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Theorem xkohmeo 22409
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 22254, 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 22185 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
52, 3, 4syl2anc 587 . . . . . 6 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
6 topontop 21507 . . . . . 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 2824 . . . . . 6 (𝐿ko (𝐽 ×t 𝐾)) = (𝐿ko (𝐽 ×t 𝐾))
109xkotopon 22194 . . . . 5 (((𝐽 ×t 𝐾) ∈ Top ∧ 𝐿 ∈ Top) → (𝐿ko (𝐽 ×t 𝐾)) ∈ (TopOn‘((𝐽 ×t 𝐾) Cn 𝐿)))
117, 8, 10syl2anc 587 . . . 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 7682 . . . . . . . 8 (𝑧 = ⟨𝑓, 𝑥⟩ → (1st𝑧) = 𝑓)
1512, 13op2ndd 7683 . . . . . . . 8 (𝑧 = ⟨𝑓, 𝑥⟩ → (2nd𝑧) = 𝑥)
16 eqidd 2825 . . . . . . . 8 (𝑧 = ⟨𝑓, 𝑥⟩ → 𝑦 = 𝑦)
1714, 15, 16oveq123d 7159 . . . . . . 7 (𝑧 = ⟨𝑓, 𝑥⟩ → ((2nd𝑧)(1st𝑧)𝑦) = (𝑥𝑓𝑦))
1817mpteq2dv 5143 . . . . . 6 (𝑧 = ⟨𝑓, 𝑥⟩ → (𝑦𝑌 ↦ ((2nd𝑧)(1st𝑧)𝑦)) = (𝑦𝑌 ↦ (𝑥𝑓𝑦)))
1918mpompt 7248 . . . . 5 (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (𝑦𝑌 ↦ ((2nd𝑧)(1st𝑧)𝑦))) = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑥𝑓𝑦)))
20 txtopon 22185 . . . . . . 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 587 . . . . . 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 7682 . . . . . . . . . 10 (𝑤 = ⟨𝑧, 𝑦⟩ → (1st𝑤) = 𝑧)
2524fveq2d 6655 . . . . . . . . 9 (𝑤 = ⟨𝑧, 𝑦⟩ → (1st ‘(1st𝑤)) = (1st𝑧))
2624fveq2d 6655 . . . . . . . . 9 (𝑤 = ⟨𝑧, 𝑦⟩ → (2nd ‘(1st𝑤)) = (2nd𝑧))
2722, 23op2ndd 7683 . . . . . . . . 9 (𝑤 = ⟨𝑧, 𝑦⟩ → (2nd𝑤) = 𝑦)
2825, 26, 27oveq123d 7159 . . . . . . . 8 (𝑤 = ⟨𝑧, 𝑦⟩ → ((2nd ‘(1st𝑤))(1st ‘(1st𝑤))(2nd𝑤)) = ((2nd𝑧)(1st𝑧)𝑦))
2928mpompt 7248 . . . . . . 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 22185 . . . . . . . . 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 587 . . . . . . . 8 (𝜑 → (((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) ∈ (TopOn‘((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)))
32 toptopon2 21512 . . . . . . . . 9 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
338, 32sylib 221 . . . . . . . 8 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
34 xkohmeo.j . . . . . . . . 9 (𝜑𝐽 ∈ 𝑛-Locally Comp)
35 xkohmeo.k . . . . . . . . 9 (𝜑𝐾 ∈ 𝑛-Locally Comp)
36 txcmp 22237 . . . . . . . . . 10 ((𝑥 ∈ Comp ∧ 𝑦 ∈ Comp) → (𝑥 ×t 𝑦) ∈ Comp)
3736txnlly 22231 . . . . . . . . 9 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐾 ∈ 𝑛-Locally Comp) → (𝐽 ×t 𝐾) ∈ 𝑛-Locally Comp)
3834, 35, 37syl2anc 587 . . . . . . . 8 (𝜑 → (𝐽 ×t 𝐾) ∈ 𝑛-Locally Comp)
3925mpompt 7248 . . . . . . . . . 10 (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (1st ‘(1st𝑤))) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌 ↦ (1st𝑧))
405adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
4133adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → 𝐿 ∈ (TopOn‘ 𝐿))
42 xp1st 7704 . . . . . . . . . . . . . . 15 (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) → (1st𝑤) ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋))
4342adantl 485 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1st𝑤) ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋))
44 xp1st 7704 . . . . . . . . . . . . . 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 21843 . . . . . . . . . . . . 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 1368 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1st ‘(1st𝑤)):(𝑋 × 𝑌)⟶ 𝐿)
