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Theorem xkohmeo 21832
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 21677, 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 ∈ ((𝐿 ^ko (𝐽 ×t 𝐾))Homeo((𝐿 ^ko 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 21608 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
52, 3, 4syl2anc 575 . . . . . 6 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
6 topontop 20931 . . . . . 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 2806 . . . . . 6 (𝐿 ^ko (𝐽 ×t 𝐾)) = (𝐿 ^ko (𝐽 ×t 𝐾))
109xkotopon 21617 . . . . 5 (((𝐽 ×t 𝐾) ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ^ko (𝐽 ×t 𝐾)) ∈ (TopOn‘((𝐽 ×t 𝐾) Cn 𝐿)))
117, 8, 10syl2anc 575 . . . 4 (𝜑 → (𝐿 ^ko (𝐽 ×t 𝐾)) ∈ (TopOn‘((𝐽 ×t 𝐾) Cn 𝐿)))
12 vex 3394 . . . . . . . . 9 𝑓 ∈ V
13 vex 3394 . . . . . . . . 9 𝑥 ∈ V
1412, 13op1std 7408 . . . . . . . 8 (𝑧 = ⟨𝑓, 𝑥⟩ → (1st𝑧) = 𝑓)
1512, 13op2ndd 7409 . . . . . . . 8 (𝑧 = ⟨𝑓, 𝑥⟩ → (2nd𝑧) = 𝑥)
16 eqidd 2807 . . . . . . . 8 (𝑧 = ⟨𝑓, 𝑥⟩ → 𝑦 = 𝑦)
1714, 15, 16oveq123d 6895 . . . . . . 7 (𝑧 = ⟨𝑓, 𝑥⟩ → ((2nd𝑧)(1st𝑧)𝑦) = (𝑥𝑓𝑦))
1817mpteq2dv 4939 . . . . . 6 (𝑧 = ⟨𝑓, 𝑥⟩ → (𝑦𝑌 ↦ ((2nd𝑧)(1st𝑧)𝑦)) = (𝑦𝑌 ↦ (𝑥𝑓𝑦)))
1918mpt2mpt 6982 . . . . 5 (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (𝑦𝑌 ↦ ((2nd𝑧)(1st𝑧)𝑦))) = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑥𝑓𝑦)))
20 txtopon 21608 . . . . . . 7 (((𝐿 ^ko (𝐽 ×t 𝐾)) ∈ (TopOn‘((𝐽 ×t 𝐾) Cn 𝐿)) ∧ 𝐽 ∈ (TopOn‘𝑋)) → ((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ∈ (TopOn‘(((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋)))
2111, 2, 20syl2anc 575 . . . . . 6 (𝜑 → ((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ∈ (TopOn‘(((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋)))
22 vex 3394 . . . . . . . . . . 11 𝑧 ∈ V
23 vex 3394 . . . . . . . . . . 11 𝑦 ∈ V
2422, 23op1std 7408 . . . . . . . . . 10 (𝑤 = ⟨𝑧, 𝑦⟩ → (1st𝑤) = 𝑧)
2524fveq2d 6412 . . . . . . . . 9 (𝑤 = ⟨𝑧, 𝑦⟩ → (1st ‘(1st𝑤)) = (1st𝑧))
2624fveq2d 6412 . . . . . . . . 9 (𝑤 = ⟨𝑧, 𝑦⟩ → (2nd ‘(1st𝑤)) = (2nd𝑧))
2722, 23op2ndd 7409 . . . . . . . . 9 (𝑤 = ⟨𝑧, 𝑦⟩ → (2nd𝑤) = 𝑦)
2825, 26, 27oveq123d 6895 . . . . . . . 8 (𝑤 = ⟨𝑧, 𝑦⟩ → ((2nd ‘(1st𝑤))(1st ‘(1st𝑤))(2nd𝑤)) = ((2nd𝑧)(1st𝑧)𝑦))
2928mpt2mpt 6982 . . . . . . 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 21608 . . . . . . . . 9 ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ∈ (TopOn‘(((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋)) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) ∈ (TopOn‘((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)))
3121, 3, 30syl2anc 575 . . . . . . . 8 (𝜑 → (((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) ∈ (TopOn‘((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)))
32 eqid 2806 . . . . . . . . . 10 𝐿 = 𝐿
3332toptopon 20935 . . . . . . . . 9 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
348, 33sylib 209 . . . . . . . 8 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
35 xkohmeo.j . . . . . . . . 9 (𝜑𝐽 ∈ 𝑛-Locally Comp)
36 xkohmeo.k . . . . . . . . 9 (𝜑𝐾 ∈ 𝑛-Locally Comp)
37 txcmp 21660 . . . . . . . . . 10 ((𝑥 ∈ Comp ∧ 𝑦 ∈ Comp) → (𝑥 ×t 𝑦) ∈ Comp)
3837txnlly 21654 . . . . . . . . 9 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐾 ∈ 𝑛-Locally Comp) → (𝐽 ×t 𝐾) ∈ 𝑛-Locally Comp)
