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Theorem xkohmeo 23839
Description: The Exponential Law for topological spaces. The "currying" function 𝐹 is a homeomorphism on function spaces when 𝐽 and 𝐾 are exponentiable spaces (by xkococn 23684, it is sufficient to assume that 𝐽, 𝐾 are locally compact to ensure exponentiability). (Contributed by Mario Carneiro, 13-Apr-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
Hypotheses
Ref Expression
xkohmeo.x (𝜑𝐽 ∈ (TopOn‘𝑋))
xkohmeo.y (𝜑𝐾 ∈ (TopOn‘𝑌))
xkohmeo.f 𝐹 = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑥𝑓𝑦))))
xkohmeo.j (𝜑𝐽 ∈ 𝑛-Locally Comp)
xkohmeo.k (𝜑𝐾 ∈ 𝑛-Locally Comp)
xkohmeo.l (𝜑𝐿 ∈ Top)
Assertion
Ref Expression
xkohmeo (𝜑𝐹 ∈ ((𝐿ko (𝐽 ×t 𝐾))Homeo((𝐿ko 𝐾) ↑ko 𝐽)))
Distinct variable groups:   𝑥,𝑓,𝑦,𝐽   𝑓,𝐾,𝑥,𝑦   𝜑,𝑓,𝑥,𝑦   𝑓,𝐿,𝑥,𝑦   𝑓,𝑋,𝑥,𝑦   𝑓,𝑌,𝑥,𝑦   𝑓,𝐹,𝑥,𝑦

Proof of Theorem xkohmeo
Dummy variables 𝑔 𝑡 𝑢 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xkohmeo.f . . 3 𝐹 = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑥𝑓𝑦))))
2 xkohmeo.x . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘𝑋))
3 xkohmeo.y . . . . . . 7 (𝜑𝐾 ∈ (TopOn‘𝑌))
4 txtopon 23615 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
52, 3, 4syl2anc 584 . . . . . 6 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
6 topontop 22935 . . . . . 6 ((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) → (𝐽 ×t 𝐾) ∈ Top)
75, 6syl 17 . . . . 5 (𝜑 → (𝐽 ×t 𝐾) ∈ Top)
8 xkohmeo.l . . . . 5 (𝜑𝐿 ∈ Top)
9 eqid 2735 . . . . . 6 (𝐿ko (𝐽 ×t 𝐾)) = (𝐿ko (𝐽 ×t 𝐾))
109xkotopon 23624 . . . . 5 (((𝐽 ×t 𝐾) ∈ Top ∧ 𝐿 ∈ Top) → (𝐿ko (𝐽 ×t 𝐾)) ∈ (TopOn‘((𝐽 ×t 𝐾) Cn 𝐿)))
117, 8, 10syl2anc 584 . . . 4 (𝜑 → (𝐿ko (𝐽 ×t 𝐾)) ∈ (TopOn‘((𝐽 ×t 𝐾) Cn 𝐿)))
12 vex 3482 . . . . . . . . 9 𝑓 ∈ V
13 vex 3482 . . . . . . . . 9 𝑥 ∈ V
1412, 13op1std 8023 . . . . . . . 8 (𝑧 = ⟨𝑓, 𝑥⟩ → (1st𝑧) = 𝑓)
1512, 13op2ndd 8024 . . . . . . . 8 (𝑧 = ⟨𝑓, 𝑥⟩ → (2nd𝑧) = 𝑥)
16 eqidd 2736 . . . . . . . 8 (𝑧 = ⟨𝑓, 𝑥⟩ → 𝑦 = 𝑦)
1714, 15, 16oveq123d 7452 . . . . . . 7 (𝑧 = ⟨𝑓, 𝑥⟩ → ((2nd𝑧)(1st𝑧)𝑦) = (𝑥𝑓𝑦))
1817mpteq2dv 5250 . . . . . 6 (𝑧 = ⟨𝑓, 𝑥⟩ → (𝑦𝑌 ↦ ((2nd𝑧)(1st𝑧)𝑦)) = (𝑦𝑌 ↦ (𝑥𝑓𝑦)))
1918mpompt 7547 . . . . 5 (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (𝑦𝑌 ↦ ((2nd𝑧)(1st𝑧)𝑦))) = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑥𝑓𝑦)))
20 txtopon 23615 . . . . . . 7 (((𝐿ko (𝐽 ×t 𝐾)) ∈ (TopOn‘((𝐽 ×t 𝐾) Cn 𝐿)) ∧ 𝐽 ∈ (TopOn‘𝑋)) → ((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) ∈ (TopOn‘(((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋)))
2111, 2, 20syl2anc 584 . . . . . 6 (𝜑 → ((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) ∈ (TopOn‘(((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋)))
22 vex 3482 . . . . . . . . . . 11 𝑧 ∈ V
23 vex 3482 . . . . . . . . . . 11 𝑦 ∈ V
2422, 23op1std 8023 . . . . . . . . . 10 (𝑤 = ⟨𝑧, 𝑦⟩ → (1st𝑤) = 𝑧)
2524fveq2d 6911 . . . . . . . . 9 (𝑤 = ⟨𝑧, 𝑦⟩ → (1st ‘(1st𝑤)) = (1st𝑧))
2624fveq2d 6911 . . . . . . . . 9 (𝑤 = ⟨𝑧, 𝑦⟩ → (2nd ‘(1st𝑤)) = (2nd𝑧))
2722, 23op2ndd 8024 . . . . . . . . 9 (𝑤 = ⟨𝑧, 𝑦⟩ → (2nd𝑤) = 𝑦)
