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Theorem xpstopnlem1 22136
Description: The function 𝐹 used in xpsvalcda 16711 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 21918 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
41, 2, 3syl2anc 576 . . . . . . . . 9 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
5 eqid 2771 . . . . . . . . . . . . 13 (∏t‘{⟨∅, 𝐽⟩}) = (∏t‘{⟨∅, 𝐽⟩})
6 0ex 5064 . . . . . . . . . . . . . 14 ∅ ∈ V
76a1i 11 . . . . . . . . . . . . 13 (𝜑 → ∅ ∈ V)
85, 7, 1pt1hmeo 22133 . . . . . . . . . . . 12 (𝜑 → (𝑧𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽Homeo(∏t‘{⟨∅, 𝐽⟩})))
9 hmeocn 22087 . . . . . . . . . . . 12 ((𝑧𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽Homeo(∏t‘{⟨∅, 𝐽⟩})) → (𝑧𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽 Cn (∏t‘{⟨∅, 𝐽⟩})))
10 cntop2 21568 . . . . . . . . . . . 12 ((𝑧𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽 Cn (∏t‘{⟨∅, 𝐽⟩})) → (∏t‘{⟨∅, 𝐽⟩}) ∈ Top)
118, 9, 103syl 18 . . . . . . . . . . 11 (𝜑 → (∏t‘{⟨∅, 𝐽⟩}) ∈ Top)
12 toptopon2 21245 . . . . . . . . . . 11 ((∏t‘{⟨∅, 𝐽⟩}) ∈ Top ↔ (∏t‘{⟨∅, 𝐽⟩}) ∈ (TopOn‘ (∏t‘{⟨∅, 𝐽⟩})))
1311, 12sylib 210 . . . . . . . . . 10 (𝜑 → (∏t‘{⟨∅, 𝐽⟩}) ∈ (TopOn‘ (∏t‘{⟨∅, 𝐽⟩})))
14 eqid 2771 . . . . . . . . . . . . 13 (∏t‘{⟨1o, 𝐾⟩}) = (∏t‘{⟨1o, 𝐾⟩})
15 1on 7910 . . . . . . . . . . . . . 14 1o ∈ On
1615a1i 11 . . . . . . . . . . . . 13 (𝜑 → 1o ∈ On)
1714, 16, 2pt1hmeo 22133 . . . . . . . . . . . 12 (𝜑 → (𝑧𝑌 ↦ {⟨1o, 𝑧⟩}) ∈ (𝐾Homeo(∏t‘{⟨1o, 𝐾⟩})))
18 hmeocn 22087 . . . . . . . . . . . 12 ((𝑧𝑌 ↦ {⟨1o, 𝑧⟩}) ∈ (𝐾Homeo(∏t‘{⟨1o, 𝐾⟩})) → (𝑧𝑌 ↦ {⟨1o, 𝑧⟩}) ∈ (𝐾 Cn (∏t‘{⟨1o, 𝐾⟩})))
19 cntop2 21568 . . . . . . . . . . . 12 ((𝑧𝑌 ↦ {⟨1o, 𝑧⟩}) ∈ (𝐾 Cn (∏t‘{⟨1o, 𝐾⟩})) → (∏t‘{⟨1o, 𝐾⟩}) ∈ Top)
2017, 18, 193syl 18 . . . . . . . . . . 11 (𝜑 → (∏t‘{⟨1o, 𝐾⟩}) ∈ Top)
21 toptopon2 21245 . . . . . . . . . . 11 ((∏t‘{⟨1o, 𝐾⟩}) ∈ Top ↔ (∏t‘{⟨1o, 𝐾⟩}) ∈ (TopOn‘ (∏t‘{⟨1o, 𝐾⟩})))
2220, 21sylib 210 . . . . . . . . . 10 (𝜑 → (∏t‘{⟨1o, 𝐾⟩}) ∈ (TopOn‘ (∏t‘{⟨1o, 𝐾⟩})))
23 txtopon 21918 . . . . . . . . . 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⟩}))))
2413, 22, 23syl2anc 576 . . . . . . . . 9 (𝜑 → ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩})) ∈ (TopOn‘( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1o, 𝐾⟩}))))
25 opeq2 4674 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑥 → ⟨∅, 𝑧⟩ = ⟨∅, 𝑥⟩)
2625sneqd 4447 . . . . . . . . . . . . . . 15 (𝑧 = 𝑥 → {⟨∅, 𝑧⟩} = {⟨∅, 𝑥⟩})
27 eqid 2771 . . . . . . . . . . . . . . 15 (𝑧𝑋 ↦ {⟨∅, 𝑧⟩}) = (𝑧𝑋 ↦ {⟨∅, 𝑧⟩})
28 snex 5184 . . . . . . . . . . . . . . 15 {⟨∅, 𝑥⟩} ∈ V
2926, 27, 28fvmpt 6593 . . . . . . . . . . . . . 14 (𝑥𝑋 → ((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥) = {⟨∅, 𝑥⟩})
30 opeq2 4674 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑦 → ⟨1o, 𝑧⟩ = ⟨1o, 𝑦⟩)
3130sneqd 4447 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → {⟨1o, 𝑧⟩} = {⟨1o, 𝑦⟩})
32 eqid 2771 . . . . . . . . . . . . . . 15 (𝑧𝑌 ↦ {⟨1o, 𝑧⟩}) = (𝑧𝑌 ↦ {⟨1o, 𝑧⟩})
