Theorem txhmeo 13113

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 13099	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐿) → 𝐹 ∈ (𝐽 Cn 𝐿))
3	1, 2	syl 14	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐿))
4		cntop1 12995	. . . . 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 12810	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
8	5, 7	sylib 121	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
9		txhmeo.4	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (𝐾Homeo𝑀))
10		hmeocn 13099	. . . . . 6 ⊢ (𝐺 ∈ (𝐾Homeo𝑀) → 𝐺 ∈ (𝐾 Cn 𝑀))
11	9, 10	syl 14	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝑀))
12		cntop1 12995	. . . . 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 12810	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
16	13, 15	sylib 121	. . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
17	8, 16	cnmpt1st 13082	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
18	8, 16, 17, 3	cnmpt21f 13086	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐹‘𝑥)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
19	8, 16	cnmpt2nd 13083	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
20	8, 16, 19, 11	cnmpt21f 13086	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐺‘𝑦)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
21	8, 16, 18, 20	cnmpt2t 13087	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
22		vex 2733	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
23		vex 2733	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
24	22, 23	op1std 6127	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑢) = 𝑥)
25	24	fveq2d 5500	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st ‘𝑢)) = (𝐹‘𝑥))
26	22, 23	op2ndd 6128	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑢) = 𝑦)
27	26	fveq2d 5500	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝐺‘(2nd ‘𝑢)) = (𝐺‘𝑦))
28	25, 27	opeq12d 3773	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → ⟨(𝐹‘(1st ‘𝑢)), (𝐺‘(2nd ‘𝑢))⟩ = ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
29	28	mpompt 5945	. . . . . . 7 ⊢ (𝑢 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐹‘(1st ‘𝑢)), (𝐺‘(2nd ‘𝑢))⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
30	29	eqcomi 2174	. . . . . 6 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = (𝑢 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐹‘(1st ‘𝑢)), (𝐺‘(2nd ‘𝑢))⟩)
31		eqid 2170	. . . . . . . . . 10 ⊢ ∪ 𝐿 = ∪ 𝐿
32	6, 31	cnf 12998	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽 Cn 𝐿) → 𝐹:𝑋⟶∪ 𝐿)
33	3, 32	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐿)
34		xp1st 6144	. . . . . . . 8 ⊢ (𝑢 ∈ (𝑋 × 𝑌) → (1st ‘𝑢) ∈ 𝑋)
35		ffvelrn 5629	. . . . . . . 8 ⊢ ((𝐹:𝑋⟶∪ 𝐿 ∧ (1st ‘𝑢) ∈ 𝑋) → (𝐹‘(1st ‘𝑢)) ∈ ∪ 𝐿)
36	33, 34, 35	syl2an 287	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑋 × 𝑌)) → (𝐹‘(1st ‘𝑢)) ∈ ∪ 𝐿)
37		eqid 2170	. . . . . . . . . 10 ⊢ ∪ 𝑀 = ∪ 𝑀
38	14, 37	cnf 12998	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝐾 Cn 𝑀) → 𝐺:𝑌⟶∪ 𝑀)
39	11, 38	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑌⟶∪ 𝑀)
40		xp2nd 6145	. . . . . . . 8 ⊢ (𝑢 ∈ (𝑋 × 𝑌) → (2nd ‘𝑢) ∈ 𝑌)
41		ffvelrn 5629	. . . . . . . 8 ⊢ ((𝐺:𝑌⟶∪ 𝑀 ∧ (2nd ‘𝑢) ∈ 𝑌) → (𝐺‘(2nd ‘𝑢)) ∈ ∪ 𝑀)
42	39, 40, 41	syl2an 287	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑋 × 𝑌)) → (𝐺‘(2nd ‘𝑢)) ∈ ∪ 𝑀)
43	36, 42	opelxpd 4644	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑋 × 𝑌)) → ⟨(𝐹‘(1st ‘𝑢)), (𝐺‘(2nd ‘𝑢))⟩ ∈ (∪ 𝐿 × ∪ 𝑀))
