Theorem xpstopnlem1 21612

Description: The function 𝐹 used in xpsval 16232 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 21394	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
4	1, 2, 3	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
5		eqid 2622	. . . . . . . . . . . . 13 ⊢ (∏t‘{⟨∅, 𝐽⟩}) = (∏t‘{⟨∅, 𝐽⟩})
6		0ex 4790	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
7	6	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∅ ∈ V)
8	5, 7, 1	pt1hmeo 21609	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽Homeo(∏t‘{⟨∅, 𝐽⟩})))
9		hmeocn 21563	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽Homeo(∏t‘{⟨∅, 𝐽⟩})) → (𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽 Cn (∏t‘{⟨∅, 𝐽⟩})))
10		cntop2 21045	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽 Cn (∏t‘{⟨∅, 𝐽⟩})) → (∏t‘{⟨∅, 𝐽⟩}) ∈ Top)
11	8, 9, 10	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (∏t‘{⟨∅, 𝐽⟩}) ∈ Top)
12		eqid 2622	. . . . . . . . . . . 12 ⊢ ∪ (∏t‘{⟨∅, 𝐽⟩}) = ∪ (∏t‘{⟨∅, 𝐽⟩})
13	12	toptopon 20722	. . . . . . . . . . 11 ⊢ ((∏t‘{⟨∅, 𝐽⟩}) ∈ Top ↔ (∏t‘{⟨∅, 𝐽⟩}) ∈ (TopOn‘∪ (∏t‘{⟨∅, 𝐽⟩})))
14	11, 13	sylib 208	. . . . . . . . . 10 ⊢ (𝜑 → (∏t‘{⟨∅, 𝐽⟩}) ∈ (TopOn‘∪ (∏t‘{⟨∅, 𝐽⟩})))
15		eqid 2622	. . . . . . . . . . . . 13 ⊢ (∏t‘{⟨1𝑜, 𝐾⟩}) = (∏t‘{⟨1𝑜, 𝐾⟩})
16		1on 7567	. . . . . . . . . . . . . 14 ⊢ 1𝑜 ∈ On
17	16	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1𝑜 ∈ On)
18	15, 17, 2	pt1hmeo 21609	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩}) ∈ (𝐾Homeo(∏t‘{⟨1𝑜, 𝐾⟩})))
19		hmeocn 21563	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩}) ∈ (𝐾Homeo(∏t‘{⟨1𝑜, 𝐾⟩})) → (𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩}) ∈ (𝐾 Cn (∏t‘{⟨1𝑜, 𝐾⟩})))
20		cntop2 21045	. . . . . . . . . . . 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 2622	. . . . . . . . . . . 12 ⊢ ∪ (∏t‘{⟨1𝑜, 𝐾⟩}) = ∪ (∏t‘{⟨1𝑜, 𝐾⟩})
23	22	toptopon 20722	. . . . . . . . . . 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 21394	. . . . . . . . . 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 693	. . . . . . . . 9 ⊢ (𝜑 → ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩})) ∈ (TopOn‘(∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1𝑜, 𝐾⟩}))))
27		opeq2 4403	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑥 → ⟨∅, 𝑧⟩ = ⟨∅, 𝑥⟩)
28	27	sneqd 4189	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → {⟨∅, 𝑧⟩} = {⟨∅, 𝑥⟩})
29		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩}) = (𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})
30		snex 4908	. . . . . . . . . . . . . . 15 ⊢ {⟨∅, 𝑥⟩} ∈ V
31	28, 29, 30	fvmpt 6282	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 → ((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥) = {⟨∅, 𝑥⟩})
32		opeq2 4403	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑦 → ⟨1𝑜, 𝑧⟩ = ⟨1𝑜, 𝑦⟩)
33	32	sneqd 4189	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑦 → {⟨1𝑜, 𝑧⟩} = {⟨1𝑜, 𝑦⟩})
34		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩}) = (𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩})
35		snex 4908	. . . . . . . . . . . . . . 15 ⊢ {⟨1𝑜, 𝑦⟩} ∈ V
