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Theorem xpstopnlem1 21660
Description: The function 𝐹 used in xpsval 16279 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 21442 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
41, 2, 3syl2anc 694 . . . . . . . . 9 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
5 eqid 2651 . . . . . . . . . . . . 13 (∏t‘{⟨∅, 𝐽⟩}) = (∏t‘{⟨∅, 𝐽⟩})
6 0ex 4823 . . . . . . . . . . . . . 14 ∅ ∈ V
76a1i 11 . . . . . . . . . . . . 13 (𝜑 → ∅ ∈ V)
85, 7, 1pt1hmeo 21657 . . . . . . . . . . . 12 (𝜑 → (𝑧𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽Homeo(∏t‘{⟨∅, 𝐽⟩})))
9 hmeocn 21611 . . . . . . . . . . . 12 ((𝑧𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽Homeo(∏t‘{⟨∅, 𝐽⟩})) → (𝑧𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽 Cn (∏t‘{⟨∅, 𝐽⟩})))
10 cntop2 21093 . . . . . . . . . . . 12 ((𝑧𝑋 ↦ {⟨∅, 𝑧⟩}) ∈ (𝐽 Cn (∏t‘{⟨∅, 𝐽⟩})) → (∏t‘{⟨∅, 𝐽⟩}) ∈ Top)
118, 9, 103syl 18 . . . . . . . . . . 11 (𝜑 → (∏t‘{⟨∅, 𝐽⟩}) ∈ Top)
12 eqid 2651 . . . . . . . . . . . 12 (∏t‘{⟨∅, 𝐽⟩}) = (∏t‘{⟨∅, 𝐽⟩})
1312toptopon 20770 . . . . . . . . . . 11 ((∏t‘{⟨∅, 𝐽⟩}) ∈ Top ↔ (∏t‘{⟨∅, 𝐽⟩}) ∈ (TopOn‘ (∏t‘{⟨∅, 𝐽⟩})))
1411, 13sylib 208 . . . . . . . . . 10 (𝜑 → (∏t‘{⟨∅, 𝐽⟩}) ∈ (TopOn‘ (∏t‘{⟨∅, 𝐽⟩})))
15 eqid 2651 . . . . . . . . . . . . 13 (∏t‘{⟨1𝑜, 𝐾⟩}) = (∏t‘{⟨1𝑜, 𝐾⟩})
16 1on 7612 . . . . . . . . . . . . . 14 1𝑜 ∈ On
1716a1i 11 . . . . . . . . . . . . 13 (𝜑 → 1𝑜 ∈ On)
1815, 17, 2pt1hmeo 21657 . . . . . . . . . . . 12 (𝜑 → (𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩}) ∈ (𝐾Homeo(∏t‘{⟨1𝑜, 𝐾⟩})))
19 hmeocn 21611 . . . . . . . . . . . 12 ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩}) ∈ (𝐾Homeo(∏t‘{⟨1𝑜, 𝐾⟩})) → (𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩}) ∈ (𝐾 Cn (∏t‘{⟨1𝑜, 𝐾⟩})))
20 cntop2 21093 . . . . . . . . . . . 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 2651 . . . . . . . . . . . 12 (∏t‘{⟨1𝑜, 𝐾⟩}) = (∏t‘{⟨1𝑜, 𝐾⟩})
2322toptopon 20770 . . . . . . . . . . 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 21442 . . . . . . . . . 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 694 . . . . . . . . 9 (𝜑 → ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩})) ∈ (TopOn‘( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1𝑜, 𝐾⟩}))))
27 opeq2 4434 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑥 → ⟨∅, 𝑧⟩ = ⟨∅, 𝑥⟩)
2827sneqd 4222 . . . . . . . . . . . . . . 15 (𝑧 = 𝑥 → {⟨∅, 𝑧⟩} = {⟨∅, 𝑥⟩})
29 eqid 2651 . . . . . . . . . . . . . . 15 (𝑧𝑋 ↦ {⟨∅, 𝑧⟩}) = (𝑧𝑋 ↦ {⟨∅, 𝑧⟩})
30 snex 4938 . . . . . . . . . . . . . . 15 {⟨∅, 𝑥⟩} ∈ V
3128, 29, 30fvmpt 6321 . . . . . . . . . . . . . 14 (𝑥𝑋 → ((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥) = {⟨∅, 𝑥⟩})
32 opeq2 4434 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑦 → ⟨1𝑜, 𝑧⟩ = ⟨1𝑜, 𝑦⟩)
3332sneqd 4222 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → {⟨1𝑜, 𝑧⟩} = {⟨1𝑜, 𝑦⟩})
34 eqid 2651 . . . . . . . . . . . . . . 15 (𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩}) = (𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩})
35 snex 4938 . . . . . . . . . . . . . . 15 {⟨1𝑜, 𝑦⟩} ∈ V
3633, 34, 35fvmpt 6321 . . . . . . . . . . . . . 14 (𝑦𝑌 → ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦) = {⟨1𝑜, 𝑦⟩})