4847feqmptd 6714 . . . . . . . . . . 11 ((𝜑𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1st ‘(1st𝑤)) = (𝑢 ∈ (𝑋 × 𝑌) ↦ ((1st ‘(1st𝑤))‘𝑢)))
4948mpteq2dva 5142 . . . . . . . . . 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, 49syl5eqr 2873 . . . . . . . . 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 22262 . . . . . . . . . 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 6651 . . . . . . . . . . . 12 (𝑡 = 𝑧 → (1st𝑡) = (1st𝑧))
5352cbvmptv 5150 . . . . . . . . . . 11 (𝑡 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st𝑡)) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st𝑧))
5414mpompt 7248 . . . . . . . . . . . 12 (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st𝑧)) = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥𝑋𝑓)
5511, 2cnmpt1st 22262 . . . . . . . . . . . 12 (𝜑 → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥𝑋𝑓) ∈ (((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿ko (𝐽 ×t 𝐾))))
5654, 55eqeltrid 2920 . . . . . . . . . . 11 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st𝑧)) ∈ (((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿ko (𝐽 ×t 𝐾))))
5753, 56eqeltrid 2920 . . . . . . . . . 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 22265 . . . . . . . . 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 2917 . . . . . . . 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 7248 . . . . . . . . . 10 (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2nd ‘(1st𝑤))) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌 ↦ (2nd𝑧))
61 fveq2 6651 . . . . . . . . . . . . 13 (𝑡 = 𝑧 → (2nd𝑡) = (2nd𝑧))
6261cbvmptv 5150 . . . . . . . . . . . 12 (𝑡 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd𝑡)) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd𝑧))
6315mpompt 7248 . . . . . . . . . . . . 13 (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd𝑧)) = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥𝑋𝑥)
6411, 2cnmpt2nd 22263 . . . . . . . . . . . . 13 (𝜑 → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥𝑋𝑥) ∈ (((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn 𝐽))
6563, 64eqeltrid 2920 . . . . . . . . . . . 12 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd𝑧)) ∈ (((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn 𝐽))
6662, 65eqeltrid 2920 . . . . . . . . . . 11 (𝜑 → (𝑡 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd𝑡)) ∈ (((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn 𝐽))
6721, 3, 51, 21, 66, 61cnmpt21 22265 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌 ↦ (2nd𝑧)) ∈ ((((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐽))
6860, 67eqeltrid 2920 . . . . . . . . 9 (𝜑 → (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2nd ‘(1st𝑤))) ∈ ((((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐽))
6927mpompt 7248 . . . . . . . . . 10 (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2nd𝑤)) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌𝑦)
7021, 3cnmpt2nd 22263 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌𝑦) ∈ ((((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐾))
7169, 70eqeltrid 2920 . . . . . . . . 9 (𝜑 → (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2nd𝑤)) ∈ ((((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐾))
7231, 68, 71cnmpt1t 22259 . . . . . . . 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 6651 . . . . . . . . 9 (𝑢 = ⟨(2nd ‘(1st𝑤)), (2nd𝑤)⟩ → ((1st ‘(1st𝑤))‘𝑢) = ((1st ‘(1st𝑤))‘⟨(2nd ‘(1st𝑤)), (2nd𝑤)⟩))