36	33, 34, 35	fvmpt 6282	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝑌 → ((𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦) = {⟨1𝑜, 𝑦⟩})
37		opeq12 4404	. . . . . . . . . . . . . 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 494	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩ = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)
39	38	mpt2eq3ia 6720	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)
40		toponuni 20719	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
41	1, 40	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
42		toponuni 20719	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
43	2, 42	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 = ∪ 𝐾)
44		mpt2eq12 6715	. . . . . . . . . . . . 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 693	. . . . . . . . . . . 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 2670	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) = (𝑥 ∈ ∪ 𝐽, 𝑦 ∈ ∪ 𝐾 ↦ ⟨((𝑧 ∈ 𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧 ∈ 𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩))
47		eqid 2622	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
48		eqid 2622	. . . . . . . . . . . 12 ⊢ ∪ 𝐾 = ∪ 𝐾
49	47, 48, 8, 18	txhmeo 21606	. . . . . . . . . . 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 2701	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾)Homeo((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩}))))
51		hmeocn 21563	. . . . . . . . . 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 21053	. . . . . . . . 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 1326	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩):(𝑋 × 𝑌)⟶(∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1𝑜, 𝐾⟩})))
55		eqid 2622	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)
56	55	fmpt2 7237	. . . . . . . 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 2932	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1𝑜, 𝐾⟩})))
59	58	r19.21bi 2932	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1𝑜, 𝐾⟩})))
60	59	anasss 679	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1𝑜, 𝐾⟩})))
61		eqidd 2623	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩))
62		vex 3203	. . . . . . . . 9 ⊢ 𝑥 ∈ V
63		vex 3203	. . . . . . . . 9 ⊢ 𝑦 ∈ V
64	62, 63	op1std 7178	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
65	62, 63	op2ndd 7179	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
66	64, 65	uneq12d 3768	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((1st ‘𝑧) ∪ (2nd ‘𝑧)) = (𝑥 ∪ 𝑦))
67	66	mpt2mpt 6752	. . . . . 6 ⊢ (𝑧 ∈ (∪ (∏t‘{⟨∅, 𝐽⟩}) × ∪ (∏t‘{⟨1𝑜, 𝐾⟩})) ↦ ((1st ‘𝑧) ∪ (2nd ‘𝑧))) = (𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦))
68	67	eqcomi 2631	. . . . 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 7178	. . . . . 6 ⊢ (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ → (1st ‘𝑧) = {⟨∅, 𝑥⟩})
71	30, 35	op2ndd 7179	. . . . . 6 ⊢ (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ → (2nd ‘𝑧) = {⟨1𝑜, 𝑦⟩})
72	70, 71	uneq12d 3768	. . . . 5 ⊢ (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ → ((1st ‘𝑧) ∪ (2nd ‘𝑧)) = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩}))
