Theorem xpstopnlem1 21833

Description: The function 𝐹 used in xpsval 16440 is a homeomorphism from the binary product topology to the indexed product topology. (Contributed by Mario Carneiro, 2-Sep-2015.)

Hypotheses

Ref	Expression
xpstopnlem1.f	⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))
xpstopnlem1.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
xpstopnlem1.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))

Assertion

Ref	Expression
xpstopnlem1	⊢ (𝜑 → 𝐹 ∈ ((𝐽 ×t 𝐾)Homeo(∏t‘◡({𝐽} +𝑐 {𝐾}))))

Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝐾,𝑦   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem xpstopnlem1

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpstopnlem1.j	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		xpstopnlem1.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
3		txtopon 21615	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
4	1, 2, 3	syl2anc 573	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
5		eqid 2771	. . . . . . . . . . . . 13 ⊢ (∏t‘{⟨∅, 𝐽⟩}) = (∏t‘{⟨∅, 𝐽⟩})
6		0ex 4924	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
7	6	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∅ ∈ V)
8	5, 7, 1	pt1hmeo 21830	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽Homeo(∏t‘{⟨∅, 𝐽⟩})))
9		hmeocn 21784	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽Homeo(∏t‘{⟨∅, 𝐽⟩})) → (𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽 Cn (∏t‘{⟨∅, 𝐽⟩})))
10		cntop2 21266	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽 Cn (∏t‘{⟨∅, 𝐽⟩})) → (∏t‘{⟨∅, 𝐽⟩}) ∈ Top)
11	8, 9, 10	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (∏t‘{⟨∅, 𝐽⟩}) ∈ Top)
12		eqid 2771	. . . . . . . . . . . 12 ⊢ ∪ (∏t‘{⟨∅, 𝐽⟩}) = ∪ (∏t‘{⟨∅, 𝐽⟩})
13	12	toptopon 20942	. . . . . . . . . . 11 ⊢ ((∏t‘{⟨∅, 𝐽⟩}) ∈ Top ↔ (∏t‘{⟨∅, 𝐽⟩}) ∈ (TopOn‘∪ (∏t‘{⟨∅, 𝐽⟩})))
14	11, 13	sylib 208	. . . . . . . . . 10 ⊢ (𝜑 → (∏t‘{⟨∅, 𝐽⟩}) ∈ (TopOn‘∪ (∏t‘{⟨∅, 𝐽⟩})))
15		eqid 2771	. . . . . . . . . . . . 13 ⊢ (∏t‘{⟨1𝑜, 𝐾⟩}) = (∏t‘{⟨1𝑜, 𝐾⟩})
16		1on 7720	. . . . . . . . . . . . . 14 ⊢ 1𝑜 ∈ On
17	16	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1𝑜 ∈ On)
18	15, 17, 2	pt1hmeo 21830	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩}) ∈ (𝐾Homeo(∏t‘{⟨1𝑜, 𝐾⟩})))
19		hmeocn 21784	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩}) ∈ (𝐾Homeo(∏t‘{⟨1𝑜, 𝐾⟩})) → (𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩}) ∈ (𝐾 Cn (∏t‘{⟨1𝑜, 𝐾⟩})))
20		cntop2 21266	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩}) ∈ (𝐾 Cn (∏t‘{⟨1𝑜, 𝐾⟩})) → (∏t‘{⟨1𝑜, 𝐾⟩}) ∈ Top)
21	18, 19, 20	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (∏t‘{⟨1𝑜, 𝐾⟩}) ∈ Top)
22		eqid 2771	. . . . . . . . . . . 12 ⊢ ∪ (∏t‘{⟨1𝑜, 𝐾⟩}) = ∪ (∏t‘{⟨1𝑜, 𝐾⟩})
23	22	toptopon 20942	. . . . . . . . . . 11 ⊢ ((∏t‘{⟨1𝑜, 𝐾⟩}) ∈ Top ↔ (∏t‘{⟨1𝑜, 𝐾⟩}) ∈ (TopOn‘∪ (∏t‘{⟨1𝑜, 𝐾⟩})))
24	21, 23	sylib 208	. . . . . . . . . 10 ⊢ (𝜑 → (∏t‘{⟨1𝑜, 𝐾⟩}) ∈ (TopOn‘∪ (∏t‘{⟨1𝑜, 𝐾⟩})))
25		txtopon 21615	. . . . . . . . . 10 ⊢ (((∏t‘{⟨∅, 𝐽⟩}) ∈ (TopOn‘∪ (∏t‘{⟨∅, 𝐽⟩})) ∧ (∏t‘{⟨1𝑜, 𝐾⟩}) ∈ (TopOn‘∪ (∏t‘{⟨1𝑜, 𝐾⟩}))) → ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩})) ∈ (TopOn‘(∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1𝑜, 𝐾⟩}))))
26	14, 24, 25	syl2anc 573	. . . . . . . . 9 ⊢ (𝜑 → ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩})) ∈ (TopOn‘(∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1𝑜, 𝐾⟩}))))