2825, 26, 27oveq123d 7452 . . . . . . . 8 (𝑤 = ⟨𝑧, 𝑦⟩ → ((2nd ‘(1st𝑤))(1st ‘(1st𝑤))(2nd𝑤)) = ((2nd𝑧)(1st𝑧)𝑦))
2928mpompt 7547 . . . . . . 7 (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ ((2nd ‘(1st𝑤))(1st ‘(1st𝑤))(2nd𝑤))) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌 ↦ ((2nd𝑧)(1st𝑧)𝑦))
30 txtopon 23615 . . . . . . . . 9 ((((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) ∈ (TopOn‘(((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋)) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) ∈ (TopOn‘((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)))
3121, 3, 30syl2anc 584 . . . . . . . 8 (𝜑 → (((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) ∈ (TopOn‘((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)))
32 toptopon2 22940 . . . . . . . . 9 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
338, 32sylib 218 . . . . . . . 8 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
34 xkohmeo.j . . . . . . . . 9 (𝜑𝐽 ∈ 𝑛-Locally Comp)
35 xkohmeo.k . . . . . . . . 9 (𝜑𝐾 ∈ 𝑛-Locally Comp)
36 txcmp 23667 . . . . . . . . . 10 ((𝑥 ∈ Comp ∧ 𝑦 ∈ Comp) → (𝑥 ×t 𝑦) ∈ Comp)
3736txnlly 23661 . . . . . . . . 9 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐾 ∈ 𝑛-Locally Comp) → (𝐽 ×t 𝐾) ∈ 𝑛-Locally Comp)
3834, 35, 37syl2anc 584 . . . . . . . 8 (𝜑 → (𝐽 ×t 𝐾) ∈ 𝑛-Locally Comp)
3925mpompt 7547 . . . . . . . . . 10 (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (1st ‘(1st𝑤))) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌 ↦ (1st𝑧))
405adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
4133adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → 𝐿 ∈ (TopOn‘ 𝐿))
42 xp1st 8045 . . . . . . . . . . . . . . 15 (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) → (1st𝑤) ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋))
4342adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1st𝑤) ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋))
44 xp1st 8045 . . . . . . . . . . . . . 14 ((1st𝑤) ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) → (1st ‘(1st𝑤)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
4543, 44syl 17 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1st ‘(1st𝑤)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
46 cnf2 23273 . . . . . . . . . . . . 13 (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (1st ‘(1st𝑤)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (1st ‘(1st𝑤)):(𝑋 × 𝑌)⟶ 𝐿)
4740, 41, 45, 46syl3anc 1370 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1st ‘(1st𝑤)):(𝑋 × 𝑌)⟶ 𝐿)
4847feqmptd 6977 . . . . . . . . . . 11 ((𝜑𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1st ‘(1st𝑤)) = (𝑢 ∈ (𝑋 × 𝑌) ↦ ((1st ‘(1st𝑤))‘𝑢)))
4948mpteq2dva 5248 . . . . . . . . . 10 (𝜑 → (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (1st ‘(1st𝑤))) = (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (𝑢 ∈ (𝑋 × 𝑌) ↦ ((1st ‘(1st𝑤))‘𝑢))))
5039, 49eqtr3id 2789 . . . . . . . . 9 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌 ↦ (1st𝑧)) = (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (𝑢 ∈ (𝑋 × 𝑌) ↦ ((1st ‘(1st𝑤))‘𝑢))))
