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Theorem xpstopnlem1 21525
Description: The function 𝐹 used in xpsval 16156 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 = (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))
xpstopnlem1.j (𝜑𝐽 ∈ (TopOn‘𝑋))
xpstopnlem1.k (𝜑𝐾 ∈ (TopOn‘𝑌))
Assertion
Ref Expression
xpstopnlem1 (𝜑𝐹 ∈ ((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.
StepHypRef Expression
1 xpstopnlem1.j . . . . . . . . . 10 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 xpstopnlem1.k . . . . . . . . . 10 (𝜑𝐾 ∈ (TopOn‘𝑌))
3 txtopon 21307 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
41, 2, 3syl2anc 692 . . . . . . . . 9 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
5 eqid 2621 . . . . . . . . . . . . 13 (∏t‘{⟨∅, 𝐽⟩}) = (∏t‘{⟨∅, 𝐽⟩})
6 0ex 4752 . . . . . . . . . . . . . 14 ∅ ∈ V
76a1i 11 . . . . . . . . . . . . 13 (𝜑 → ∅ ∈ V)
85, 7, 1pt1hmeo 21522 . . . . . . . . . . . 12 (𝜑 → (𝑧𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽Homeo(∏t‘{⟨∅, 𝐽⟩})))
9 hmeocn 21476 . . . . . . . . . . . 12 ((𝑧𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽Homeo(∏t‘{⟨∅, 𝐽⟩})) → (𝑧𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽 Cn (∏t‘{⟨∅, 𝐽⟩})))
10 cntop2 20958 . . . . . . . . . . . 12 ((𝑧𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽 Cn (∏t‘{⟨∅, 𝐽⟩})) → (∏t‘{⟨∅, 𝐽⟩}) ∈ Top)
118, 9, 103syl 18 . . . . . . . . . . 11 (𝜑 → (∏t‘{⟨∅, 𝐽⟩}) ∈ Top)
12 eqid 2621 . . . . . . . . . . . 12 (∏t‘{⟨∅, 𝐽⟩}) = (∏t‘{⟨∅, 𝐽⟩})
1312toptopon 20647 . . . . . . . . . . 11 ((∏t‘{⟨∅, 𝐽⟩}) ∈ Top ↔ (∏t‘{⟨∅, 𝐽⟩}) ∈ (TopOn‘ (∏t‘{⟨∅, 𝐽⟩})))
1411, 13sylib 208 . . . . . . . . . 10 (𝜑 → (∏t‘{⟨∅, 𝐽⟩}) ∈ (TopOn‘ (∏t‘{⟨∅, 𝐽⟩})))
15 eqid 2621 . . . . . . . . . . . . 13 (∏t‘{⟨1𝑜, 𝐾⟩}) = (∏t‘{⟨1𝑜, 𝐾⟩})
16 1on 7515 . . . . . . . . . . . . . 14 1𝑜 ∈ On
1716a1i 11 . . . . . . . . . . . . 13 (𝜑 → 1𝑜 ∈ On)
1815, 17, 2pt1hmeo 21522 . . . . . . . . . . . 12 (𝜑 → (𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩}) ∈ (𝐾Homeo(∏t‘{⟨1𝑜, 𝐾⟩})))
19 hmeocn 21476 . . . . . . . . . . . 12 ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩}) ∈ (𝐾Homeo(∏t‘{⟨1𝑜, 𝐾⟩})) → (𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩}) ∈ (𝐾 Cn (∏t‘{⟨1𝑜, 𝐾⟩})))
20 cntop2 20958 . . . . . . . . . . . 12 ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩}) ∈ (𝐾 Cn (∏t‘{⟨1𝑜, 𝐾⟩})) → (∏t‘{⟨1𝑜, 𝐾⟩}) ∈ Top)
2118, 19, 203syl 18 . . . . . . . . . . 11 (𝜑 → (∏t‘{⟨1𝑜, 𝐾⟩}) ∈ Top)
22 eqid 2621 . . . . . . . . . . . 12 (∏t‘{⟨1𝑜, 𝐾⟩}) = (∏t‘{⟨1𝑜, 𝐾⟩})
2322toptopon 20647 . . . . . . . . . . 11 ((∏t‘{⟨1𝑜, 𝐾⟩}) ∈ Top ↔ (∏t‘{⟨1𝑜, 𝐾⟩}) ∈ (TopOn‘ (∏t‘{⟨1𝑜, 𝐾⟩})))
2421, 23sylib 208 . . . . . . . . . 10 (𝜑 → (∏t‘{⟨1𝑜, 𝐾⟩}) ∈ (TopOn‘ (∏t‘{⟨1𝑜, 𝐾⟩})))
25 txtopon 21307 . . . . . . . . . 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⟩}))))
2614, 24, 25syl2anc 692 . . . . . . . . 9 (𝜑 → ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩})) ∈ (TopOn‘( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1𝑜, 𝐾⟩}))))
27 opeq2 4373 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑥 → ⟨∅, 𝑧⟩ = ⟨∅, 𝑥⟩)
2827sneqd 4162 . . . . . . . . . . . . . . 15 (𝑧 = 𝑥 → {⟨∅, 𝑧⟩} = {⟨∅, 𝑥⟩})
29 eqid 2621 . . . . . . . . . . . . . . 15 (𝑧𝑋 ↦ {⟨∅, 𝑧⟩}) = (𝑧𝑋 ↦ {⟨∅, 𝑧⟩})
30 snex 4871 . . . . . . . . . . . . . . 15 {⟨∅, 𝑥⟩} ∈ V
3128, 29, 30fvmpt 6241 . . . . . . . . . . . . . 14 (𝑥𝑋 → ((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥) = {⟨∅, 𝑥⟩})