27		opeq2 4540	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑥 → ⟨∅, 𝑧⟩ = ⟨∅, 𝑥⟩)
28	27	sneqd 4328	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → {⟨∅, 𝑧⟩} = {⟨∅, 𝑥⟩})
29		eqid 2771	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩}) = (𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})
30		snex 5036	. . . . . . . . . . . . . . 15 ⊢ {⟨∅, 𝑥⟩} ∈ V
31	28, 29, 30	fvmpt 6424	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 → ((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥) = {⟨∅, 𝑥⟩})
32		opeq2 4540	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑦 → ⟨1𝑜, 𝑧⟩ = ⟨1𝑜, 𝑦⟩)
33	32	sneqd 4328	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑦 → {⟨1𝑜, 𝑧⟩} = {⟨1𝑜, 𝑦⟩})
34		eqid 2771	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩}) = (𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩})
35		snex 5036	. . . . . . . . . . . . . . 15 ⊢ {⟨1𝑜, 𝑦⟩} ∈ V
36	33, 34, 35	fvmpt 6424	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝑌 → ((𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦) = {⟨1𝑜, 𝑦⟩})
37		opeq12 4541	. . . . . . . . . . . . . 14 ⊢ ((((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥) = {⟨∅, 𝑥⟩} ∧ ((𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦) = {⟨1𝑜, 𝑦⟩}) → ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩ = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)
38	31, 36, 37	syl2an 583	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩ = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)
39	38	mpt2eq3ia 6867	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)
40		toponuni 20939	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
41	1, 40	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
42		toponuni 20939	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
43	2, 42	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 = ∪ 𝐾)
44		mpt2eq12 6862	. . . . . . . . . . . . 13 ⊢ ((𝑋 = ∪ 𝐽 ∧ 𝑌 = ∪ 𝐾) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩) = (𝑥 ∈ ∪ 𝐽, 𝑦 ∈ ∪ 𝐾 ↦ ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩))
45	41, 43, 44	syl2anc 573	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩) = (𝑥 ∈ ∪ 𝐽, 𝑦 ∈ ∪ 𝐾 ↦ ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩))
46	39, 45	syl5eqr 2819	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) = (𝑥 ∈ ∪ 𝐽, 𝑦 ∈ ∪ 𝐾 ↦ ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩))
47		eqid 2771	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
48		eqid 2771	. . . . . . . . . . . 12 ⊢ ∪ 𝐾 = ∪ 𝐾
49	47, 48, 8, 18	txhmeo 21827	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ∪ 𝐽, 𝑦 ∈ ∪ 𝐾 ↦ ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩) ∈ ((𝐽 ×t 𝐾)Homeo((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩}))))
50	46, 49	eqeltrd 2850	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾)Homeo((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩}))))
51		hmeocn 21784	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾)Homeo((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩}))) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾) Cn ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩}))))
52	50, 51	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾) Cn ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩}))))