5121, 3cnmpt1st 23692 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌𝑧) ∈ ((((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn ((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽)))
52 fveq2 6907 . . . . . . . . . . . 12 (𝑡 = 𝑧 → (1st𝑡) = (1st𝑧))
5352cbvmptv 5261 . . . . . . . . . . 11 (𝑡 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st𝑡)) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st𝑧))
5414mpompt 7547 . . . . . . . . . . . 12 (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st𝑧)) = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥𝑋𝑓)
5511, 2cnmpt1st 23692 . . . . . . . . . . . 12 (𝜑 → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥𝑋𝑓) ∈ (((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿ko (𝐽 ×t 𝐾))))
5654, 55eqeltrid 2843 . . . . . . . . . . 11 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st𝑧)) ∈ (((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿ko (𝐽 ×t 𝐾))))
5753, 56eqeltrid 2843 . . . . . . . . . 10 (𝜑 → (𝑡 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st𝑡)) ∈ (((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿ko (𝐽 ×t 𝐾))))
5821, 3, 51, 21, 57, 52cnmpt21 23695 . . . . . . . . 9 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌 ↦ (1st𝑧)) ∈ ((((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn (𝐿ko (𝐽 ×t 𝐾))))
5950, 58eqeltrrd 2840 . . . . . . . 8 (𝜑 → (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (𝑢 ∈ (𝑋 × 𝑌) ↦ ((1st ‘(1st𝑤))‘𝑢))) ∈ ((((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn (𝐿ko (𝐽 ×t 𝐾))))
6026mpompt 7547 . . . . . . . . . 10 (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2nd ‘(1st𝑤))) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌 ↦ (2nd𝑧))
61 fveq2 6907 . . . . . . . . . . . . 13 (𝑡 = 𝑧 → (2nd𝑡) = (2nd𝑧))
6261cbvmptv 5261 . . . . . . . . . . . 12 (𝑡 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd𝑡)) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd𝑧))
6315mpompt 7547 . . . . . . . . . . . . 13 (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd𝑧)) = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥𝑋𝑥)
6411, 2cnmpt2nd 23693 . . . . . . . . . . . . 13 (𝜑 → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥𝑋𝑥) ∈ (((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn 𝐽))
6563, 64eqeltrid 2843 . . . . . . . . . . . 12 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd𝑧)) ∈ (((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn 𝐽))
6662, 65eqeltrid 2843 . . . . . . . . . . 11 (𝜑 → (𝑡 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd𝑡)) ∈ (((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn 𝐽))
6721, 3, 51, 21, 66, 61cnmpt21 23695 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌 ↦ (2nd𝑧)) ∈ ((((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐽))
6860, 67eqeltrid 2843 . . . . . . . . 9 (𝜑 → (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2nd ‘(1st𝑤))) ∈ ((((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐽))
6927mpompt 7547 . . . . . . . . . 10 (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2nd𝑤)) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌𝑦)