44	6, 31	hmeof1o 13103	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐽Homeo𝐿) → 𝐹:𝑋–1-1-onto→∪ 𝐿)
45	1, 44	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→∪ 𝐿)
46		f1ocnv 5455	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐿 → ◡𝐹:∪ 𝐿–1-1-onto→𝑋)
47		f1of 5442	. . . . . . . . 9 ⊢ (◡𝐹:∪ 𝐿–1-1-onto→𝑋 → ◡𝐹:∪ 𝐿⟶𝑋)
48	45, 46, 47	3syl 17	. . . . . . . 8 ⊢ (𝜑 → ◡𝐹:∪ 𝐿⟶𝑋)
49		xp1st 6144	. . . . . . . 8 ⊢ (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) → (1st ‘𝑣) ∈ ∪ 𝐿)
50		ffvelrn 5629	. . . . . . . 8 ⊢ ((◡𝐹:∪ 𝐿⟶𝑋 ∧ (1st ‘𝑣) ∈ ∪ 𝐿) → (◡𝐹‘(1st ‘𝑣)) ∈ 𝑋)
51	48, 49, 50	syl2an 287	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀)) → (◡𝐹‘(1st ‘𝑣)) ∈ 𝑋)
52	14, 37	hmeof1o 13103	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝐾Homeo𝑀) → 𝐺:𝑌–1-1-onto→∪ 𝑀)
53	9, 52	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝑌–1-1-onto→∪ 𝑀)
54		f1ocnv 5455	. . . . . . . . 9 ⊢ (𝐺:𝑌–1-1-onto→∪ 𝑀 → ◡𝐺:∪ 𝑀–1-1-onto→𝑌)
55		f1of 5442	. . . . . . . . 9 ⊢ (◡𝐺:∪ 𝑀–1-1-onto→𝑌 → ◡𝐺:∪ 𝑀⟶𝑌)
56	53, 54, 55	3syl 17	. . . . . . . 8 ⊢ (𝜑 → ◡𝐺:∪ 𝑀⟶𝑌)
57		xp2nd 6145	. . . . . . . 8 ⊢ (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) → (2nd ‘𝑣) ∈ ∪ 𝑀)
58		ffvelrn 5629	. . . . . . . 8 ⊢ ((◡𝐺:∪ 𝑀⟶𝑌 ∧ (2nd ‘𝑣) ∈ ∪ 𝑀) → (◡𝐺‘(2nd ‘𝑣)) ∈ 𝑌)
59	56, 57, 58	syl2an 287	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀)) → (◡𝐺‘(2nd ‘𝑣)) ∈ 𝑌)
60	51, 59	opelxpd 4644	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀)) → ⟨(◡𝐹‘(1st ‘𝑣)), (◡𝐺‘(2nd ‘𝑣))⟩ ∈ (𝑋 × 𝑌))
61	45	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → 𝐹:𝑋–1-1-onto→∪ 𝐿)
62	34	ad2antrl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (1st ‘𝑢) ∈ 𝑋)
63	49	ad2antll 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (1st ‘𝑣) ∈ ∪ 𝐿)
64		f1ocnvfvb 5759	. . . . . . . . . 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 1233	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → ((𝐹‘(1st ‘𝑢)) = (1st ‘𝑣) ↔ (◡𝐹‘(1st ‘𝑣)) = (1st ‘𝑢)))
66		eqcom 2172	. . . . . . . . 9 ⊢ ((1st ‘𝑣) = (𝐹‘(1st ‘𝑢)) ↔ (𝐹‘(1st ‘𝑢)) = (1st ‘𝑣))
67		eqcom 2172	. . . . . . . . 9 ⊢ ((1st ‘𝑢) = (◡𝐹‘(1st ‘𝑣)) ↔ (◡𝐹‘(1st ‘𝑣)) = (1st ‘𝑢))
68	65, 66, 67	3bitr4g 222	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → ((1st ‘𝑣) = (𝐹‘(1st ‘𝑢)) ↔ (1st ‘𝑢) = (◡𝐹‘(1st ‘𝑣))))
69	53	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → 𝐺:𝑌–1-1-onto→∪ 𝑀)
70	40	ad2antrl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (2nd ‘𝑢) ∈ 𝑌)
71	57	ad2antll 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (2nd ‘𝑣) ∈ ∪ 𝑀)
72		f1ocnvfvb 5759	. . . . . . . . . 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 1233	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → ((𝐺‘(2nd ‘𝑢)) = (2nd ‘𝑣) ↔ (◡𝐺‘(2nd ‘𝑣)) = (2nd ‘𝑢)))
74		eqcom 2172	. . . . . . . . 9 ⊢ ((2nd ‘𝑣) = (𝐺‘(2nd ‘𝑢)) ↔ (𝐺‘(2nd ‘𝑢)) = (2nd ‘𝑣))
75		eqcom 2172	. . . . . . . . 9 ⊢ ((2nd ‘𝑢) = (◡𝐺‘(2nd ‘𝑣)) ↔ (◡𝐺‘(2nd ‘𝑣)) = (2nd ‘𝑢))
76	73, 74, 75	3bitr4g 222	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → ((2nd ‘𝑣) = (𝐺‘(2nd ‘𝑢)) ↔ (2nd ‘𝑢) = (◡𝐺‘(2nd ‘𝑣))))
77	68, 76	anbi12d 470	. . . . . . 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 6156	. . . . . . . 8 ⊢ (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) → (𝑣 = ⟨(𝐹‘(1st ‘𝑢)), (𝐺‘(2nd ‘𝑢))⟩ ↔ ((1st ‘𝑣) = (𝐹‘(1st ‘𝑢)) ∧ (2nd ‘𝑣) = (𝐺‘(2nd ‘𝑢)))))
79	78	ad2antll 488	. . . . . . 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 6156	. . . . . . . 8 ⊢ (𝑢 ∈ (𝑋 × 𝑌) → (𝑢 = ⟨(◡𝐹‘(1st ‘𝑣)), (◡𝐺‘(2nd ‘𝑣))⟩ ↔ ((1st ‘𝑢) = (◡𝐹‘(1st ‘𝑣)) ∧ (2nd ‘𝑢) = (◡𝐺‘(2nd ‘𝑣)))))
