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Theorem txhmeo 22405

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 22362	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐿) → 𝐹 ∈ (𝐽 Cn 𝐿))
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐿))
4		cntop1 21842	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐿) → 𝐽 ∈ Top)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Top)
6		txhmeo.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
7	6	toptopon 21519	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
8	5, 7	sylib 220	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
9		txhmeo.4	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (𝐾Homeo𝑀))
10		hmeocn 22362	. . . . . 6 ⊢ (𝐺 ∈ (𝐾Homeo𝑀) → 𝐺 ∈ (𝐾 Cn 𝑀))
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝑀))
12		cntop1 21842	. . . . 5 ⊢ (𝐺 ∈ (𝐾 Cn 𝑀) → 𝐾 ∈ Top)
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Top)
14		txhmeo.2	. . . . 5 ⊢ 𝑌 = ∪ 𝐾
15	14	toptopon 21519	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
16	13, 15	sylib 220	. . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
17	8, 16	cnmpt1st 22270	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
18	8, 16, 17, 3	cnmpt21f 22274	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐹‘𝑥)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐿))
19	8, 16	cnmpt2nd 22271	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐽 ×_t 𝐾) Cn 𝐾))
20	8, 16, 19, 11	cnmpt21f 22274	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐺‘𝑦)) ∈ ((𝐽 ×_t 𝐾) Cn 𝑀))
21	8, 16, 18, 20	cnmpt2t 22275	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐽 ×_t 𝐾) Cn (𝐿 ×_t 𝑀)))
22		vex 3497	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
23		vex 3497	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
24	22, 23	op1std 7693	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝑢) = 𝑥)
25	24	fveq2d 6668	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1^st ‘𝑢)) = (𝐹‘𝑥))
26	22, 23	op2ndd 7694	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝑢) = 𝑦)
27	26	fveq2d 6668	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝐺‘(2^nd ‘𝑢)) = (𝐺‘𝑦))
28	25, 27	opeq12d 4804	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → ⟨(𝐹‘(1^st ‘𝑢)), (𝐺‘(2^nd ‘𝑢))⟩ = ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
29	28	mpompt 7260	. . . . . . 7 ⊢ (𝑢 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐹‘(1^st ‘𝑢)), (𝐺‘(2^nd ‘𝑢))⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
30	29	eqcomi 2830	. . . . . 6 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = (𝑢 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐹‘(1^st ‘𝑢)), (𝐺‘(2^nd ‘𝑢))⟩)
31		eqid 2821	. . . . . . . . . 10 ⊢ ∪ 𝐿 = ∪ 𝐿
32	6, 31	cnf 21848	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽 Cn 𝐿) → 𝐹:𝑋⟶∪ 𝐿)
33	3, 32	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐿)
34		xp1st 7715	. . . . . . . 8 ⊢ (𝑢 ∈ (𝑋 × 𝑌) → (1^st ‘𝑢) ∈ 𝑋)