33 snex 5184 . . . . . . . . . . . . . . 15 {⟨1o, 𝑦⟩} ∈ V
3431, 32, 33fvmpt 6593 . . . . . . . . . . . . . 14 (𝑦𝑌 → ((𝑧𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦) = {⟨1o, 𝑦⟩})
35 opeq12 4675 . . . . . . . . . . . . . 14 ((((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥) = {⟨∅, 𝑥⟩} ∧ ((𝑧𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦) = {⟨1o, 𝑦⟩}) → ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦)⟩ = ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩)
3629, 34, 35syl2an 587 . . . . . . . . . . . . 13 ((𝑥𝑋𝑦𝑌) → ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦)⟩ = ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩)
3736mpoeq3ia 7048 . . . . . . . . . . . 12 (𝑥𝑋, 𝑦𝑌 ↦ ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦)⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩)
38 toponuni 21241 . . . . . . . . . . . . . 14 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
391, 38syl 17 . . . . . . . . . . . . 13 (𝜑𝑋 = 𝐽)
40 toponuni 21241 . . . . . . . . . . . . . 14 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
412, 40syl 17 . . . . . . . . . . . . 13 (𝜑𝑌 = 𝐾)
42 mpoeq12 7043 . . . . . . . . . . . . 13 ((𝑋 = 𝐽𝑌 = 𝐾) → (𝑥𝑋, 𝑦𝑌 ↦ ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦)⟩) = (𝑥 𝐽, 𝑦 𝐾 ↦ ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦)⟩))
4339, 41, 42syl2anc 576 . . . . . . . . . . . 12 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦)⟩) = (𝑥 𝐽, 𝑦 𝐾 ↦ ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦)⟩))
4437, 43syl5eqr 2821 . . . . . . . . . . 11 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩) = (𝑥 𝐽, 𝑦 𝐾 ↦ ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦)⟩))
45 eqid 2771 . . . . . . . . . . . 12 𝐽 = 𝐽
46 eqid 2771 . . . . . . . . . . . 12 𝐾 = 𝐾
4745, 46, 8, 17txhmeo 22130 . . . . . . . . . . 11 (𝜑 → (𝑥 𝐽, 𝑦 𝐾 ↦ ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1o, 𝑧⟩})‘𝑦)⟩) ∈ ((𝐽 ×t 𝐾)Homeo((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩}))))
4844, 47eqeltrd 2859 . . . . . . . . . 10 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾)Homeo((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩}))))
49 hmeocn 22087 . . . . . . . . . 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⟩}))))
5048, 49syl 17 . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾) Cn ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩}))))
51 cnf2 21576 . . . . . . . . 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⟩})))
524, 24, 50, 51syl3anc 1352 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩):(𝑋 × 𝑌)⟶( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1o, 𝐾⟩})))
53 eqid 2771 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩)
5453fmpo 7572 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ ∈ ( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1o, 𝐾⟩})) ↔ (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩):(𝑋 × 𝑌)⟶( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1o, 𝐾⟩})))
