢ ((1st ‘𝑣) = (𝐹‘(1st ‘𝑢)) ↔ (𝐹‘(1st ‘𝑢)) = (1st ‘𝑣))
67		eqcom 2179	. . . . . . . . 9 ⊢ ((1st ‘𝑢) = (◡𝐹‘(1st ‘𝑣)) ↔ (◡𝐹‘(1st ‘𝑣)) = (1st ‘𝑢))
68	65, 66, 67	3bitr4g 223	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → ((1st ‘𝑣) = (𝐹‘(1st ‘𝑢)) ↔ (1st ‘𝑢) = (◡𝐹‘(1st ‘𝑣))))
69	53	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → 𝐺:𝑌–1-1-onto→∪ 𝑀)
70	40	ad2antrl 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (2nd ‘𝑢) ∈ 𝑌)
71	57	ad2antll 491	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (2nd ‘𝑣) ∈ ∪ 𝑀)
72		f1ocnvfvb 5783	. . . . . . . . . 10 ⊢ ((𝐺:𝑌–1-1-onto→∪ 𝑀 ∧ (2nd ‘𝑢) ∈ 𝑌 ∧ (2nd ‘𝑣) ∈ ∪ 𝑀) → ((𝐺‘(2nd ‘𝑢)) = (2nd ‘𝑣) ↔ (◡𝐺‘(2nd ‘𝑣)) = (2nd ‘𝑢)))
73	69, 70, 71, 72	syl3anc 1238	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → ((𝐺‘(2nd ‘𝑢)) = (2nd ‘𝑣) ↔ (◡𝐺‘(2nd ‘𝑣)) = (2nd ‘𝑢)))
74		eqcom 2179	. . . . . . . . 9 ⊢ ((2nd ‘𝑣) = (𝐺‘(2nd ‘𝑢)) ↔ (𝐺‘(2nd ‘𝑢)) = (2nd ‘𝑣))
75		eqcom 2179	. . . . . . . . 9 ⊢ ((2nd ‘𝑢) = (◡𝐺‘(2nd ‘𝑣)) ↔ (◡𝐺‘(2nd ‘𝑣)) = (2nd ‘𝑢))
76	73, 74, 75	3bitr4g 223	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → ((2nd ‘𝑣) = (𝐺‘(2nd ‘𝑢)) ↔ (2nd ‘𝑢) = (◡𝐺‘(2nd ‘𝑣))))
77	68, 76	anbi12d 473	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (((1st ‘𝑣) = (𝐹‘(1st ‘𝑢)) ∧ (2nd ‘𝑣) = (𝐺‘(2nd ‘𝑢))) ↔ ((1st ‘𝑢) = (◡𝐹‘(1st ‘𝑣)) ∧ (2nd ‘𝑢) = (◡𝐺‘(2nd ‘𝑣)))))
78		eqop 6180	. . . . . . . 8 ⊢ (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) → (𝑣 = ⟨(𝐹‘(1st ‘𝑢)), (𝐺‘(2nd ‘𝑢))⟩ ↔ ((1st ‘𝑣) = (𝐹‘(1st ‘𝑢)) ∧ (2nd ‘𝑣) = (𝐺‘(2nd ‘𝑢)))))
79	78	ad2antll 491	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (𝑣 = ⟨(𝐹‘(1st ‘𝑢)), (𝐺‘(2nd ‘𝑢))⟩ ↔ ((1st ‘𝑣) = (𝐹‘(1st ‘𝑢)) ∧ (2nd ‘𝑣) = (𝐺‘(2nd ‘𝑢)))))
80		eqop 6180	. . . . . . . 8 ⊢ (𝑢 ∈ (𝑋 × 𝑌) → (𝑢 = ⟨(◡𝐹‘(1st ‘𝑣)), (◡𝐺‘(2nd ‘𝑣))⟩ ↔ ((1st ‘𝑢) = (◡𝐹‘(1st ‘𝑣)) ∧ (2nd ‘𝑢) = (◡𝐺‘(2nd ‘𝑣)))))
81	80	ad2antrl 490	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (𝑢 = ⟨(◡𝐹‘(1st ‘𝑣)), (◡𝐺‘(2nd ‘𝑣))⟩ ↔ ((1st ‘𝑢) = (◡𝐹‘(1st ‘𝑣)) ∧ (2nd ‘𝑢) = (◡𝐺‘(2nd ‘𝑣)))))
82	77, 79, 81	3bitr4rd 221	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (𝑢 = ⟨(◡𝐹‘(1st ‘𝑣)), (◡𝐺‘(2nd ‘𝑣))⟩ ↔ 𝑣 = ⟨(𝐹‘(1st ‘𝑢)), (𝐺‘(2nd ‘𝑢))⟩))
83	30, 43, 60, 82	f1ocnv2d 6077	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩):(𝑋 × 𝑌)–1-1-onto→(∪ 𝐿 × ∪ 𝑀) ∧ ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) ↦ ⟨(◡𝐹‘(1st ‘𝑣)), (◡𝐺‘(2nd ‘𝑣))⟩)))
84	83	simprd 114	. . . 4 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) ↦ ⟨(◡𝐹‘(1st ‘𝑣)), (◡𝐺‘(2nd ‘𝑣))⟩))
85		vex 2742	. . . . . . . 8 ⊢ 𝑧 ∈ V
86		vex 2742	. . . . . . . 8 ⊢ 𝑤 ∈ V
87	85, 86	op1std 6151	. . . . . . 7 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (1st ‘𝑣) = 𝑧)
88	87	fveq2d 5521	. . . . . 6 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (◡𝐹‘(1st ‘𝑣)) = (◡𝐹‘𝑧))
89	85, 86	op2ndd 6152	. . . . . . 7 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (2nd ‘𝑣) = 𝑤)
90	89	fveq2d 5521	. . . . . 6 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (◡𝐺‘(2nd ‘𝑣)) = (◡𝐺‘𝑤))