81	80	ad2antrl 487	. . . . . . 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 220	. . . . . 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 6053	. . . . 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 113	. . . 4 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) ↦ ⟨(◡𝐹‘(1st ‘𝑣)), (◡𝐺‘(2nd ‘𝑣))⟩))
85		vex 2733	. . . . . . . 8 ⊢ 𝑧 ∈ V
86		vex 2733	. . . . . . . 8 ⊢ 𝑤 ∈ V
87	85, 86	op1std 6127	. . . . . . 7 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (1st ‘𝑣) = 𝑧)
88	87	fveq2d 5500	. . . . . 6 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (◡𝐹‘(1st ‘𝑣)) = (◡𝐹‘𝑧))
89	85, 86	op2ndd 6128	. . . . . . 7 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (2nd ‘𝑣) = 𝑤)
90	89	fveq2d 5500	. . . . . 6 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (◡𝐺‘(2nd ‘𝑣)) = (◡𝐺‘𝑤))
91	88, 90	opeq12d 3773	. . . . 5 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ⟨(◡𝐹‘(1st ‘𝑣)), (◡𝐺‘(2nd ‘𝑣))⟩ = ⟨(◡𝐹‘𝑧), (◡𝐺‘𝑤)⟩)
92	91	mpompt 5945	. . . 4 ⊢ (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) ↦ ⟨(◡𝐹‘(1st ‘𝑣)), (◡𝐺‘(2nd ‘𝑣))⟩) = (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ ⟨(◡𝐹‘𝑧), (◡𝐺‘𝑤)⟩)
93	84, 92	eqtrdi 2219	. . 3 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ ⟨(◡𝐹‘𝑧), (◡𝐺‘𝑤)⟩))
94		cntop2 12996	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top)
95	3, 94	syl 14	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ Top)
96	31	toptopon 12810	. . . . 5 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
97	95, 96	sylib 121	. . . 4 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
98		cntop2 12996	. . . . . 6 ⊢ (𝐺 ∈ (𝐾 Cn 𝑀) → 𝑀 ∈ Top)
99	11, 98	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ Top)
100	37	toptopon 12810	. . . . 5 ⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀))
101	99, 100	sylib 121	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀))
102	97, 101	cnmpt1st 13082	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ 𝑧) ∈ ((𝐿 ×t 𝑀) Cn 𝐿))
103		hmeocnvcn 13100	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐿) → ◡𝐹 ∈ (𝐿 Cn 𝐽))
104	1, 103	syl 14	. . . . 5 ⊢ (𝜑 → ◡𝐹 ∈ (𝐿 Cn 𝐽))
105	97, 101, 102, 104	cnmpt21f 13086	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (◡𝐹‘𝑧)) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
106	97, 101	cnmpt2nd 13083	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ 𝑤) ∈ ((𝐿 ×t 𝑀) Cn 𝑀))
107		hmeocnvcn 13100	. . . . . 6 ⊢ (𝐺 ∈ (𝐾Homeo𝑀) → ◡𝐺 ∈ (𝑀 Cn 𝐾))
108	9, 107	syl 14	. . . . 5 ⊢ (𝜑 → ◡𝐺 ∈ (𝑀 Cn 𝐾))
109	97, 101, 106, 108	cnmpt21f 13086	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (◡𝐺‘𝑤)) ∈ ((𝐿 ×t 𝑀) Cn 𝐾))
110	97, 101, 105, 109	cnmpt2t 13087	. . 3 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ ⟨(◡𝐹‘𝑧), (◡𝐺‘𝑤)⟩) ∈ ((𝐿 ×t 𝑀) Cn (𝐽 ×t 𝐾)))
111	93, 110	eqeltrd 2247	. 2 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐿 ×t 𝑀) Cn (𝐽 ×t 𝐾)))
112		ishmeo 13098	. 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 415	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐽 ×t 𝐾)Homeo(𝐿 ×t 𝑀)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141  ⟨cop 3586  ∪ cuni 3796   ↦ cmpt 4050   × cxp 4609  ^◡ccnv 4610  ⟶wf 5194  –1-1-onto→wf1o 5197  ‘cfv 5198  (class class class)co 5853   ∈ cmpo 5855  1^st c1st 6117  2^nd c2nd 6118  Topctop 12789  TopOnctopon 12802   Cn ccn 12979   ×_t ctx 13046  Homeochmeo 13094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-topgen 12600  df-top 12790  df-topon 12803  df-bases 12835  df-cn 12982  df-tx 13047  df-hmeo 13095

This theorem is referenced by: (None)