74 df-ov 7141 . . . . . . . . 9 ((2nd ‘(1st𝑤))(1st ‘(1st𝑤))(2nd𝑤)) = ((1st ‘(1st𝑤))‘⟨(2nd ‘(1st𝑤)), (2nd𝑤)⟩)
7573, 74syl6eqr 2877 . . . . . . . 8 (𝑢 = ⟨(2nd ‘(1st𝑤)), (2nd𝑤)⟩ → ((1st ‘(1st𝑤))‘𝑢) = ((2nd ‘(1st𝑤))(1st ‘(1st𝑤))(2nd𝑤)))
7631, 5, 33, 38, 59, 72, 75cnmptk1p 22279 . . . . . . 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 2921 . . . . . 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 22282 . . . . 5 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (𝑦𝑌 ↦ ((2nd𝑧)(1st𝑧)𝑦))) ∈ (((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿ko 𝐾)))
7919, 78eqeltrrid 2921 . . . 4 (𝜑 → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑥𝑓𝑦))) ∈ (((𝐿ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿ko 𝐾)))
8011, 2, 79cnmpt2k 22282 . . 3 (𝜑 → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑥𝑓𝑦)))) ∈ ((𝐿ko (𝐽 ×t 𝐾)) Cn ((𝐿ko 𝐾) ↑ko 𝐽)))
811, 80eqeltrid 2920 . 2 (𝜑𝐹 ∈ ((𝐿ko (𝐽 ×t 𝐾)) Cn ((𝐿ko 𝐾) ↑ko 𝐽)))
822, 3, 1, 34, 35, 8xkocnv 22408 . . . 4 (𝜑𝐹 = (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)) ↦ (𝑥𝑋, 𝑦𝑌 ↦ ((𝑔𝑥)‘𝑦))))
8313, 23op1std 7682 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
8483fveq2d 6655 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑔‘(1st𝑧)) = (𝑔𝑥))
8513, 23op2ndd 7683 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
8684, 85fveq12d 6658 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑔‘(1st𝑧))‘(2nd𝑧)) = ((𝑔𝑥)‘𝑦))
8786mpompt 7248 . . . . 5 (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st𝑧))‘(2nd𝑧))) = (𝑥𝑋, 𝑦𝑌 ↦ ((𝑔𝑥)‘𝑦))
8887mpteq2i 5139 . . . 4 (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)) ↦ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st𝑧))‘(2nd𝑧)))) = (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)) ↦ (𝑥𝑋, 𝑦𝑌 ↦ ((𝑔𝑥)‘𝑦)))
8982, 88syl6eqr 2877 . . 3 (𝜑𝐹 = (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)) ↦ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st𝑧))‘(2nd𝑧)))))
90 nllytop 22067 . . . . . 6 (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ Top)
9134, 90syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
92 nllytop 22067 . . . . . . 7 (𝐾 ∈ 𝑛-Locally Comp → 𝐾 ∈ Top)
9335, 92syl 17 . . . . . 6 (𝜑𝐾 ∈ Top)
94 xkotop 22182 . . . . . 6 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿ko 𝐾) ∈ Top)
9593, 8, 94syl2anc 587 . . . . 5 (𝜑 → (𝐿ko 𝐾) ∈ Top)
96 eqid 2824 . . . . . 6 ((𝐿ko 𝐾) ↑ko 𝐽) = ((𝐿ko 𝐾) ↑ko 𝐽)
9796xkotopon 22194 . . . . 5 ((𝐽 ∈ Top ∧ (𝐿ko 𝐾) ∈ Top) → ((𝐿ko 𝐾) ↑ko 𝐽) ∈ (TopOn‘(𝐽 Cn (𝐿ko 𝐾))))
9891, 95, 97syl2anc 587 . . . 4 (𝜑 → ((𝐿ko 𝐾) ↑ko 𝐽) ∈ (TopOn‘(𝐽 Cn (𝐿ko 𝐾))))
99 vex 3482 . . . . . . . . 9 𝑔 ∈ V
10099, 22op1std 7682 . . . . . . . 8 (𝑤 = ⟨𝑔, 𝑧⟩ → (1st𝑤) = 𝑔)
10199, 22op2ndd 7683 . . . . . . . . 9 (𝑤 = ⟨𝑔, 𝑧⟩ → (2nd𝑤) = 𝑧)
102101fveq2d 6655 . . . . . . . 8 (𝑤 = ⟨𝑔, 𝑧⟩ → (1st ‘(2nd𝑤)) = (1st𝑧))
103100, 102fveq12d 6658 . . . . . . 7 (𝑤 = ⟨𝑔, 𝑧⟩ → ((1st𝑤)‘(1st ‘(2nd𝑤))) = (𝑔‘(1st𝑧)))
104101fveq2d 6655 . . . . . . 7 (𝑤 = ⟨𝑔, 𝑧⟩ → (2nd ‘(2nd𝑤)) = (2nd𝑧))
105103, 104fveq12d 6658 . . . . . 6 (𝑤 = ⟨𝑔, 𝑧⟩ → (((1st𝑤)‘(1st ‘(2nd𝑤)))‘(2nd ‘(2nd𝑤))) = ((𝑔‘(1st𝑧))‘(2nd𝑧)))
106105mpompt 7248 . . . . 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 22185 . . . . . . 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 587 . . . . . 6 (𝜑 → (((𝐿ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) ∈ (TopOn‘((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌))))