37 opeq12 4435 . . . . . . . . . . . . . 14 ((((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥) = {⟨∅, 𝑥⟩} ∧ ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦) = {⟨1𝑜, 𝑦⟩}) → ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩ = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)
3831, 36, 37syl2an 493 . . . . . . . . . . . . 13 ((𝑥𝑋𝑦𝑌) → ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩ = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)
3938mpt2eq3ia 6762 . . . . . . . . . . . 12 (𝑥𝑋, 𝑦𝑌 ↦ ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)
40 toponuni 20767 . . . . . . . . . . . . . 14 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
411, 40syl 17 . . . . . . . . . . . . 13 (𝜑𝑋 = 𝐽)
42 toponuni 20767 . . . . . . . . . . . . . 14 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
432, 42syl 17 . . . . . . . . . . . . 13 (𝜑𝑌 = 𝐾)
44 mpt2eq12 6757 . . . . . . . . . . . . 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 694 . . . . . . . . . . . 12 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩) = (𝑥 𝐽, 𝑦 𝐾 ↦ ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩))
4639, 45syl5eqr 2699 . . . . . . . . . . 11 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) = (𝑥 𝐽, 𝑦 𝐾 ↦ ⟨((𝑧𝑋 ↦ {⟨∅, 𝑧⟩})‘𝑥), ((𝑧𝑌 ↦ {⟨1𝑜, 𝑧⟩})‘𝑦)⟩))
47 eqid 2651 . . . . . . . . . . . 12 𝐽 = 𝐽
48 eqid 2651 . . . . . . . . . . . 12 𝐾 = 𝐾
4947, 48, 8, 18txhmeo 21654 . . . . . . . . . . 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 2730 . . . . . . . . . 10 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) ∈ ((𝐽 ×t 𝐾)Homeo((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩}))))
51 hmeocn 21611 . . . . . . . . . 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 21101 . . . . . . . . 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 1366 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩):(𝑋 × 𝑌)⟶( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1𝑜, 𝐾⟩})))
55 eqid 2651 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)
5655fmpt2 7282 . . . . . . . 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 2961 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑌 ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ ∈ ( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1𝑜, 𝐾⟩})))
5958r19.21bi 2961 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ ∈ ( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1𝑜, 𝐾⟩})))
6059anasss 680 . . . 4 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ ∈ ( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1𝑜, 𝐾⟩})))
61 eqidd 2652 . . . 4 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩))
62 vex 3234 . . . . . . . . 9 𝑥 ∈ V
63 vex 3234 . . . . . . . . 9 𝑦 ∈ V
6462, 63op1std 7220 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
6562, 63op2ndd 7221 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
6664, 65uneq12d 3801 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → ((1st𝑧) ∪ (2nd𝑧)) = (𝑥𝑦))
6766mpt2mpt 6794 . . . . . 6 (𝑧 ∈ ( (∏t‘{⟨∅, 𝐽⟩}) × (∏t‘{⟨1𝑜, 𝐾⟩})) ↦ ((1st𝑧) ∪ (2nd𝑧))) = (𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥𝑦))
6867eqcomi 2660 . . . . 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 7220 . . . . . 6 (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ → (1st𝑧) = {⟨∅, 𝑥⟩})