53		cnf2 21274	. . . . . . . . 9 ⊢ (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩})) ∈ (TopOn‘(∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1𝑜, 𝐾⟩}))) ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾) Cn ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩})))) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩):(𝑋 × 𝑌)⟶(∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1𝑜, 𝐾⟩})))
54	4, 26, 52, 53	syl3anc 1476	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩):(𝑋 × 𝑌)⟶(∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1𝑜, 𝐾⟩})))
55		eqid 2771	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)
56	55	fmpt2 7387	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1𝑜, 𝐾⟩})) ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩):(𝑋 × 𝑌)⟶(∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1𝑜, 𝐾⟩})))
57	54, 56	sylibr 224	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1𝑜, 𝐾⟩})))
58	57	r19.21bi 3081	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1𝑜, 𝐾⟩})))
59	58	r19.21bi 3081	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1𝑜, 𝐾⟩})))
60	59	anasss 457	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1𝑜, 𝐾⟩})))
61		eqidd 2772	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩))
62		vex 3354	. . . . . . . . 9 ⊢ 𝑥 ∈ V
63		vex 3354	. . . . . . . . 9 ⊢ 𝑦 ∈ V
64	62, 63	op1std 7325	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
65	62, 63	op2ndd 7326	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
66	64, 65	uneq12d 3919	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((1st ‘𝑧) ∪ (2nd ‘𝑧)) = (𝑥 ∪ 𝑦))
67	66	mpt2mpt 6899	. . . . . 6 ⊢ (𝑧 ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1𝑜, 𝐾⟩})) ↦ ((1st ‘𝑧) ∪ (2nd ‘𝑧))) = (𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦))
68	67	eqcomi 2780	. . . . 5 ⊢ (𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)) = (𝑧 ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1𝑜, 𝐾⟩})) ↦ ((1st ‘𝑧) ∪ (2nd ‘𝑧)))
69	68	a1i 11	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)) = (𝑧 ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1𝑜, 𝐾⟩})) ↦ ((1st ‘𝑧) ∪ (2nd ‘𝑧))))
70	30, 35	op1std 7325	. . . . . 6 ⊢ (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ → (1st ‘𝑧) = {⟨∅, 𝑥⟩})
71	30, 35	op2ndd 7326	. . . . . 6 ⊢ (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ → (2nd ‘𝑧) = {⟨1𝑜, 𝑦⟩})
72	70, 71	uneq12d 3919	. . . . 5 ⊢ (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ → ((1st ‘𝑧) ∪ (2nd ‘𝑧)) = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩}))
73		xpscg 16426	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ◡({𝑥} +𝑐 {𝑦}) = {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩})
74	62, 63, 73	mp2an 672	. . . . . 6 ⊢ ◡({𝑥} +𝑐 {𝑦}) = {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}
75		df-pr 4319	. . . . . 6 ⊢ {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩})
76	74, 75	eqtri 2793	. . . . 5 ⊢ ◡({𝑥} +𝑐 {𝑦}) = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩})
77	72, 76	syl6eqr 2823	. . . 4 ⊢ (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ → ((1st ‘𝑧) ∪ (2nd ‘𝑧)) = ◡({𝑥} +𝑐 {𝑦}))
78	60, 61, 69, 77	fmpt2co 7411	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
79		xpstopnlem1.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))
80	78, 79	syl6reqr 2824	. 2 ⊢ (𝜑 → 𝐹 = ((𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)))
81		eqid 2771	. . . . 5 ⊢ ∪ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {∅})) = ∪ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {∅}))