32 opeq2 4373 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑦 → ⟨1𝑜, 𝑧⟩ = ⟨1𝑜, 𝑦⟩)
3332sneqd 4162 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → {⟨1𝑜, 𝑧⟩} = {⟨1𝑜, 𝑦⟩})
34 eqid 2621 . . . . . . . . . . . . . . 15 (𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩}) = (𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩})
35 snex 4871 . . . . . . . . . . . . . . 15 {⟨1𝑜, 𝑦⟩} ∈ V
3633, 34, 35fvmpt 6241 . . . . . . . . . . . . . 14 (𝑦𝑌 → ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦) = {⟨1𝑜, 𝑦⟩})
37 opeq12 4374 . . . . . . . . . . . . . 14 ((((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥) = {⟨∅, 𝑥⟩} ∧ ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦) = {⟨1𝑜, 𝑦⟩}) → ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩ = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)
3831, 36, 37syl2an 494 . . . . . . . . . . . . 13 ((𝑥𝑋𝑦𝑌) → ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩ = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)
3938mpt2eq3ia 6676 . . . . . . . . . . . 12 (𝑥𝑋, 𝑦𝑌 ↦ ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)
40 toponuni 20641 . . . . . . . . . . . . . 14 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
411, 40syl 17 . . . . . . . . . . . . 13 (𝜑𝑋 = 𝐽)
42 toponuni 20641 . . . . . . . . . . . . . 14 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
432, 42syl 17 . . . . . . . . . . . . 13 (𝜑𝑌 = 𝐾)
44 mpt2eq12 6671 . . . . . . . . . . . . 13 ((𝑋 = 𝐽𝑌 = 𝐾) → (𝑥𝑋, 𝑦𝑌 ↦ ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩) = (𝑥 𝐽, 𝑦 𝐾 ↦ ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩))
4541, 43, 44syl2anc 692 . . . . . . . . . . . 12 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩) = (𝑥 𝐽, 𝑦 𝐾 ↦ ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩))
4639, 45syl5eqr 2669 . . . . . . . . . . 11 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) = (𝑥 𝐽, 𝑦 𝐾 ↦ ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩))
47 eqid 2621 . . . . . . . . . . . 12 𝐽 = 𝐽
48 eqid 2621 . . . . . . . . . . . 12 𝐾 = 𝐾
4947, 48, 8, 18txhmeo 21519 . . . . . . . . . . 11 (𝜑 → (𝑥 𝐽, 𝑦 𝐾 ↦ ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩) ∈ ((𝐽 ×t 𝐾)Homeo((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩}))))
5046, 49eqeltrd 2698 . . . . . . . . . 10 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾)Homeo((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩}))))
51 hmeocn 21476 . . . . . . . . . 10 ((𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾)Homeo((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩}))) → (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾) Cn ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩}))))
5250, 51syl 17 . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾) Cn ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩}))))
53 cnf2 20966 . . . . . . . . 9 (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩})) ∈ (TopOn‘( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1𝑜, 𝐾⟩}))) ∧ (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾) Cn ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩})))) → (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩):(𝑋 × 𝑌)⟶( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1𝑜, 𝐾⟩})))