7021, 3cnmpt2nd 23693 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌𝑦) ∈ ((((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐾))
7169, 70eqeltrid 2843 . . . . . . . . 9 (𝜑 → (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2nd𝑤)) ∈ ((((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐾))
7231, 68, 71cnmpt1t 23689 . . . . . . . 8 (𝜑 → (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ ⟨(2nd ‘(1st𝑤)), (2nd𝑤)⟩) ∈ ((((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn (𝐽 ×t 𝐾)))
73 fveq2 6907 . . . . . . . . 9 (𝑢 = ⟨(2nd ‘(1st𝑤)), (2nd𝑤)⟩ → ((1st ‘(1st𝑤))‘𝑢) = ((1st ‘(1st𝑤))‘⟨(2nd ‘(1st𝑤)), (2nd𝑤)⟩))
74 df-ov 7434 . . . . . . . . 9 ((2nd ‘(1st𝑤))(1st ‘(1st𝑤))(2nd𝑤)) = ((1st ‘(1st𝑤))‘⟨(2nd ‘(1st𝑤)), (2nd𝑤)⟩)
7573, 74eqtr4di 2793 . . . . . . . 8 (𝑢 = ⟨(2nd ‘(1st𝑤)), (2nd𝑤)⟩ → ((1st ‘(1st𝑤))‘𝑢) = ((2nd ‘(1st𝑤))(1st ‘(1st𝑤))(2nd𝑤)))
7631, 5, 33, 38, 59, 72, 75cnmptk1p 23709 . . . . . . 7 (𝜑 → (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ ((2nd ‘(1st𝑤))(1st ‘(1st𝑤))(2nd𝑤))) ∈ ((((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐿))
7729, 76eqeltrrid 2844 . . . . . 6 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌 ↦ ((2nd𝑧)(1st𝑧)𝑦)) ∈ ((((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐿))
7821, 3, 77cnmpt2k 23712 . . . . 5 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (𝑦𝑌 ↦ ((2nd𝑧)(1st𝑧)𝑦))) ∈ (((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿ko 𝐾)))
7919, 78eqeltrrid 2844 . . . 4 (𝜑 → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑥𝑓𝑦))) ∈ (((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿ko 𝐾)))
8011, 2, 79cnmpt2k 23712 . . 3 (𝜑 → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑥𝑓𝑦)))) ∈ ((𝐿ko (𝐽 ×t 𝐾)) Cn ((𝐿ko 𝐾) ↑ko 𝐽)))
811, 80eqeltrid 2843 . 2 (𝜑𝐹 ∈ ((𝐿ko (𝐽 ×t 𝐾)) Cn ((𝐿ko 𝐾) ↑ko 𝐽)))
822, 3, 1, 34, 35, 8xkocnv 23838 . . . 4 (𝜑𝐹 = (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)) ↦ (𝑥𝑋, 𝑦𝑌 ↦ ((𝑔𝑥)‘𝑦))))
8313, 23op1std 8023 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
8483fveq2d 6911 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑔‘(1st𝑧)) = (𝑔𝑥))
8513, 23op2ndd 8024 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
8684, 85fveq12d 6914 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑔‘(1st𝑧))‘(2nd𝑧)) = ((𝑔𝑥)‘𝑦))
8786mpompt 7547 . . . . 5 (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st𝑧))‘(2nd𝑧))) = (𝑥𝑋, 𝑦𝑌 ↦ ((𝑔𝑥)‘𝑦))
8887mpteq2i 5253 . . . 4 (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)) ↦ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st𝑧))‘(2nd𝑧)))) = (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)) ↦ (𝑥𝑋, 𝑦𝑌 ↦ ((𝑔𝑥)‘𝑦)))
8982, 88eqtr4di 2793 . . 3 (𝜑𝐹 = (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)) ↦ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st𝑧))‘(2nd𝑧)))))
90 nllytop 23497 . . . . . 6 (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ Top)
9134, 90syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
92 nllytop 23497 . . . . . . 7 (𝐾 ∈ 𝑛-Locally Comp → 𝐾 ∈ Top)