82		eqid 2771	. . . . 5 ⊢ ∪ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) = ∪ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}))
83		eqid 2771	. . . . 5 ⊢ (∏t‘◡({𝐽} +𝑐 {𝐾})) = (∏t‘◡({𝐽} +𝑐 {𝐾}))
84		eqid 2771	. . . . 5 ⊢ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {∅})) = (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {∅}))
85		eqid 2771	. . . . 5 ⊢ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) = (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}))
86		eqid 2771	. . . . 5 ⊢ (𝑥 ∈ ∪ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {∅})), 𝑦 ∈ ∪ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) ↦ (𝑥 ∪ 𝑦)) = (𝑥 ∈ ∪ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {∅})), 𝑦 ∈ ∪ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) ↦ (𝑥 ∪ 𝑦))
87		2on 7722	. . . . . 6 ⊢ 2𝑜 ∈ On
88	87	a1i 11	. . . . 5 ⊢ (𝜑 → 2𝑜 ∈ On)
89		topontop 20938	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
90	1, 89	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Top)
91		topontop 20938	. . . . . . 7 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
92	2, 91	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Top)
93		xpscf 16434	. . . . . 6 ⊢ (◡({𝐽} +𝑐 {𝐾}):2𝑜⟶Top ↔ (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
94	90, 92, 93	sylanbrc 572	. . . . 5 ⊢ (𝜑 → ◡({𝐽} +𝑐 {𝐾}):2𝑜⟶Top)
95		df2o3 7727	. . . . . . 7 ⊢ 2𝑜 = {∅, 1𝑜}
96		df-pr 4319	. . . . . . 7 ⊢ {∅, 1𝑜} = ({∅} ∪ {1𝑜})
97	95, 96	eqtri 2793	. . . . . 6 ⊢ 2𝑜 = ({∅} ∪ {1𝑜})
98	97	a1i 11	. . . . 5 ⊢ (𝜑 → 2𝑜 = ({∅} ∪ {1𝑜}))
99		1n0 7729	. . . . . . 7 ⊢ 1𝑜 ≠ ∅
100	99	necomi 2997	. . . . . 6 ⊢ ∅ ≠ 1𝑜
101		disjsn2 4384	. . . . . 6 ⊢ (∅ ≠ 1𝑜 → ({∅} ∩ {1𝑜}) = ∅)
102	100, 101	mp1i 13	. . . . 5 ⊢ (𝜑 → ({∅} ∩ {1𝑜}) = ∅)
103	81, 82, 83, 84, 85, 86, 88, 94, 98, 102	ptunhmeo 21832	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ∪ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {∅})), 𝑦 ∈ ∪ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) ↦ (𝑥 ∪ 𝑦)) ∈ (((∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {∅})) ×t (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})))Homeo(∏t‘◡({𝐽} +𝑐 {𝐾}))))
104		xpscfn 16427	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ◡({𝐽} +𝑐 {𝐾}) Fn 2𝑜)
105	1, 2, 104	syl2anc 573	. . . . . . . . 9 ⊢ (𝜑 → ◡({𝐽} +𝑐 {𝐾}) Fn 2𝑜)
106	6	prid1 4433	. . . . . . . . . 10 ⊢ ∅ ∈ {∅, 1𝑜}
107	106, 95	eleqtrri 2849	. . . . . . . . 9 ⊢ ∅ ∈ 2𝑜
108		fnressn 6568	. . . . . . . . 9 ⊢ ((◡({𝐽} +𝑐 {𝐾}) Fn 2𝑜 ∧ ∅ ∈ 2𝑜) → (◡({𝐽} +𝑐 {𝐾}) ↾ {∅}) = {⟨∅, (◡({𝐽} +𝑐 {𝐾})‘∅)⟩})
109	105, 107, 108	sylancl 574	. . . . . . . 8 ⊢ (𝜑 → (◡({𝐽} +𝑐 {𝐾}) ↾ {∅}) = {⟨∅, (◡({𝐽} +𝑐 {𝐾})‘∅)⟩})
110		xpsc0 16428	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (◡({𝐽} +𝑐 {𝐾})‘∅) = 𝐽)
111	1, 110	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (◡({𝐽} +𝑐 {𝐾})‘∅) = 𝐽)
112	111	opeq2d 4546	. . . . . . . . 9 ⊢ (𝜑 → ⟨∅, (◡({𝐽} +𝑐 {𝐾})‘∅)⟩ = ⟨∅, 𝐽⟩)
113	112	sneqd 4328	. . . . . . . 8 ⊢ (𝜑 → {⟨∅, (◡({𝐽} +𝑐 {𝐾})‘∅)⟩} = {⟨∅, 𝐽⟩})
114	109, 113	eqtrd 2805	. . . . . . 7 ⊢ (𝜑 → (◡({𝐽} +𝑐 {𝐾}) ↾ {∅}) = {⟨∅, 𝐽⟩})
115	114	fveq2d 6336	. . . . . 6 ⊢ (𝜑 → (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {∅})) = (∏t‘{⟨∅, 𝐽⟩}))
116	115	unieqd 4584	. . . . 5 ⊢ (𝜑 → ∪ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {∅})) = ∪ (∏t‘{⟨∅, 𝐽⟩}))
117		1oex 7721	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ V
118	117	prid2 4434	. . . . . . . . . 10 ⊢ 1𝑜 ∈ {∅, 1𝑜}
119	118, 95	eleqtrri 2849	. . . . . . . . 9 ⊢ 1𝑜 ∈ 2𝑜