73		xpscg 16218	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ◡({𝑥} +𝑐 {𝑦}) = {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩})
74	62, 63, 73	mp2an 708	. . . . . 6 ⊢ ◡({𝑥} +𝑐 {𝑦}) = {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}
75		df-pr 4180	. . . . . 6 ⊢ {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩})
76	74, 75	eqtri 2644	. . . . 5 ⊢ ◡({𝑥} +𝑐 {𝑦}) = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩})
77	72, 76	syl6eqr 2674	. . . 4 ⊢ (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ → ((1st ‘𝑧) ∪ (2nd ‘𝑧)) = ◡({𝑥} +𝑐 {𝑦}))
78	60, 61, 69, 77	fmpt2co 7260	. . 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 2675	. 2 ⊢ (𝜑 → 𝐹 = ((𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)))
81		eqid 2622	. . . . 5 ⊢ ∪ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {∅})) = ∪ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {∅}))
82		eqid 2622	. . . . 5 ⊢ ∪ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) = ∪ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}))
83		eqid 2622	. . . . 5 ⊢ (∏t‘◡({𝐽} +𝑐 {𝐾})) = (∏t‘◡({𝐽} +𝑐 {𝐾}))
84		eqid 2622	. . . . 5 ⊢ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {∅})) = (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {∅}))
85		eqid 2622	. . . . 5 ⊢ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) = (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}))
86		eqid 2622	. . . . 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 7568	. . . . . 6 ⊢ 2𝑜 ∈ On
88	87	a1i 11	. . . . 5 ⊢ (𝜑 → 2𝑜 ∈ On)
89		topontop 20718	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
90	1, 89	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Top)
91		topontop 20718	. . . . . . 7 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
92	2, 91	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Top)
93		xpscf 16226	. . . . . 6 ⊢ (◡({𝐽} +𝑐 {𝐾}):2𝑜⟶Top ↔ (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
94	90, 92, 93	sylanbrc 698	. . . . 5 ⊢ (𝜑 → ◡({𝐽} +𝑐 {𝐾}):2𝑜⟶Top)
95		df2o3 7573	. . . . . . 7 ⊢ 2𝑜 = {∅, 1𝑜}
96		df-pr 4180	. . . . . . 7 ⊢ {∅, 1𝑜} = ({∅} ∪ {1𝑜})
97	95, 96	eqtri 2644	. . . . . 6 ⊢ 2𝑜 = ({∅} ∪ {1𝑜})
98	97	a1i 11	. . . . 5 ⊢ (𝜑 → 2𝑜 = ({∅} ∪ {1𝑜}))
99		1n0 7575	. . . . . . 7 ⊢ 1𝑜 ≠ ∅
100	99	necomi 2848	. . . . . 6 ⊢ ∅ ≠ 1𝑜
101		disjsn2 4247	. . . . . 6 ⊢ (∅ ≠ 1𝑜 → ({∅} ∩ {1𝑜}) = ∅)
102	100, 101	mp1i 13	. . . . 5 ⊢ (𝜑 → ({∅} ∩ {1𝑜}) = ∅)
103	81, 82, 83, 84, 85, 86, 88, 94, 98, 102	ptunhmeo 21611	. . . 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 16219	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ◡({𝐽} +𝑐 {𝐾}) Fn 2𝑜)
105	1, 2, 104	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → ◡({𝐽} +𝑐 {𝐾}) Fn 2𝑜)
106	6	prid1 4297	. . . . . . . . . 10 ⊢ ∅ ∈ {∅, 1𝑜}
107	106, 95	eleqtrri 2700	. . . . . . . . 9 ⊢ ∅ ∈ 2𝑜
108		fnressn 6425	. . . . . . . . 9 ⊢ ((◡({𝐽} +𝑐 {𝐾}) Fn 2𝑜 ∧ ∅ ∈ 2𝑜) → (◡({𝐽} +𝑐 {𝐾}) ↾ {∅}) = {⟨∅, (◡({𝐽} +𝑐 {𝐾})‘∅)⟩})
109	105, 107, 108	sylancl 694	. . . . . . . 8 ⊢ (𝜑 → (◡({𝐽} +𝑐 {𝐾}) ↾ {∅}) = {⟨∅, (◡({𝐽} +𝑐 {𝐾})‘∅)⟩})
110		xpsc0 16220	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (◡({𝐽} +𝑐 {𝐾})‘∅) = 𝐽)
111	1, 110	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (◡({𝐽} +𝑐 {𝐾})‘∅) = 𝐽)