91	88, 90	opeq12d 3788	. . . . 5 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ⟨(◡𝐹‘(1st ‘𝑣)), (◡𝐺‘(2nd ‘𝑣))⟩ = ⟨(◡𝐹‘𝑧), (◡𝐺‘𝑤)⟩)
92	91	mpompt 5969	. . . 4 ⊢ (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) ↦ ⟨(◡𝐹‘(1st ‘𝑣)), (◡𝐺‘(2nd ‘𝑣))⟩) = (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ ⟨(◡𝐹‘𝑧), (◡𝐺‘𝑤)⟩)
93	84, 92	eqtrdi 2226	. . 3 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ ⟨(◡𝐹‘𝑧), (◡𝐺‘𝑤)⟩))
94		cntop2 13787	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top)
95	3, 94	syl 14	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ Top)
96	31	toptopon 13603	. . . . 5 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
97	95, 96	sylib 122	. . . 4 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
98		cntop2 13787	. . . . . 6 ⊢ (𝐺 ∈ (𝐾 Cn 𝑀) → 𝑀 ∈ Top)
99	11, 98	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ Top)
100	37	toptopon 13603	. . . . 5 ⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀))
101	99, 100	sylib 122	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀))
102	97, 101	cnmpt1st 13873	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ 𝑧) ∈ ((𝐿 ×t 𝑀) Cn 𝐿))
103		hmeocnvcn 13891	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐿) → ◡𝐹 ∈ (𝐿 Cn 𝐽))
104	1, 103	syl 14	. . . . 5 ⊢ (𝜑 → ◡𝐹 ∈ (𝐿 Cn 𝐽))
105	97, 101, 102, 104	cnmpt21f 13877	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (◡𝐹‘𝑧)) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
106	97, 101	cnmpt2nd 13874	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ 𝑤) ∈ ((𝐿 ×t 𝑀) Cn 𝑀))
107		hmeocnvcn 13891	. . . . . 6 ⊢ (𝐺 ∈ (𝐾Homeo𝑀) → ◡𝐺 ∈ (𝑀 Cn 𝐾))
108	9, 107	syl 14	. . . . 5 ⊢ (𝜑 → ◡𝐺 ∈ (𝑀 Cn 𝐾))
109	97, 101, 106, 108	cnmpt21f 13877	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (◡𝐺‘𝑤)) ∈ ((𝐿 ×t 𝑀) Cn 𝐾))
110	97, 101, 105, 109	cnmpt2t 13878	. . 3 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ ⟨(◡𝐹‘𝑧), (◡𝐺‘𝑤)⟩) ∈ ((𝐿 ×t 𝑀) Cn (𝐽 ×t 𝐾)))
111	93, 110	eqeltrd 2254	. 2 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐿 ×t 𝑀) Cn (𝐽 ×t 𝐾)))
112		ishmeo 13889	. 2 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐽 ×t 𝐾)Homeo(𝐿 ×t 𝑀)) ↔ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)) ∧ ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐿 ×t 𝑀) Cn (𝐽 ×t 𝐾))))
113	21, 111, 112	sylanbrc 417	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐽 ×t 𝐾)Homeo(𝐿 ×t 𝑀)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  ⟨cop 3597  ∪ cuni 3811   ↦ cmpt 4066   × cxp 4626  ^◡ccnv 4627  ⟶wf 5214  –1-1-onto→wf1o 5217  ‘cfv 5218  (class class class)co 5877   ∈ cmpo 5879  1^st c1st 6141  2^nd c2nd 6142  Topctop 13582  TopOnctopon 13595   Cn ccn 13770   ×_t ctx 13837  Homeochmeo 13885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-map 6652  df-topgen 12714  df-top 13583  df-topon 13596  df-bases 13628  df-cn 13773  df-tx 13838  df-hmeo 13886

This theorem is referenced by: (None)