9335, 92syl 17 . . . . . 6 (𝜑𝐾 ∈ Top)
94 xkotop 23612 . . . . . 6 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿ko 𝐾) ∈ Top)
9593, 8, 94syl2anc 584 . . . . 5 (𝜑 → (𝐿ko 𝐾) ∈ Top)
96 eqid 2735 . . . . . 6 ((𝐿ko 𝐾) ↑ko 𝐽) = ((𝐿ko 𝐾) ↑ko 𝐽)
9796xkotopon 23624 . . . . 5 ((𝐽 ∈ Top ∧ (𝐿ko 𝐾) ∈ Top) → ((𝐿ko 𝐾) ↑ko 𝐽) ∈ (TopOn‘(𝐽 Cn (𝐿ko 𝐾))))
9891, 95, 97syl2anc 584 . . . 4 (𝜑 → ((𝐿ko 𝐾) ↑ko 𝐽) ∈ (TopOn‘(𝐽 Cn (𝐿ko 𝐾))))
99 vex 3482 . . . . . . . . 9 𝑔 ∈ V
10099, 22op1std 8023 . . . . . . . 8 (𝑤 = ⟨𝑔, 𝑧⟩ → (1st𝑤) = 𝑔)
10199, 22op2ndd 8024 . . . . . . . . 9 (𝑤 = ⟨𝑔, 𝑧⟩ → (2nd𝑤) = 𝑧)
102101fveq2d 6911 . . . . . . . 8 (𝑤 = ⟨𝑔, 𝑧⟩ → (1st ‘(2nd𝑤)) = (1st𝑧))
103100, 102fveq12d 6914 . . . . . . 7 (𝑤 = ⟨𝑔, 𝑧⟩ → ((1st𝑤)‘(1st ‘(2nd𝑤))) = (𝑔‘(1st𝑧)))
104101fveq2d 6911 . . . . . . 7 (𝑤 = ⟨𝑔, 𝑧⟩ → (2nd ‘(2nd𝑤)) = (2nd𝑧))
105103, 104fveq12d 6914 . . . . . 6 (𝑤 = ⟨𝑔, 𝑧⟩ → (((1st𝑤)‘(1st ‘(2nd𝑤)))‘(2nd ‘(2nd𝑤))) = ((𝑔‘(1st𝑧))‘(2nd𝑧)))
106105mpompt 7547 . . . . 5 (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ (((1st𝑤)‘(1st ‘(2nd𝑤)))‘(2nd ‘(2nd𝑤)))) = (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st𝑧))‘(2nd𝑧)))
107 txtopon 23615 . . . . . . 7 ((((𝐿ko 𝐾) ↑ko 𝐽) ∈ (TopOn‘(𝐽 Cn (𝐿ko 𝐾))) ∧ (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) → (((𝐿ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) ∈ (TopOn‘((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌))))
10898, 5, 107syl2anc 584 . . . . . 6 (𝜑 → (((𝐿ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) ∈ (TopOn‘((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌))))
1093adantr 480 . . . . . . . . . 10 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌))) → 𝐾 ∈ (TopOn‘𝑌))
11033adantr 480 . . . . . . . . . 10 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌))) → 𝐿 ∈ (TopOn‘ 𝐿))
111 eqid 2735 . . . . . . . . . . . . . 14 (𝐿ko 𝐾) = (𝐿ko 𝐾)
112111xkotopon 23624 . . . . . . . . . . . . 13 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
11393, 8, 112syl2anc 584 . . . . . . . . . . . 12 (𝜑 → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
114 xp1st 8045 . . . . . . . . . . . 12 (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) → (1st𝑤) ∈ (𝐽 Cn (𝐿ko 𝐾)))
115 cnf2 23273 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ (1st𝑤) ∈ (𝐽 Cn (𝐿ko 𝐾))) → (1st𝑤):𝑋⟶(𝐾 Cn 𝐿))
1162, 113, 114, 115syl2an3an 1421 . . . . . . . . . . 11 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌))) → (1st𝑤):𝑋⟶(𝐾 Cn 𝐿))
117 xp2nd 8046 . . . . . . . . . . . . 13 (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) → (2nd𝑤) ∈ (𝑋 × 𝑌))
118117adantl 481 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌))) → (2nd𝑤) ∈ (𝑋 × 𝑌))
119 xp1st 8045 . . . . . . . . . . . 12 ((2nd𝑤) ∈ (𝑋 × 𝑌) → (1st ‘(2nd𝑤)) ∈ 𝑋)
120118, 119syl 17 . . . . . . . . . . 11 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌))) → (1st ‘(2nd𝑤)) ∈ 𝑋)
121116, 120ffvelcdmd 7105 . . . . . . . . . 10 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌))) → ((1st𝑤)‘(1st ‘(2nd𝑤))) ∈ (𝐾 Cn 𝐿))