112	111	opeq2d 4409	. . . . . . . . 9 ⊢ (𝜑 → ⟨∅, (◡({𝐽} +𝑐 {𝐾})‘∅)⟩ = ⟨∅, 𝐽⟩)
113	112	sneqd 4189	. . . . . . . 8 ⊢ (𝜑 → {⟨∅, (◡({𝐽} +𝑐 {𝐾})‘∅)⟩} = {⟨∅, 𝐽⟩})
114	109, 113	eqtrd 2656	. . . . . . 7 ⊢ (𝜑 → (◡({𝐽} +𝑐 {𝐾}) ↾ {∅}) = {⟨∅, 𝐽⟩})
115	114	fveq2d 6195	. . . . . 6 ⊢ (𝜑 → (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {∅})) = (∏t‘{⟨∅, 𝐽⟩}))
116	115	unieqd 4446	. . . . 5 ⊢ (𝜑 → ∪ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {∅})) = ∪ (∏t‘{⟨∅, 𝐽⟩}))
117	16	elexi 3213	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ V
118	117	prid2 4298	. . . . . . . . . 10 ⊢ 1𝑜 ∈ {∅, 1𝑜}
119	118, 95	eleqtrri 2700	. . . . . . . . 9 ⊢ 1𝑜 ∈ 2𝑜
120		fnressn 6425	. . . . . . . . 9 ⊢ ((◡({𝐽} +𝑐 {𝐾}) Fn 2𝑜 ∧ 1𝑜 ∈ 2𝑜) → (◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}) = {⟨1𝑜, (◡({𝐽} +𝑐 {𝐾})‘1𝑜)⟩})
121	105, 119, 120	sylancl 694	. . . . . . . 8 ⊢ (𝜑 → (◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}) = {⟨1𝑜, (◡({𝐽} +𝑐 {𝐾})‘1𝑜)⟩})
122		xpsc1 16221	. . . . . . . . . . 11 ⊢ (𝐾 ∈ (TopOn‘𝑌) → (◡({𝐽} +𝑐 {𝐾})‘1𝑜) = 𝐾)
123	2, 122	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (◡({𝐽} +𝑐 {𝐾})‘1𝑜) = 𝐾)
124	123	opeq2d 4409	. . . . . . . . 9 ⊢ (𝜑 → ⟨1𝑜, (◡({𝐽} +𝑐 {𝐾})‘1𝑜)⟩ = ⟨1𝑜, 𝐾⟩)
125	124	sneqd 4189	. . . . . . . 8 ⊢ (𝜑 → {⟨1𝑜, (◡({𝐽} +𝑐 {𝐾})‘1𝑜)⟩} = {⟨1𝑜, 𝐾⟩})
126	121, 125	eqtrd 2656	. . . . . . 7 ⊢ (𝜑 → (◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}) = {⟨1𝑜, 𝐾⟩})
127	126	fveq2d 6195	. . . . . 6 ⊢ (𝜑 → (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) = (∏t‘{⟨1𝑜, 𝐾⟩}))
128	127	unieqd 4446	. . . . 5 ⊢ (𝜑 → ∪ (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) = ∪ (∏t‘{⟨1𝑜, 𝐾⟩}))
129		eqidd 2623	. . . . 5 ⊢ (𝜑 → (𝑥 ∪ 𝑦) = (𝑥 ∪ 𝑦))
130	116, 128, 129	mpt2eq123dv 6717	. . . 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 6668	. . . . 5 ⊢ (𝜑 → ((∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {∅})) ×t (∏t‘(◡({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}))) = ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩})))
132	131	oveq1d 6665	. . . 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 2715	. . 3 ⊢ (𝜑 → (𝑥 ∈ ∪ (∏t‘{⟨∅, 𝐽⟩}), 𝑦 ∈ ∪ (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥 ∪ 𝑦)) ∈ (((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩}))Homeo(∏t‘◡({𝐽} +𝑐 {𝐾}))))
134		hmeoco 21575	. . 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 693	. 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 2701	1 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ×t 𝐾)Homeo(∏t‘◡({𝐽} +𝑐 {𝐾}))))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ∪ cun 3572   ∩ cin 3573  ∅c0 3915  {csn 4177  {cpr 4179  ⟨cop 4183  ∪ cuni 4436   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113   ↾ cres 5116   ∘ ccom 5118  Oncon0 5723   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1^st c1st 7166  2^nd c2nd 7167  1_𝑜c1o 7553  2_𝑜c2o 7554   +_𝑐 ccda 8989  ∏_tcpt 16099  Topctop 20698  TopOnctopon 20715   Cn ccn 21028   ×_t ctx 21363  Homeochmeo 21556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-cda 8990  df-topgen 16104  df-pt 16105  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558

This theorem is referenced by:  xpstopnlem2  21614