544, 26, 52, 53syl3anc 1323 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩):(𝑋 × 𝑌)⟶( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1𝑜, 𝐾⟩})))
55 eqid 2621 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)
5655fmpt2 7185 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ ∈ ( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1𝑜, 𝐾⟩})) ↔ (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩):(𝑋 × 𝑌)⟶( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1𝑜, 𝐾⟩})))
5754, 56sylibr 224 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ ∈ ( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1𝑜, 𝐾⟩})))
5857r19.21bi 2927 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑌 ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ ∈ ( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1𝑜, 𝐾⟩})))
5958r19.21bi 2927 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ ∈ ( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1𝑜, 𝐾⟩})))
6059anasss 678 . . . 4 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ ∈ ( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1𝑜, 𝐾⟩})))
61 eqidd 2622 . . . 4 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩))
62 vex 3189 . . . . . . . . 9 𝑥 ∈ V
63 vex 3189 . . . . . . . . 9 𝑦 ∈ V
6462, 63op1std 7126 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
6562, 63op2ndd 7127 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
6664, 65uneq12d 3748 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → ((1st𝑧) ∪ (2nd𝑧)) = (𝑥𝑦))
6766mpt2mpt 6708 . . . . . 6 (𝑧 ∈ ( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1𝑜, 𝐾⟩})) ↦ ((1st𝑧) ∪ (2nd𝑧))) = (𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥𝑦))
6867eqcomi 2630 . . . . 5 (𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥𝑦)) = (𝑧 ∈ ( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1𝑜, 𝐾⟩})) ↦ ((1st𝑧) ∪ (2nd𝑧)))
6968a1i 11 . . . 4 (𝜑 → (𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥𝑦)) = (𝑧 ∈ ( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1𝑜, 𝐾⟩})) ↦ ((1st𝑧) ∪ (2nd𝑧))))
7030, 35op1std 7126 . . . . . 6 (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ → (1st𝑧) = {⟨∅, 𝑥⟩})
7130, 35op2ndd 7127 . . . . . 6 (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ → (2nd𝑧) = {⟨1𝑜, 𝑦⟩})
7270, 71uneq12d 3748 . . . . 5 (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ → ((1st𝑧) ∪ (2nd𝑧)) = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩}))
73 xpscg 16142 . . . . . . 7 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ({𝑥} +𝑐 {𝑦}) = {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩})
7462, 63, 73mp2an 707 . . . . . 6 ({𝑥} +𝑐 {𝑦}) = {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}
75 df-pr 4153 . . . . . 6 {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩})
7674, 75eqtri 2643 . . . . 5 ({𝑥} +𝑐 {𝑦}) = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩})
7772, 76syl6eqr 2673 . . . 4 (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ → ((1st𝑧) ∪ (2nd𝑧)) = ({𝑥} +𝑐 {𝑦}))
7860, 61, 69, 77fmpt2co 7208 . . 3 (𝜑 → ((𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥𝑦)) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)) = (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})))
79 xpstopnlem1.f . . 3 𝐹 = (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))