122 cnf2 23273 . . . . . . . . . 10 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ ((1st𝑤)‘(1st ‘(2nd𝑤))) ∈ (𝐾 Cn 𝐿)) → ((1st𝑤)‘(1st ‘(2nd𝑤))):𝑌 𝐿)
123109, 110, 121, 122syl3anc 1370 . . . . . . . . 9 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌))) → ((1st𝑤)‘(1st ‘(2nd𝑤))):𝑌 𝐿)
124123feqmptd 6977 . . . . . . . 8 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌))) → ((1st𝑤)‘(1st ‘(2nd𝑤))) = (𝑦𝑌 ↦ (((1st𝑤)‘(1st ‘(2nd𝑤)))‘𝑦)))
125124mpteq2dva 5248 . . . . . . 7 (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ ((1st𝑤)‘(1st ‘(2nd𝑤)))) = (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑦𝑌 ↦ (((1st𝑤)‘(1st ‘(2nd𝑤)))‘𝑦))))
126100mpompt 7547 . . . . . . . . . 10 (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1st𝑤)) = (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑔)
127116feqmptd 6977 . . . . . . . . . . 11 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌))) → (1st𝑤) = (𝑥𝑋 ↦ ((1st𝑤)‘𝑥)))
128127mpteq2dva 5248 . . . . . . . . . 10 (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1st𝑤)) = (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑥𝑋 ↦ ((1st𝑤)‘𝑥))))
129126, 128eqtr3id 2789 . . . . . . . . 9 (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑔) = (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑥𝑋 ↦ ((1st𝑤)‘𝑥))))
13098, 5cnmpt1st 23692 . . . . . . . . 9 (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑔) ∈ ((((𝐿ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn ((𝐿ko 𝐾) ↑ko 𝐽)))
131129, 130eqeltrrd 2840 . . . . . . . 8 (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑥𝑋 ↦ ((1st𝑤)‘𝑥))) ∈ ((((𝐿ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn ((𝐿ko 𝐾) ↑ko 𝐽)))
132102mpompt 7547 . . . . . . . . 9 (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1st ‘(2nd𝑤))) = (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧))
13398, 5cnmpt2nd 23693 . . . . . . . . . 10 (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) ∈ ((((𝐿ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn (𝐽 ×t 𝐾)))
13452cbvmptv 5261 . . . . . . . . . . 11 (𝑡 ∈ (𝑋 × 𝑌) ↦ (1st𝑡)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧))
13583mpompt 7547 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧)) = (𝑥𝑋, 𝑦𝑌𝑥)
1362, 3cnmpt1st 23692 . . . . . . . . . . . 12 (𝜑 → (𝑥𝑋, 𝑦𝑌𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
137135, 136eqeltrid 2843 . . . . . . . . . . 11 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
138134, 137eqeltrid 2843 . . . . . . . . . 10 (𝜑 → (𝑡 ∈ (𝑋 × 𝑌) ↦ (1st𝑡)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
13998, 5, 133, 5, 138, 52cnmpt21 23695 . . . . . . . . 9 (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧)) ∈ ((((𝐿ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐽))
140132, 139eqeltrid 2843 . . . . . . . 8 (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1st ‘(2nd𝑤))) ∈ ((((𝐿ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐽))
141 fveq2 6907 . . . . . . . 8 (𝑥 = (1st ‘(2nd𝑤)) → ((1st𝑤)‘𝑥) = ((1st𝑤)‘(1st ‘(2nd𝑤))))