7130, 35op2ndd 7221 . . . . . 6 (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ → (2nd𝑧) = {⟨1𝑜, 𝑦⟩})
7270, 71uneq12d 3801 . . . . 5 (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ → ((1st𝑧) ∪ (2nd𝑧)) = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩}))
73 xpscg 16265 . . . . . . 7 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ({𝑥} +𝑐 {𝑦}) = {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩})
7462, 63, 73mp2an 708 . . . . . 6 ({𝑥} +𝑐 {𝑦}) = {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}
75 df-pr 4213 . . . . . 6 {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩})
7674, 75eqtri 2673 . . . . 5 ({𝑥} +𝑐 {𝑦}) = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩})
7772, 76syl6eqr 2703 . . . 4 (𝑧 = ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩ → ((1st𝑧) ∪ (2nd𝑧)) = ({𝑥} +𝑐 {𝑦}))
7860, 61, 69, 77fmpt2co 7305 . . 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 2704 . 2 (𝜑𝐹 = ((𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥𝑦)) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨{⟨∅, 𝑥⟩}, {⟨1𝑜, 𝑦⟩}⟩)))
81 eqid 2651 . . . . 5 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})) = (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅}))
82 eqid 2651 . . . . 5 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) = (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}))
83 eqid 2651 . . . . 5 (∏t({𝐽} +𝑐 {𝐾})) = (∏t({𝐽} +𝑐 {𝐾}))
84 eqid 2651 . . . . 5 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})) = (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅}))
85 eqid 2651 . . . . 5 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) = (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}))
86 eqid 2651 . . . . 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 7613 . . . . . 6 2𝑜 ∈ On
8887a1i 11 . . . . 5 (𝜑 → 2𝑜 ∈ On)
89 topontop 20766 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
901, 89syl 17 . . . . . 6 (𝜑𝐽 ∈ Top)
91 topontop 20766 . . . . . . 7 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
922, 91syl 17 . . . . . 6 (𝜑𝐾 ∈ Top)
93 xpscf 16273 . . . . . 6 (({𝐽} +𝑐 {𝐾}):2𝑜⟶Top ↔ (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
9490, 92, 93sylanbrc 699 . . . . 5 (𝜑({𝐽} +𝑐 {𝐾}):2𝑜⟶Top)
95 df2o3 7618 . . . . . . 7 2𝑜 = {∅, 1𝑜}
96 df-pr 4213 . . . . . . 7 {∅, 1𝑜} = ({∅} ∪ {1𝑜})
9795, 96eqtri 2673 . . . . . 6 2𝑜 = ({∅} ∪ {1𝑜})
9897a1i 11 . . . . 5 (𝜑 → 2𝑜 = ({∅} ∪ {1𝑜}))
99 1n0 7620 . . . . . . 7 1𝑜 ≠ ∅
10099necomi 2877 . . . . . 6 ∅ ≠ 1𝑜
101 disjsn2 4279 . . . . . 6 (∅ ≠ 1𝑜 → ({∅} ∩ {1𝑜}) = ∅)
102100, 101mp1i 13 . . . . 5 (𝜑 → ({∅} ∩ {1𝑜}) = ∅)
10381, 82, 83, 84, 85, 86, 88, 94, 98, 102ptunhmeo 21659 . . . 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 16266 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ({𝐽} +𝑐 {𝐾}) Fn 2𝑜)
1051, 2, 104syl2anc 694 . . . . . . . . 9 (𝜑({𝐽} +𝑐 {𝐾}) Fn 2𝑜)
1066prid1 4329 . . . . . . . . . 10 ∅ ∈ {∅, 1𝑜}
107106, 95eleqtrri 2729 . . . . . . . . 9 ∅ ∈ 2𝑜
108 fnressn 6465 . . . . . . . . 9 ((({𝐽} +𝑐 {𝐾}) Fn 2𝑜 ∧ ∅ ∈ 2𝑜) → (({𝐽} +𝑐 {𝐾}) ↾ {∅}) = {⟨∅, (({𝐽} +𝑐 {𝐾})‘∅)⟩})
109105, 107, 108sylancl 695 . . . . . . . 8 (𝜑 → (({𝐽} +𝑐 {𝐾}) ↾ {∅}) = {⟨∅, (({𝐽} +𝑐 {𝐾})‘∅)⟩})