8078, 79syl6reqr 2674 . 2 (𝜑𝐹 = ((𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥𝑦)) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)))
81 eqid 2621 . . . . 5 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})) = (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅}))
82 eqid 2621 . . . . 5 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) = (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}))
83 eqid 2621 . . . . 5 (∏t({𝐽} +𝑐 {𝐾})) = (∏t({𝐽} +𝑐 {𝐾}))
84 eqid 2621 . . . . 5 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})) = (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅}))
85 eqid 2621 . . . . 5 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) = (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}))
86 eqid 2621 . . . . 5 (𝑥 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})), 𝑦 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) ↦ (𝑥𝑦)) = (𝑥 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})), 𝑦 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) ↦ (𝑥𝑦))
87 2on 7516 . . . . . 6 2𝑜 ∈ On
8887a1i 11 . . . . 5 (𝜑 → 2𝑜 ∈ On)
89 topontop 20640 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
901, 89syl 17 . . . . . 6 (𝜑𝐽 ∈ Top)
91 topontop 20640 . . . . . . 7 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
922, 91syl 17 . . . . . 6 (𝜑𝐾 ∈ Top)
93 xpscf 16150 . . . . . 6 (({𝐽} +𝑐 {𝐾}):2𝑜⟶Top ↔ (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
9490, 92, 93sylanbrc 697 . . . . 5 (𝜑({𝐽} +𝑐 {𝐾}):2𝑜⟶Top)
95 df2o3 7521 . . . . . . 7 2𝑜 = {∅, 1𝑜}
96 df-pr 4153 . . . . . . 7 {∅, 1𝑜} = ({∅} ∪ {1𝑜})
9795, 96eqtri 2643 . . . . . 6 2𝑜 = ({∅} ∪ {1𝑜})
9897a1i 11 . . . . 5 (𝜑 → 2𝑜 = ({∅} ∪ {1𝑜}))
99 1n0 7523 . . . . . . 7 1𝑜 ≠ ∅
10099necomi 2844 . . . . . 6 ∅ ≠ 1𝑜
101 disjsn2 4219 . . . . . 6 (∅ ≠ 1𝑜 → ({∅} ∩ {1𝑜}) = ∅)
102100, 101mp1i 13 . . . . 5 (𝜑 → ({∅} ∩ {1𝑜}) = ∅)
10381, 82, 83, 84, 85, 86, 88, 94, 98, 102ptunhmeo 21524 . . . 4 (𝜑 → (𝑥 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})), 𝑦 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) ↦ (𝑥𝑦)) ∈ (((∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})) ×t (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})))Homeo(∏t({𝐽} +𝑐 {𝐾}))))
104 xpscfn 16143 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ({𝐽} +𝑐 {𝐾}) Fn 2𝑜)
1051, 2, 104syl2anc 692 . . . . . . . . 9 (𝜑({𝐽} +𝑐 {𝐾}) Fn 2𝑜)
1066prid1 4269 . . . . . . . . . 10 ∅ ∈ {∅, 1𝑜}
107106, 95eleqtrri 2697 . . . . . . . . 9 ∅ ∈ 2𝑜
108 fnressn 6382 . . . . . . . . 9 ((({𝐽} +𝑐 {𝐾}) Fn 2𝑜 ∧ ∅ ∈ 2𝑜) → (({𝐽} +𝑐 {𝐾}) ↾ {∅}) = {⟨∅, (({𝐽} +𝑐 {𝐾})‘∅)⟩})
109105, 107, 108sylancl 693 . . . . . . . 8 (𝜑 → (({𝐽} +𝑐 {𝐾}) ↾ {∅}) = {⟨∅, (({𝐽} +𝑐 {𝐾})‘∅)⟩})
110 xpsc0 16144 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → (({𝐽} +𝑐 {𝐾})‘∅) = 𝐽)
1111, 110syl 17 . . . . . . . . . 10 (𝜑 → (({𝐽} +𝑐 {𝐾})‘∅) = 𝐽)
112111opeq2d 4379 . . . . . . . . 9 (𝜑 → ⟨∅, (({𝐽} +𝑐 {𝐾})‘∅)⟩ = ⟨∅, 𝐽⟩)
113112sneqd 4162 . . . . . . . 8 (𝜑 → {⟨∅, (({𝐽} +𝑐 {𝐾})‘∅)⟩} = {⟨∅, 𝐽⟩})
114109, 113eqtrd 2655 . . . . . . 7 (𝜑 → (({𝐽} +𝑐 {𝐾}) ↾ {∅}) = {⟨∅, 𝐽⟩})
115114fveq2d 6154 . . . . . 6 (𝜑 → (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})) = (∏t‘{⟨∅, 𝐽⟩}))
116115unieqd 4414 . . . . 5 (𝜑 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})) = (∏t‘{⟨∅, 𝐽⟩}))
11716elexi 3199 . . . . . . . . . . 11 1𝑜 ∈ V
118117prid2 4270 . . . . . . . . . 10 1𝑜 ∈ {∅, 1𝑜}
119118, 95eleqtrri 2697 . . . . . . . . 9 1𝑜 ∈ 2𝑜