142108, 2, 113, 34, 131, 140, 141cnmptk1p 23709 . . . . . . 7 (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ ((1st𝑤)‘(1st ‘(2nd𝑤)))) ∈ ((((𝐿ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn (𝐿ko 𝐾)))
143125, 142eqeltrrd 2840 . . . . . 6 (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑦𝑌 ↦ (((1st𝑤)‘(1st ‘(2nd𝑤)))‘𝑦))) ∈ ((((𝐿ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn (𝐿ko 𝐾)))
144104mpompt 7547 . . . . . . 7 (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ (2nd ‘(2nd𝑤))) = (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧))
14561cbvmptv 5261 . . . . . . . . 9 (𝑡 ∈ (𝑋 × 𝑌) ↦ (2nd𝑡)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧))
14685mpompt 7547 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧)) = (𝑥𝑋, 𝑦𝑌𝑦)
1472, 3cnmpt2nd 23693 . . . . . . . . . 10 (𝜑 → (𝑥𝑋, 𝑦𝑌𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
148146, 147eqeltrid 2843 . . . . . . . . 9 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
149145, 148eqeltrid 2843 . . . . . . . 8 (𝜑 → (𝑡 ∈ (𝑋 × 𝑌) ↦ (2nd𝑡)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
15098, 5, 133, 5, 149, 61cnmpt21 23695 . . . . . . 7 (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧)) ∈ ((((𝐿ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐾))
151144, 150eqeltrid 2843 . . . . . 6 (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ (2nd ‘(2nd𝑤))) ∈ ((((𝐿ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐾))
152 fveq2 6907 . . . . . 6 (𝑦 = (2nd ‘(2nd𝑤)) → (((1st𝑤)‘(1st ‘(2nd𝑤)))‘𝑦) = (((1st𝑤)‘(1st ‘(2nd𝑤)))‘(2nd ‘(2nd𝑤))))
153108, 3, 33, 35, 143, 151, 152cnmptk1p 23709 . . . . 5 (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ (((1st𝑤)‘(1st ‘(2nd𝑤)))‘(2nd ‘(2nd𝑤)))) ∈ ((((𝐿ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐿))
154106, 153eqeltrrid 2844 . . . 4 (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st𝑧))‘(2nd𝑧))) ∈ ((((𝐿ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐿))
15598, 5, 154cnmpt2k 23712 . . 3 (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)) ↦ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st𝑧))‘(2nd𝑧)))) ∈ (((𝐿ko 𝐾) ↑ko 𝐽) Cn (𝐿ko (𝐽 ×t 𝐾))))
15689, 155eqeltrd 2839 . 2 (𝜑𝐹 ∈ (((𝐿ko 𝐾) ↑ko 𝐽) Cn (𝐿ko (𝐽 ×t 𝐾))))
157 ishmeo 23783 . 2 (𝐹 ∈ ((𝐿ko (𝐽 ×t 𝐾))Homeo((𝐿ko 𝐾) ↑ko 𝐽)) ↔ (𝐹 ∈ ((𝐿ko (𝐽 ×t 𝐾)) Cn ((𝐿ko 𝐾) ↑ko 𝐽)) ∧ 𝐹 ∈ (((𝐿ko 𝐾) ↑ko 𝐽) Cn (𝐿ko (𝐽 ×t 𝐾)))))
15881, 156, 157sylanbrc 583 1 (𝜑𝐹 ∈ ((𝐿ko (𝐽 ×t 𝐾))Homeo((𝐿ko 𝐾) ↑ko 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cop 4637   cuni 4912  cmpt 5231   × cxp 5687  ccnv 5688  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  Topctop 22915  TopOnctopon 22932   Cn ccn 23248  Compccmp 23410  𝑛-Locally cnlly 23489   ×t ctx 23584  ko cxko 23585  Homeochmeo 23777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-2o 8506  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-pt 17491  df-top 22916  df-topon 22933  df-bases 22969  df-ntr 23044  df-nei 23122  df-cn 23251  df-cnp 23252  df-cmp 23411  df-nlly 23491  df-tx 23586  df-xko 23587  df-hmeo 23779
This theorem is referenced by: (None)
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