35		ffvelrn 6843	. . . . . . . 8 ⊢ ((𝐹:𝑋⟶∪ 𝐿 ∧ (1^st ‘𝑢) ∈ 𝑋) → (𝐹‘(1^st ‘𝑢)) ∈ ∪ 𝐿)
36	33, 34, 35	syl2an 597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑋 × 𝑌)) → (𝐹‘(1^st ‘𝑢)) ∈ ∪ 𝐿)
37		eqid 2821	. . . . . . . . . 10 ⊢ ∪ 𝑀 = ∪ 𝑀
38	14, 37	cnf 21848	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝐾 Cn 𝑀) → 𝐺:𝑌⟶∪ 𝑀)
39	11, 38	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑌⟶∪ 𝑀)
40		xp2nd 7716	. . . . . . . 8 ⊢ (𝑢 ∈ (𝑋 × 𝑌) → (2^nd ‘𝑢) ∈ 𝑌)
41		ffvelrn 6843	. . . . . . . 8 ⊢ ((𝐺:𝑌⟶∪ 𝑀 ∧ (2^nd ‘𝑢) ∈ 𝑌) → (𝐺‘(2^nd ‘𝑢)) ∈ ∪ 𝑀)
42	39, 40, 41	syl2an 597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑋 × 𝑌)) → (𝐺‘(2^nd ‘𝑢)) ∈ ∪ 𝑀)
43	36, 42	opelxpd 5587	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑋 × 𝑌)) → ⟨(𝐹‘(1^st ‘𝑢)), (𝐺‘(2^nd ‘𝑢))⟩ ∈ (∪ 𝐿 × ∪ 𝑀))
44	6, 31	hmeof1o 22366	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐽Homeo𝐿) → 𝐹:𝑋–1-1-onto→∪ 𝐿)
45	1, 44	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→∪ 𝐿)
46		f1ocnv 6621	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐿 → ^◡𝐹:∪ 𝐿–1-1-onto→𝑋)
47		f1of 6609	. . . . . . . . 9 ⊢ (^◡𝐹:∪ 𝐿–1-1-onto→𝑋 → ^◡𝐹:∪ 𝐿⟶𝑋)
48	45, 46, 47	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ^◡𝐹:∪ 𝐿⟶𝑋)
49		xp1st 7715	. . . . . . . 8 ⊢ (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) → (1^st ‘𝑣) ∈ ∪ 𝐿)
50		ffvelrn 6843	. . . . . . . 8 ⊢ ((^◡𝐹:∪ 𝐿⟶𝑋 ∧ (1^st ‘𝑣) ∈ ∪ 𝐿) → (^◡𝐹‘(1^st ‘𝑣)) ∈ 𝑋)
51	48, 49, 50	syl2an 597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀)) → (^◡𝐹‘(1^st ‘𝑣)) ∈ 𝑋)
52	14, 37	hmeof1o 22366	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝐾Homeo𝑀) → 𝐺:𝑌–1-1-onto→∪ 𝑀)
53	9, 52	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝑌–1-1-onto→∪ 𝑀)
54		f1ocnv 6621	. . . . . . . . 9 ⊢ (𝐺:𝑌–1-1-onto→∪ 𝑀 → ^◡𝐺:∪ 𝑀–1-1-onto→𝑌)
55		f1of 6609	. . . . . . . . 9 ⊢ (^◡𝐺:∪ 𝑀–1-1-onto→𝑌 → ^◡𝐺:∪ 𝑀⟶𝑌)
56	53, 54, 55	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ^◡𝐺:∪ 𝑀⟶𝑌)
57		xp2nd 7716	. . . . . . . 8 ⊢ (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) → (2^nd ‘𝑣) ∈ ∪ 𝑀)
58		ffvelrn 6843	. . . . . . . 8 ⊢ ((^◡𝐺:∪ 𝑀⟶𝑌 ∧ (2^nd ‘𝑣) ∈ ∪ 𝑀) → (^◡𝐺‘(2^nd ‘𝑣)) ∈ 𝑌)
59	56, 57, 58	syl2an 597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀)) → (^◡𝐺‘(2^nd ‘𝑣)) ∈ 𝑌)
60	51, 59	opelxpd 5587	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀)) → ⟨(^◡𝐹‘(1^st ‘𝑣)), (^◡𝐺‘(2^nd ‘𝑣))⟩ ∈ (𝑋 × 𝑌))
61	45	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → 𝐹:𝑋–1-1-onto→∪ 𝐿)
62	34	ad2antrl 726	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (1^st ‘𝑢) ∈ 𝑋)
63	49	ad2antll 727	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (1^st ‘𝑣) ∈ ∪ 𝐿)
64		f1ocnvfvb 7030	. . . . . . . . . 10 ⊢ ((𝐹:𝑋–1-1-onto→∪ 𝐿 ∧ (1^st ‘𝑢) ∈ 𝑋 ∧ (1^st ‘𝑣) ∈ ∪ 𝐿) → ((𝐹‘(1^st ‘𝑢)) = (1^st ‘𝑣) ↔ (^◡𝐹‘(1^st ‘𝑣)) = (1^st ‘𝑢)))
65	61, 62, 63, 64	syl3anc 1367	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → ((𝐹‘(1^st ‘𝑢)) = (1^st ‘𝑣) ↔ (^◡𝐹‘(1^st ‘𝑣)) = (1^st ‘𝑢)))