3935, 36, 38syl2anc 575 . . . . . . . 8 (𝜑 → (𝐽 ×t 𝐾) ∈ 𝑛-Locally Comp)
4025mpt2mpt 6982 . . . . . . . . . 10 (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (1st ‘(1st𝑤))) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌 ↦ (1st𝑧))
415adantr 468 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
4234adantr 468 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → 𝐿 ∈ (TopOn‘ 𝐿))
43 xp1st 7430 . . . . . . . . . . . . . . 15 (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) → (1st𝑤) ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋))
4443adantl 469 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1st𝑤) ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋))
45 xp1st 7430 . . . . . . . . . . . . . 14 ((1st𝑤) ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) → (1st ‘(1st𝑤)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
4644, 45syl 17 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1st ‘(1st𝑤)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
47 cnf2 21267 . . . . . . . . . . . . 13 (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (1st ‘(1st𝑤)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (1st ‘(1st𝑤)):(𝑋 × 𝑌)⟶ 𝐿)
4841, 42, 46, 47syl3anc 1483 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1st ‘(1st𝑤)):(𝑋 × 𝑌)⟶ 𝐿)
4948feqmptd 6470 . . . . . . . . . . 11 ((𝜑𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1st ‘(1st𝑤)) = (𝑢 ∈ (𝑋 × 𝑌) ↦ ((1st ‘(1st𝑤))‘𝑢)))
5049mpteq2dva 4938 . . . . . . . . . 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))))
5140, 50syl5eqr 2854 . . . . . . . . 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))))
5221, 3cnmpt1st 21685 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌𝑧) ∈ ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn ((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽)))
53 fveq2 6408 . . . . . . . . . . . 12 (𝑡 = 𝑧 → (1st𝑡) = (1st𝑧))
5453cbvmptv 4944 . . . . . . . . . . 11 (𝑡 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st𝑡)) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st𝑧))
5514mpt2mpt 6982 . . . . . . . . . . . 12 (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st𝑧)) = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥𝑋𝑓)
5611, 2cnmpt1st 21685 . . . . . . . . . . . 12 (𝜑 → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥𝑋𝑓) ∈ (((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿 ^ko (𝐽 ×t 𝐾))))
5755, 56syl5eqel 2889 . . . . . . . . . . 11 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st𝑧)) ∈ (((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿 ^ko (𝐽 ×t 𝐾))))
5854, 57syl5eqel 2889 . . . . . . . . . 10 (𝜑 → (𝑡 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st𝑡)) ∈ (((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿 ^ko (𝐽 ×t 𝐾))))
5921, 3, 52, 21, 58, 53cnmpt21 21688 . . . . . . . . 9 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌 ↦ (1st𝑧)) ∈ ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn (𝐿 ^ko (𝐽 ×t 𝐾))))
6051, 59eqeltrrd 2886 . . . . . . . 8 (𝜑 → (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (𝑢 ∈ (𝑋 × 𝑌) ↦ ((1st ‘(1st𝑤))‘𝑢))) ∈ ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn (𝐿 ^ko (𝐽 ×t 𝐾))))