120		fnressn 6568	. . . . . . . . 9 ⊢ ((◡({𝐽} +𝑐 {𝐾}) Fn 2𝑜 ∧ 1𝑜 ∈ 2𝑜) → (◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}) = {⟨1𝑜, (◡({𝐽} +𝑐 {𝐾})‘1𝑜)⟩})
121	105, 119, 120	sylancl 574	. . . . . . . 8 ⊢ (𝜑 → (◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}) = {⟨1𝑜, (◡({𝐽} +𝑐 {𝐾})‘1𝑜)⟩})
122		xpsc1 16429	. . . . . . . . . . 11 ⊢ (𝐾 ∈ (TopOn‘𝑌) → (◡({𝐽} +𝑐 {𝐾})‘1𝑜) = 𝐾)
123	2, 122	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (◡({𝐽} +𝑐 {𝐾})‘1𝑜) = 𝐾)
124	123	opeq2d 4546	. . . . . . . . 9 ⊢ (𝜑 → ⟨1𝑜, (◡({𝐽} +𝑐 {𝐾})‘1𝑜)⟩ = ⟨1𝑜, 𝐾⟩)
125	124	sneqd 4328	. . . . . . . 8 ⊢ (𝜑 → {⟨1𝑜, (◡({𝐽} +𝑐 {𝐾})‘1𝑜)⟩} = {⟨1𝑜, 𝐾⟩})
126	121, 125	eqtrd 2805	. . . . . . 7 ⊢ (𝜑 → (◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}) = {⟨1𝑜, 𝐾⟩})
127	126	fveq2d 6336	. . . . . 6 ⊢ (𝜑 → (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) = (∏t‘{⟨1𝑜, 𝐾⟩}))
128	127	unieqd 4584	. . . . 5 ⊢ (𝜑 → ∪ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) = ∪ (∏t‘{⟨1𝑜, 𝐾⟩}))
129		eqidd 2772	. . . . 5 ⊢ (𝜑 → (𝑥 ∪ 𝑦) = (𝑥 ∪ 𝑦))
130	116, 128, 129	mpt2eq123dv 6864	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ∪ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {∅})), 𝑦 ∈ ∪ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) ↦ (𝑥 ∪ 𝑦)) = (𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)))
131	115, 127	oveq12d 6811	. . . . 5 ⊢ (𝜑 → ((∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {∅})) ×t (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}))) = ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩})))
132	131	oveq1d 6808	. . . 4 ⊢ (𝜑 → (((∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {∅})) ×t (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})))Homeo(∏t‘◡({𝐽} +𝑐 {𝐾}))) = (((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩}))Homeo(∏t‘◡({𝐽} +𝑐 {𝐾}))))
133	103, 130, 132	3eltr3d 2864	. . 3 ⊢ (𝜑 → (𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)) ∈ (((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩}))Homeo(∏t‘◡({𝐽} +𝑐 {𝐾}))))
134		hmeoco 21796	. . 3 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾)Homeo((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩}))) ∧ (𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)) ∈ (((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩}))Homeo(∏t‘◡({𝐽} +𝑐 {𝐾})))) → ((𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)) ∈ ((𝐽 ×t 𝐾)Homeo(∏t‘◡({𝐽} +𝑐 {𝐾}))))
135	50, 133, 134	syl2anc 573	. 2 ⊢ (𝜑 → ((𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)) ∈ ((𝐽 ×t 𝐾)Homeo(∏t‘◡({𝐽} +𝑐 {𝐾}))))
136	80, 135	eqeltrd 2850	1 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ×t 𝐾)Homeo(∏t‘◡({𝐽} +𝑐 {𝐾}))))

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145   ≠ wne 2943  ∀wral 3061  Vcvv 3351   ∪ cun 3721   ∩ cin 3722  ∅c0 4063  {csn 4316  {cpr 4318  ⟨cop 4322  ∪ cuni 4574   ↦ cmpt 4863   × cxp 5247  ^◡ccnv 5248   ↾ cres 5251   ∘ ccom 5253  Oncon0 5866   Fn wfn 6026  ⟶wf 6027  ‘cfv 6031  (class class class)co 6793   ↦ cmpt2 6795  1^st c1st 7313  2^nd c2nd 7314  1_𝑜c1o 7706  2_𝑜c2o 7707   +_𝑐 ccda 9191  ∏_tcpt 16307  Topctop 20918  TopOnctopon 20935   Cn ccn 21249   ×_t ctx 21584  Homeochmeo 21777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fi 8473  df-cda 9192  df-topgen 16312  df-pt 16313  df-top 20919  df-topon 20936  df-bases 20971  df-cn 21252  df-cnp 21253  df-tx 21586  df-hmeo 21779

This theorem is referenced by:  xpstopnlem2  21835