66		eqcom 2828	. . . . . . . . 9 ⊢ ((1^st ‘𝑣) = (𝐹‘(1^st ‘𝑢)) ↔ (𝐹‘(1^st ‘𝑢)) = (1^st ‘𝑣))
67		eqcom 2828	. . . . . . . . 9 ⊢ ((1^st ‘𝑢) = (^◡𝐹‘(1^st ‘𝑣)) ↔ (^◡𝐹‘(1^st ‘𝑣)) = (1^st ‘𝑢))
68	65, 66, 67	3bitr4g 316	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → ((1^st ‘𝑣) = (𝐹‘(1^st ‘𝑢)) ↔ (1^st ‘𝑢) = (^◡𝐹‘(1^st ‘𝑣))))
69	53	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → 𝐺:𝑌–1-1-onto→∪ 𝑀)
70	40	ad2antrl 726	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (2^nd ‘𝑢) ∈ 𝑌)
71	57	ad2antll 727	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (2^nd ‘𝑣) ∈ ∪ 𝑀)
72		f1ocnvfvb 7030	. . . . . . . . . 10 ⊢ ((𝐺:𝑌–1-1-onto→∪ 𝑀 ∧ (2^nd ‘𝑢) ∈ 𝑌 ∧ (2^nd ‘𝑣) ∈ ∪ 𝑀) → ((𝐺‘(2^nd ‘𝑢)) = (2^nd ‘𝑣) ↔ (^◡𝐺‘(2^nd ‘𝑣)) = (2^nd ‘𝑢)))
73	69, 70, 71, 72	syl3anc 1367	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → ((𝐺‘(2^nd ‘𝑢)) = (2^nd ‘𝑣) ↔ (^◡𝐺‘(2^nd ‘𝑣)) = (2^nd ‘𝑢)))
74		eqcom 2828	. . . . . . . . 9 ⊢ ((2^nd ‘𝑣) = (𝐺‘(2^nd ‘𝑢)) ↔ (𝐺‘(2^nd ‘𝑢)) = (2^nd ‘𝑣))
75		eqcom 2828	. . . . . . . . 9 ⊢ ((2^nd ‘𝑢) = (^◡𝐺‘(2^nd ‘𝑣)) ↔ (^◡𝐺‘(2^nd ‘𝑣)) = (2^nd ‘𝑢))
76	73, 74, 75	3bitr4g 316	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → ((2^nd ‘𝑣) = (𝐺‘(2^nd ‘𝑢)) ↔ (2^nd ‘𝑢) = (^◡𝐺‘(2^nd ‘𝑣))))
77	68, 76	anbi12d 632	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (((1^st ‘𝑣) = (𝐹‘(1^st ‘𝑢)) ∧ (2^nd ‘𝑣) = (𝐺‘(2^nd ‘𝑢))) ↔ ((1^st ‘𝑢) = (^◡𝐹‘(1^st ‘𝑣)) ∧ (2^nd ‘𝑢) = (^◡𝐺‘(2^nd ‘𝑣)))))
78		eqop 7725	. . . . . . . 8 ⊢ (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) → (𝑣 = ⟨(𝐹‘(1^st ‘𝑢)), (𝐺‘(2^nd ‘𝑢))⟩ ↔ ((1^st ‘𝑣) = (𝐹‘(1^st ‘𝑢)) ∧ (2^nd ‘𝑣) = (𝐺‘(2^nd ‘𝑢)))))
79	78	ad2antll 727	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (𝑣 = ⟨(𝐹‘(1^st ‘𝑢)), (𝐺‘(2^nd ‘𝑢))⟩ ↔ ((1^st ‘𝑣) = (𝐹‘(1^st ‘𝑢)) ∧ (2^nd ‘𝑣) = (𝐺‘(2^nd ‘𝑢)))))
80		eqop 7725	. . . . . . . 8 ⊢ (𝑢 ∈ (𝑋 × 𝑌) → (𝑢 = ⟨(^◡𝐹‘(1^st ‘𝑣)), (^◡𝐺‘(2^nd ‘𝑣))⟩ ↔ ((1^st ‘𝑢) = (^◡𝐹‘(1^st ‘𝑣)) ∧ (2^nd ‘𝑢) = (^◡𝐺‘(2^nd ‘𝑣)))))
81	80	ad2antrl 726	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (𝑢 = ⟨(^◡𝐹‘(1^st ‘𝑣)), (^◡𝐺‘(2^nd ‘𝑣))⟩ ↔ ((1^st ‘𝑢) = (^◡𝐹‘(1^st ‘𝑣)) ∧ (2^nd ‘𝑢) = (^◡𝐺‘(2^nd ‘𝑣)))))
82	77, 79, 81	3bitr4rd 314	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (𝑢 = ⟨(^◡𝐹‘(1^st ‘𝑣)), (^◡𝐺‘(2^nd ‘𝑣))⟩ ↔ 𝑣 = ⟨(𝐹‘(1^st ‘𝑢)), (𝐺‘(2^nd ‘𝑢))⟩))
83	30, 43, 60, 82	f1ocnv2d 7392	. . . . 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‘(1^st ‘𝑣)), (^◡𝐺‘(2^nd ‘𝑣))⟩)))
84	83	simprd 498	. . . 4 ⊢ (𝜑 → ^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) ↦ ⟨(^◡𝐹‘(1^st ‘𝑣)), (^◡𝐺‘(2^nd ‘𝑣))⟩))
85		vex 3497	. . . . . . . 8 ⊢ 𝑧 ∈ V
86		vex 3497	. . . . . . . 8 ⊢ 𝑤 ∈ V
87	85, 86	op1std 7693	. . . . . . 7 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (1^st ‘𝑣) = 𝑧)
88	87	fveq2d 6668	. . . . . 6 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (^◡𝐹‘(1^st ‘𝑣)) = (^◡𝐹‘𝑧))