1093adantr 484 . . . . . . . . . 10 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌))) → 𝐾 ∈ (TopOn‘𝑌))
11033adantr 484 . . . . . . . . . 10 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌))) → 𝐿 ∈ (TopOn‘ 𝐿))
111 eqid 2824 . . . . . . . . . . . . . 14 (𝐿ko 𝐾) = (𝐿ko 𝐾)
112111xkotopon 22194 . . . . . . . . . . . . 13 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
11393, 8, 112syl2anc 587 . . . . . . . . . . . 12 (𝜑 → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
114 xp1st 7704 . . . . . . . . . . . 12 (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) → (1st𝑤) ∈ (𝐽 Cn (𝐿ko 𝐾)))
115 cnf2 21843 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ (1st𝑤) ∈ (𝐽 Cn (𝐿ko 𝐾))) → (1st𝑤):𝑋⟶(𝐾 Cn 𝐿))
1162, 113, 114, 115syl2an3an 1419 . . . . . . . . . . 11 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌))) → (1st𝑤):𝑋⟶(𝐾 Cn 𝐿))
117 xp2nd 7705 . . . . . . . . . . . . 13 (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) → (2nd𝑤) ∈ (𝑋 × 𝑌))
118117adantl 485 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌))) → (2nd𝑤) ∈ (𝑋 × 𝑌))
119 xp1st 7704 . . . . . . . . . . . 12 ((2nd𝑤) ∈ (𝑋 × 𝑌) → (1st ‘(2nd𝑤)) ∈ 𝑋)
120118, 119syl 17 . . . . . . . . . . 11 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌))) → (1st ‘(2nd𝑤)) ∈ 𝑋)
121116, 120ffvelrnd 6833 . . . . . . . . . 10 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌))) → ((1st𝑤)‘(1st ‘(2nd𝑤))) ∈ (𝐾 Cn 𝐿))
122 cnf2 21843 . . . . . . . . . 10 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ ((1st𝑤)‘(1st ‘(2nd𝑤))) ∈ (𝐾 Cn 𝐿)) → ((1st𝑤)‘(1st ‘(2nd𝑤))):𝑌 𝐿)
123109, 110, 121, 122syl3anc 1368 . . . . . . . . 9 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌))) → ((1st𝑤)‘(1st ‘(2nd𝑤))):𝑌 𝐿)
124123feqmptd 6714 . . . . . . . 8 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌))) → ((1st𝑤)‘(1st ‘(2nd𝑤))) = (𝑦𝑌 ↦ (((1st𝑤)‘(1st ‘(2nd𝑤)))‘𝑦)))
125124mpteq2dva 5142 . . . . . . 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 7248 . . . . . . . . . 10 (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1st𝑤)) = (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑔)
127116feqmptd 6714 . . . . . . . . . . 11 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌))) → (1st𝑤) = (𝑥𝑋 ↦ ((1st𝑤)‘𝑥)))
128127mpteq2dva 5142 . . . . . . . . . 10 (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1st𝑤)) = (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑥𝑋 ↦ ((1st𝑤)‘𝑥))))
129126, 128syl5eqr 2873 . . . . . . . . 9 (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑔) = (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑥𝑋 ↦ ((1st𝑤)‘𝑥))))
13098, 5cnmpt1st 22262 . . . . . . . . 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 2917 . . . . . . . 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 7248 . . . . . . . . 9 (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1st ‘(2nd𝑤))) = (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧))
13398, 5cnmpt2nd 22263 . . . . . . . . . 10 (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) ∈ ((((𝐿ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn (𝐽 ×t 𝐾)))
13452cbvmptv 5150 . . . . . . . . . . 11 (𝑡 ∈ (𝑋 × 𝑌) ↦ (1st𝑡)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧))
13583mpompt 7248 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧)) = (𝑥𝑋, 𝑦𝑌𝑥)
1362, 3cnmpt1st 22262 . . . . . . . . . . . 12 (𝜑 → (𝑥𝑋, 𝑦𝑌𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