5552, 54sylibr 226 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ ∈ ( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1o, 𝐾⟩})))
5655r19.21bi 3151 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑌 ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ ∈ ( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1o, 𝐾⟩})))
5756r19.21bi 3151 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ ∈ ( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1o, 𝐾⟩})))
5857anasss 459 . . . 4 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ ∈ ( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1o, 𝐾⟩})))
59 eqidd 2772 . . . 4 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩))
60 vex 3411 . . . . . . . . 9 𝑥 ∈ V
61 vex 3411 . . . . . . . . 9 𝑦 ∈ V
6260, 61op1std 7509 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
6360, 61op2ndd 7510 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
6462, 63uneq12d 4022 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → ((1st𝑧) ∪ (2nd𝑧)) = (𝑥𝑦))
6564mpompt 7080 . . . . . 6 (𝑧 ∈ ( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1o, 𝐾⟩})) ↦ ((1st𝑧) ∪ (2nd𝑧))) = (𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1o, 𝐾⟩}) ↦ (𝑥𝑦))
6665eqcomi 2780 . . . . 5 (𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1o, 𝐾⟩}) ↦ (𝑥𝑦)) = (𝑧 ∈ ( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1o, 𝐾⟩})) ↦ ((1st𝑧) ∪ (2nd𝑧)))
6766a1i 11 . . . 4 (𝜑 → (𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1o, 𝐾⟩}) ↦ (𝑥𝑦)) = (𝑧 ∈ ( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1o, 𝐾⟩})) ↦ ((1st𝑧) ∪ (2nd𝑧))))
6828, 33op1std 7509 . . . . . 6 (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ → (1st𝑧) = {⟨∅, 𝑥⟩})
6928, 33op2ndd 7510 . . . . . 6 (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ → (2nd𝑧) = {⟨1o, 𝑦⟩})
7068, 69uneq12d 4022 . . . . 5 (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ → ((1st𝑧) ∪ (2nd𝑧)) = ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩}))
71 xpscg 16685 . . . . . . 7 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ({𝑥} +𝑐 {𝑦}) = {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
7271el2v 3415 . . . . . 6 ({𝑥} +𝑐 {𝑦}) = {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}
73 df-pr 4438 . . . . . 6 {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
7472, 73eqtri 2795 . . . . 5 ({𝑥} +𝑐 {𝑦}) = ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
7570, 74syl6eqr 2825 . . . 4 (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩ → ((1st𝑧) ∪ (2nd𝑧)) = ({𝑥} +𝑐 {𝑦}))
7658, 59, 67, 75fmpoco 7596 . . 3 (𝜑 → ((𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1o, 𝐾⟩}) ↦ (𝑥𝑦)) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩)) = (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})))
77 xpstopnlem1.f . . 3 𝐹 = (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))
7876, 77syl6reqr 2826 . 2 (𝜑𝐹 = ((𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1o, 𝐾⟩}) ↦ (𝑥𝑦)) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩)))