89	85, 86	op2ndd 7694	. . . . . . 7 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (2^nd ‘𝑣) = 𝑤)
90	89	fveq2d 6668	. . . . . 6 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (^◡𝐺‘(2^nd ‘𝑣)) = (^◡𝐺‘𝑤))
91	88, 90	opeq12d 4804	. . . . 5 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ⟨(^◡𝐹‘(1^st ‘𝑣)), (^◡𝐺‘(2^nd ‘𝑣))⟩ = ⟨(^◡𝐹‘𝑧), (^◡𝐺‘𝑤)⟩)
92	91	mpompt 7260	. . . 4 ⊢ (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) ↦ ⟨(^◡𝐹‘(1^st ‘𝑣)), (^◡𝐺‘(2^nd ‘𝑣))⟩) = (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ ⟨(^◡𝐹‘𝑧), (^◡𝐺‘𝑤)⟩)
93	84, 92	syl6eq 2872	. . 3 ⊢ (𝜑 → ^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ ⟨(^◡𝐹‘𝑧), (^◡𝐺‘𝑤)⟩))
94		cntop2 21843	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top)
95	3, 94	syl 17	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ Top)
96	31	toptopon 21519	. . . . 5 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
97	95, 96	sylib 220	. . . 4 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
98		cntop2 21843	. . . . . 6 ⊢ (𝐺 ∈ (𝐾 Cn 𝑀) → 𝑀 ∈ Top)
99	11, 98	syl 17	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ Top)
100	37	toptopon 21519	. . . . 5 ⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀))
101	99, 100	sylib 220	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀))
102	97, 101	cnmpt1st 22270	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ 𝑧) ∈ ((𝐿 ×_t 𝑀) Cn 𝐿))
103		hmeocnvcn 22363	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐿) → ^◡𝐹 ∈ (𝐿 Cn 𝐽))
104	1, 103	syl 17	. . . . 5 ⊢ (𝜑 → ^◡𝐹 ∈ (𝐿 Cn 𝐽))
105	97, 101, 102, 104	cnmpt21f 22274	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (^◡𝐹‘𝑧)) ∈ ((𝐿 ×_t 𝑀) Cn 𝐽))
106	97, 101	cnmpt2nd 22271	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ 𝑤) ∈ ((𝐿 ×_t 𝑀) Cn 𝑀))
107		hmeocnvcn 22363	. . . . . 6 ⊢ (𝐺 ∈ (𝐾Homeo𝑀) → ^◡𝐺 ∈ (𝑀 Cn 𝐾))
108	9, 107	syl 17	. . . . 5 ⊢ (𝜑 → ^◡𝐺 ∈ (𝑀 Cn 𝐾))
109	97, 101, 106, 108	cnmpt21f 22274	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (^◡𝐺‘𝑤)) ∈ ((𝐿 ×_t 𝑀) Cn 𝐾))
110	97, 101, 105, 109	cnmpt2t 22275	. . 3 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ ⟨(^◡𝐹‘𝑧), (^◡𝐺‘𝑤)⟩) ∈ ((𝐿 ×_t 𝑀) Cn (𝐽 ×_t 𝐾)))
111	93, 110	eqeltrd 2913	. 2 ⊢ (𝜑 → ^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐿 ×_t 𝑀) Cn (𝐽 ×_t 𝐾)))
112		ishmeo 22361	. 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 585	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐽 ×_t 𝐾)Homeo(𝐿 ×_t 𝑀)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⟨cop 4566 ∪ cuni 4831 ↦ cmpt 5138 × cxp 5547 ^◡ccnv 5548 ⟶wf 6345 –1-1-onto→wf1o 6348 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 1^st c1st 7681 2^nd c2nd 7682 Topctop 21495 TopOnctopon 21512 Cn ccn 21826 ×_t ctx 22162 Homeochmeo 22355

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-map 8402 df-topgen 16711 df-top 21496 df-topon 21513 df-bases 21548 df-cn 21829 df-tx 22164 df-hmeo 22357

This theorem is referenced by: xpstopnlem1 22411

W3C validator