110 xpsc0 16267 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → (({𝐽} +𝑐 {𝐾})‘∅) = 𝐽)
1111, 110syl 17 . . . . . . . . . 10 (𝜑 → (({𝐽} +𝑐 {𝐾})‘∅) = 𝐽)
112111opeq2d 4440 . . . . . . . . 9 (𝜑 → ⟨∅, (({𝐽} +𝑐 {𝐾})‘∅)⟩ = ⟨∅, 𝐽⟩)
113112sneqd 4222 . . . . . . . 8 (𝜑 → {⟨∅, (({𝐽} +𝑐 {𝐾})‘∅)⟩} = {⟨∅, 𝐽⟩})
114109, 113eqtrd 2685 . . . . . . 7 (𝜑 → (({𝐽} +𝑐 {𝐾}) ↾ {∅}) = {⟨∅, 𝐽⟩})
115114fveq2d 6233 . . . . . 6 (𝜑 → (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})) = (∏t‘{⟨∅, 𝐽⟩}))
116115unieqd 4478 . . . . 5 (𝜑 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})) = (∏t‘{⟨∅, 𝐽⟩}))
11716elexi 3244 . . . . . . . . . . 11 1𝑜 ∈ V
118117prid2 4330 . . . . . . . . . 10 1𝑜 ∈ {∅, 1𝑜}
119118, 95eleqtrri 2729 . . . . . . . . 9 1𝑜 ∈ 2𝑜
120 fnressn 6465 . . . . . . . . 9 ((({𝐽} +𝑐 {𝐾}) Fn 2𝑜 ∧ 1𝑜 ∈ 2𝑜) → (({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}) = {⟨1𝑜, (({𝐽} +𝑐 {𝐾})‘1𝑜)⟩})
121105, 119, 120sylancl 695 . . . . . . . 8 (𝜑 → (({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}) = {⟨1𝑜, (({𝐽} +𝑐 {𝐾})‘1𝑜)⟩})
122 xpsc1 16268 . . . . . . . . . . 11 (𝐾 ∈ (TopOn‘𝑌) → (({𝐽} +𝑐 {𝐾})‘1𝑜) = 𝐾)
1232, 122syl 17 . . . . . . . . . 10 (𝜑 → (({𝐽} +𝑐 {𝐾})‘1𝑜) = 𝐾)
124123opeq2d 4440 . . . . . . . . 9 (𝜑 → ⟨1𝑜, (({𝐽} +𝑐 {𝐾})‘1𝑜)⟩ = ⟨1𝑜, 𝐾⟩)
125124sneqd 4222 . . . . . . . 8 (𝜑 → {⟨1𝑜, (({𝐽} +𝑐 {𝐾})‘1𝑜)⟩} = {⟨1𝑜, 𝐾⟩})
126121, 125eqtrd 2685 . . . . . . 7 (𝜑 → (({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}) = {⟨1𝑜, 𝐾⟩})
127126fveq2d 6233 . . . . . 6 (𝜑 → (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) = (∏t‘{⟨1𝑜, 𝐾⟩}))
128127unieqd 4478 . . . . 5 (𝜑 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) = (∏t‘{⟨1𝑜, 𝐾⟩}))
129 eqidd 2652 . . . . 5 (𝜑 → (𝑥𝑦) = (𝑥𝑦))
130116, 128, 129mpt2eq123dv 6759 . . . 4 (𝜑 → (𝑥 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})), 𝑦 (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1𝑜})) ↦ (𝑥𝑦)) = (𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥𝑦)))
131115, 127oveq12d 6708 . . . . 5 (𝜑 → ((∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {∅})) ×t (∏t‘(({𝐽} +𝑐 {𝐾}) ↾ {1𝑜}))) = ((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩})))
132131oveq1d 6705 . . . 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 2744 . . 3 (𝜑 → (𝑥 (∏t‘{⟨∅, 𝐽⟩}), 𝑦 (∏t‘{⟨1𝑜, 𝐾⟩}) ↦ (𝑥𝑦)) ∈ (((∏t‘{⟨∅, 𝐽⟩}) ×t (∏t‘{⟨1𝑜, 𝐾⟩}))Homeo(∏t({𝐽} +𝑐 {𝐾}))))
134 hmeoco 21623 . . 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 694 . 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 2730 1 (𝜑𝐹 ∈ ((𝐽 ×t 𝐾)Homeo(∏t({𝐽} +𝑐 {𝐾}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  Vcvv 3231  cun 3605  cin 3606  c0 3948  {csn 4210  {cpr 4212  cop 4216   cuni 4468  cmpt 4762   × cxp 5141  ccnv 5142  cres 5145  ccom 5147  Oncon0 5761   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  1st c1st 7208  2nd c2nd 7209  1𝑜c1o 7598  2𝑜c2o 7599   +𝑐 ccda 9027  tcpt 16146  Topctop 20746  TopOnctopon 20763   Cn ccn 21076   ×t ctx 21411  Homeochmeo 21604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-cda 9028  df-topgen 16151  df-pt 16152  df-top 20747  df-topon 20764  df-bases 20798  df-cn 21079  df-cnp 21080  df-tx 21413  df-hmeo 21606
This theorem is referenced by:  xpstopnlem2  21662
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