79 eqid 2771 . . . . 5 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})) = (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅}))
80 eqid 2771 . . . . 5 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1o})) = (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1o}))
81 eqid 2771 . . . . 5 (∏t({𝐽} +𝑐 {𝐾})) = (∏t({𝐽} +𝑐 {𝐾}))
82 eqid 2771 . . . . 5 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})) = (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅}))
83 eqid 2771 . . . . 5 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1o})) = (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1o}))
84 eqid 2771 . . . . 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))
85 2on 7912 . . . . . 6 2o ∈ On
8685a1i 11 . . . . 5 (𝜑 → 2o ∈ On)
87 topontop 21240 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
881, 87syl 17 . . . . . 6 (𝜑𝐽 ∈ Top)
89 topontop 21240 . . . . . . 7 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
902, 89syl 17 . . . . . 6 (𝜑𝐾 ∈ Top)
91 xpscfcda 16699 . . . . . 6 (({𝐽} +𝑐 {𝐾}):2o⟶Top ↔ (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
9288, 90, 91sylanbrc 575 . . . . 5 (𝜑({𝐽} +𝑐 {𝐾}):2o⟶Top)
93 df2o3 7917 . . . . . . 7 2o = {∅, 1o}
94 df-pr 4438 . . . . . . 7 {∅, 1o} = ({∅} ∪ {1o})
9593, 94eqtri 2795 . . . . . 6 2o = ({∅} ∪ {1o})
9695a1i 11 . . . . 5 (𝜑 → 2o = ({∅} ∪ {1o}))
97 1n0 7919 . . . . . . 7 1o ≠ ∅
9897necomi 3014 . . . . . 6 ∅ ≠ 1o
99 disjsn2 4518 . . . . . 6 (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
10098, 99mp1i 13 . . . . 5 (𝜑 → ({∅} ∩ {1o}) = ∅)
10179, 80, 81, 82, 83, 84, 86, 92, 96, 100ptunhmeo 22135 . . . 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}))))
102 xpscfn 16686 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ({𝐽} +𝑐 {𝐾}) Fn 2o)
1031, 2, 102syl2anc 576 . . . . . . . . 9 (𝜑({𝐽} +𝑐 {𝐾}) Fn 2o)
1046prid1 4568 . . . . . . . . . 10 ∅ ∈ {∅, 1o}
105104, 93eleqtrri 2858 . . . . . . . . 9 ∅ ∈ 2o
106 fnressn 6741 . . . . . . . . 9 ((({𝐽} +𝑐 {𝐾}) Fn 2o ∧ ∅ ∈ 2o) → (({𝐽} +𝑐 {𝐾}) ↾ {∅}) = {⟨∅, (({𝐽} +𝑐 {𝐾})‘∅)⟩})
107103, 105, 106sylancl 578 . . . . . . . 8 (𝜑 → (({𝐽} +𝑐 {𝐾}) ↾ {∅}) = {⟨∅, (({𝐽} +𝑐 {𝐾})‘∅)⟩})
108 xpsc0 16689 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → (({𝐽} +𝑐 {𝐾})‘∅) = 𝐽)
1091, 108syl 17 . . . . . . . . . 10 (𝜑 → (({𝐽} +𝑐 {𝐾})‘∅) = 𝐽)
110109opeq2d 4680 . . . . . . . . 9 (𝜑 → ⟨∅, (({𝐽} +𝑐 {𝐾})‘∅)⟩ = ⟨∅, 𝐽⟩)
111110sneqd 4447 . . . . . . . 8 (𝜑 → {⟨∅, (({𝐽} +𝑐 {𝐾})‘∅)⟩} = {⟨∅, 𝐽⟩})
112107, 111eqtrd 2807 . . . . . . 7 (𝜑 → (({𝐽} +𝑐 {𝐾}) ↾ {∅}) = {⟨∅, 𝐽⟩})
113112fveq2d 6500 . . . . . 6 (𝜑 → (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})) = (∏t‘{⟨∅, 𝐽⟩}))
114113unieqd 4718 . . . . 5 (𝜑 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})) = (∏t‘{⟨∅, 𝐽⟩}))
115 1oex 7911 . . . . . . . . . . 11 1o ∈ V
116115prid2 4569 . . . . . . . . . 10 1o ∈ {∅, 1o}
117116, 93eleqtrri 2858 . . . . . . . . 9 1o ∈ 2o