137135, 136eqeltrid 2920 . . . . . . . . . . 11 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
138134, 137eqeltrid 2920 . . . . . . . . . 10 (𝜑 → (𝑡 ∈ (𝑋 × 𝑌) ↦ (1st𝑡)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
13998, 5, 133, 5, 138, 52cnmpt21 22265 . . . . . . . . 9 (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧)) ∈ ((((𝐿ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐽))
140132, 139eqeltrid 2920 . . . . . . . 8 (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1st ‘(2nd𝑤))) ∈ ((((𝐿ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐽))
141 fveq2 6651 . . . . . . . 8 (𝑥 = (1st ‘(2nd𝑤)) → ((1st𝑤)‘𝑥) = ((1st𝑤)‘(1st ‘(2nd𝑤))))
142108, 2, 113, 34, 131, 140, 141cnmptk1p 22279 . . . . . . 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 2917 . . . . . 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 7248 . . . . . . 7 (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ (2nd ‘(2nd𝑤))) = (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧))
14561cbvmptv 5150 . . . . . . . . 9 (𝑡 ∈ (𝑋 × 𝑌) ↦ (2nd𝑡)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧))
14685mpompt 7248 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧)) = (𝑥𝑋, 𝑦𝑌𝑦)
1472, 3cnmpt2nd 22263 . . . . . . . . . 10 (𝜑 → (𝑥𝑋, 𝑦𝑌𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
148146, 147eqeltrid 2920 . . . . . . . . 9 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
149145, 148eqeltrid 2920 . . . . . . . 8 (𝜑 → (𝑡 ∈ (𝑋 × 𝑌) ↦ (2nd𝑡)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
15098, 5, 133, 5, 149, 61cnmpt21 22265 . . . . . . 7 (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧)) ∈ ((((𝐿ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐾))
151144, 150eqeltrid 2920 . . . . . 6 (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿ko 𝐾)) × (𝑋 × 𝑌)) ↦ (2nd ‘(2nd𝑤))) ∈ ((((𝐿ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐾))
152 fveq2 6651 . . . . . 6 (𝑦 = (2nd ‘(2nd𝑤)) → (((1st𝑤)‘(1st ‘(2nd𝑤)))‘𝑦) = (((1st𝑤)‘(1st ‘(2nd𝑤)))‘(2nd ‘(2nd𝑤))))
153108, 3, 33, 35, 143, 151, 152cnmptk1p 22279 . . . . 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 2921 . . . 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 22282 . . 3 (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)) ↦ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st𝑧))‘(2nd𝑧)))) ∈ (((𝐿ko 𝐾) ↑ko 𝐽) Cn (𝐿ko (𝐽 ×t 𝐾))))
15689, 155eqeltrd 2916 . 2 (𝜑𝐹 ∈ (((𝐿ko 𝐾) ↑ko 𝐽) Cn (𝐿ko (𝐽 ×t 𝐾))))
157 ishmeo 22353 . 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 586 1 (𝜑𝐹 ∈ ((𝐿ko (𝐽 ×t 𝐾))Homeo((𝐿ko 𝐾) ↑ko 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  cop 4554   cuni 4819  cmpt 5127   × cxp 5534  ccnv 5535  wf 6332  cfv 6336  (class class class)co 7138  cmpo 7140  1st c1st 7670  2nd c2nd 7671  Topctop 21487  TopOnctopon 21504   Cn ccn 21818  Compccmp 21980  𝑛-Locally cnlly 22059   ×t ctx 22154  ko cxko 22155  Homeochmeo 22347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-int 4858  df-iun 4902  df-iin 4903  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-om 7564  df-1st 7672  df-2nd 7673  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fi 8859  df-rest 16685  df-topgen 16706  df-pt 16707  df-top 21488  df-topon 21505  df-bases 21540  df-ntr 21614  df-nei 21692  df-cn 21821  df-cnp 21822  df-cmp 21981  df-nlly 22061  df-tx 22156  df-xko 22157  df-hmeo 22349
This theorem is referenced by: (None)
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