120 fnressn 6382 . . . . . . . . 9 ((({𝐽} +𝑐 {𝐾}) Fn 2𝑜 ∧ 1𝑜 ∈ 2𝑜) → (({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}) = {⟨1𝑜, (({𝐽} +𝑐 {𝐾})‘1𝑜)⟩})
121105, 119, 120sylancl 693 . . . . . . . 8 (𝜑 → (({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}) = {⟨1𝑜, (({𝐽} +𝑐 {𝐾})‘1𝑜)⟩})
122 xpsc1 16145 . . . . . . . . . . 11 (𝐾 ∈ (TopOn‘𝑌) → (({𝐽} +𝑐 {𝐾})‘1𝑜) = 𝐾)
1232, 122syl 17 . . . . . . . . . 10 (𝜑 → (({𝐽} +𝑐 {𝐾})‘1𝑜) = 𝐾)
124123opeq2d 4379 . . . . . . . . 9 (𝜑 → ⟨1𝑜, (({𝐽} +𝑐 {𝐾})‘1𝑜)⟩ = ⟨1𝑜, 𝐾⟩)
125124sneqd 4162 . . . . . . . 8 (𝜑 → {⟨1𝑜, (({𝐽} +𝑐 {𝐾})‘1𝑜)⟩} = {⟨1𝑜, 𝐾⟩})
126121, 125eqtrd 2655 . . . . . . 7 (𝜑 → (({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}) = {⟨1𝑜, 𝐾⟩})
127126fveq2d 6154 . . . . . 6 (𝜑 → (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) = (∏t‘{⟨1𝑜, 𝐾⟩}))
128127unieqd 4414 . . . . 5 (𝜑 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) = (∏t‘{⟨1𝑜, 𝐾⟩}))
129 eqidd 2622 . . . . 5 (𝜑 → (𝑥𝑦) = (𝑥𝑦))
130116, 128, 129mpt2eq123dv 6673 . . . 4 (𝜑 → (𝑥 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})), 𝑦 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) ↦ (𝑥𝑦)) = (𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥𝑦)))
131115, 127oveq12d 6625 . . . . 5 (𝜑 → ((∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})) ×t (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}))) = ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩})))
132131oveq1d 6622 . . . 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}))))
133103, 130, 1323eltr3d 2712 . . 3 (𝜑 → (𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥𝑦)) ∈ (((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩}))Homeo(∏t({𝐽} +𝑐 {𝐾}))))
134 hmeoco 21488 . . 3 (((𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾)Homeo((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩}))) ∧ (𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥𝑦)) ∈ (((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩}))Homeo(∏t({𝐽} +𝑐 {𝐾})))) → ((𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥𝑦)) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)) ∈ ((𝐽 ×t 𝐾)Homeo(∏t({𝐽} +𝑐 {𝐾}))))
13550, 133, 134syl2anc 692 . 2 (𝜑 → ((𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥𝑦)) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)) ∈ ((𝐽 ×t 𝐾)Homeo(∏t({𝐽} +𝑐 {𝐾}))))
13680, 135eqeltrd 2698 1 (𝜑𝐹 ∈ ((𝐽 ×t 𝐾)Homeo(∏t({𝐽} +𝑐 {𝐾}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  Vcvv 3186  cun 3554  cin 3555  c0 3893  {csn 4150  {cpr 4152  cop 4156   cuni 4404  cmpt 4675   × cxp 5074  ccnv 5075  cres 5078  ccom 5080  Oncon0 5684   Fn wfn 5844  wf 5845  cfv 5849  (class class class)co 6607  cmpt2 6609  1st c1st 7114  2nd c2nd 7115  1𝑜c1o 7501  2𝑜c2o 7502   +𝑐 ccda 8936  tcpt 16023  Topctop 20620  TopOnctopon 20637   Cn ccn 20941   ×t ctx 21276  Homeochmeo 21469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-iin 4490  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-2o 7509  df-oadd 7512  df-er 7690  df-map 7807  df-ixp 7856  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-fi 8264  df-cda 8937  df-topgen 16028  df-pt 16029  df-top 20621  df-topon 20638  df-bases 20664  df-cn 20944  df-cnp 20945  df-tx 21278  df-hmeo 21471
This theorem is referenced by:  xpstopnlem2  21527
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