118 fnressn 6741 . . . . . . . . 9 ((({𝐽} +𝑐 {𝐾}) Fn 2o ∧ 1o ∈ 2o) → (({𝐽} +𝑐 {𝐾}) ↾ {1o}) = {⟨1o, (({𝐽} +𝑐 {𝐾})‘1o)⟩})
119103, 117, 118sylancl 578 . . . . . . . 8 (𝜑 → (({𝐽} +𝑐 {𝐾}) ↾ {1o}) = {⟨1o, (({𝐽} +𝑐 {𝐾})‘1o)⟩})
120 xpsc1 16690 . . . . . . . . . . 11 (𝐾 ∈ (TopOn‘𝑌) → (({𝐽} +𝑐 {𝐾})‘1o) = 𝐾)
1212, 120syl 17 . . . . . . . . . 10 (𝜑 → (({𝐽} +𝑐 {𝐾})‘1o) = 𝐾)
122121opeq2d 4680 . . . . . . . . 9 (𝜑 → ⟨1o, (({𝐽} +𝑐 {𝐾})‘1o)⟩ = ⟨1o, 𝐾⟩)
123122sneqd 4447 . . . . . . . 8 (𝜑 → {⟨1o, (({𝐽} +𝑐 {𝐾})‘1o)⟩} = {⟨1o, 𝐾⟩})
124119, 123eqtrd 2807 . . . . . . 7 (𝜑 → (({𝐽} +𝑐 {𝐾}) ↾ {1o}) = {⟨1o, 𝐾⟩})
125124fveq2d 6500 . . . . . 6 (𝜑 → (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1o})) = (∏t‘{⟨1o, 𝐾⟩}))
126125unieqd 4718 . . . . 5 (𝜑 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1o})) = (∏t‘{⟨1o, 𝐾⟩}))
127 eqidd 2772 . . . . 5 (𝜑 → (𝑥𝑦) = (𝑥𝑦))
128114, 126, 127mpoeq123dv 7045 . . . 4 (𝜑 → (𝑥 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})), 𝑦 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1o})) ↦ (𝑥𝑦)) = (𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1o, 𝐾⟩}) ↦ (𝑥𝑦)))
129113, 125oveq12d 6992 . . . . 5 (𝜑 → ((∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})) ×t (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1o}))) = ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩})))
130129oveq1d 6989 . . . 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}))))
131101, 128, 1303eltr3d 2873 . . 3 (𝜑 → (𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1o, 𝐾⟩}) ↦ (𝑥𝑦)) ∈ (((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1o, 𝐾⟩}))Homeo(∏t({𝐽} +𝑐 {𝐾}))))
132 hmeoco 22099 . . 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}))))
13348, 131, 132syl2anc 576 . 2 (𝜑 → ((𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1o, 𝐾⟩}) ↦ (𝑥𝑦)) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1o, 𝑦⟩}⟩)) ∈ ((𝐽 ×t 𝐾)Homeo(∏t({𝐽} +𝑐 {𝐾}))))
13478, 133eqeltrd 2859 1 (𝜑𝐹 ∈ ((𝐽 ×t 𝐾)Homeo(∏t({𝐽} +𝑐 {𝐾}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1508  wcel 2051  wne 2960  wral 3081  Vcvv 3408  cun 3820  cin 3821  c0 4172  {csn 4435  {cpr 4437  cop 4441   cuni 4708  cmpt 5004   × cxp 5401  ccnv 5402  cres 5405  ccom 5407  Oncon0 6026   Fn wfn 6180  wf 6181  cfv 6185  (class class class)co 6974  cmpo 6976  1st c1st 7497  2nd c2nd 7498  1oc1o 7896  2oc2o 7897   +𝑐 ccda 9385  tcpt 16566  Topctop 21220  TopOnctopon 21237   Cn ccn 21551   ×t ctx 21887  Homeochmeo 22080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-iin 4791  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7499  df-2nd 7500  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-1o 7903  df-2o 7904  df-oadd 7907  df-er 8087  df-map 8206  df-ixp 8258  df-en 8305  df-dom 8306  df-sdom 8307  df-fin 8308  df-fi 8668  df-cda 9386  df-topgen 16571  df-pt 16572  df-top 21221  df-topon 21238  df-bases 21273  df-cn 21554  df-cnp 21555  df-tx 21889  df-hmeo 22082
This theorem is referenced by:  xpstopnlem2  22138
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