6126mpt2mpt 6982 . . . . . . . . . 10 (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2nd ‘(1st𝑤))) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌 ↦ (2nd𝑧))
62 fveq2 6408 . . . . . . . . . . . . 13 (𝑡 = 𝑧 → (2nd𝑡) = (2nd𝑧))
6362cbvmptv 4944 . . . . . . . . . . . 12 (𝑡 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd𝑡)) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd𝑧))
6415mpt2mpt 6982 . . . . . . . . . . . . 13 (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd𝑧)) = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥𝑋𝑥)
6511, 2cnmpt2nd 21686 . . . . . . . . . . . . 13 (𝜑 → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥𝑋𝑥) ∈ (((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn 𝐽))
6664, 65syl5eqel 2889 . . . . . . . . . . . 12 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd𝑧)) ∈ (((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn 𝐽))
6763, 66syl5eqel 2889 . . . . . . . . . . 11 (𝜑 → (𝑡 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd𝑡)) ∈ (((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn 𝐽))
6821, 3, 52, 21, 67, 62cnmpt21 21688 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌 ↦ (2nd𝑧)) ∈ ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐽))
6961, 68syl5eqel 2889 . . . . . . . . 9 (𝜑 → (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2nd ‘(1st𝑤))) ∈ ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐽))
7027mpt2mpt 6982 . . . . . . . . . 10 (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2nd𝑤)) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌𝑦)
7121, 3cnmpt2nd 21686 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌𝑦) ∈ ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐾))
7270, 71syl5eqel 2889 . . . . . . . . 9 (𝜑 → (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2nd𝑤)) ∈ ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐾))
7331, 69, 72cnmpt1t 21682 . . . . . . . 8 (𝜑 → (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ ⟨(2nd ‘(1st𝑤)), (2nd𝑤)⟩) ∈ ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn (𝐽 ×t 𝐾)))
74 fveq2 6408 . . . . . . . . 9 (𝑢 = ⟨(2nd ‘(1st𝑤)), (2nd𝑤)⟩ → ((1st ‘(1st𝑤))‘𝑢) = ((1st ‘(1st𝑤))‘⟨(2nd ‘(1st𝑤)), (2nd𝑤)⟩))
75 df-ov 6877 . . . . . . . . 9 ((2nd ‘(1st𝑤))(1st ‘(1st𝑤))(2nd𝑤)) = ((1st ‘(1st𝑤))‘⟨(2nd ‘(1st𝑤)), (2nd𝑤)⟩)
7674, 75syl6eqr 2858 . . . . . . . 8 (𝑢 = ⟨(2nd ‘(1st𝑤)), (2nd𝑤)⟩ → ((1st ‘(1st𝑤))‘𝑢) = ((2nd ‘(1st𝑤))(1st ‘(1st𝑤))(2nd𝑤)))
7731, 5, 34, 39, 60, 73, 76cnmptk1p 21702 . . . . . . 7 (𝜑 → (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ ((2nd ‘(1st𝑤))(1st ‘(1st𝑤))(2nd𝑤))) ∈ ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐿))
7829, 77syl5eqelr 2890 . . . . . 6 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦𝑌 ↦ ((2nd𝑧)(1st𝑧)𝑦)) ∈ ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐿))
7921, 3, 78cnmpt2k 21705 . . . . 5 (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (𝑦𝑌 ↦ ((2nd𝑧)(1st𝑧)𝑦))) ∈ (((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿 ^ko 𝐾)))
8019, 79syl5eqelr 2890 . . . 4 (𝜑 → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑥𝑓𝑦))) ∈ (((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿 ^ko 𝐾)))
8111, 2, 80cnmpt2k 21705 . . 3 (𝜑 → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑥𝑓𝑦)))) ∈ ((𝐿 ^ko (𝐽 ×t 𝐾)) Cn ((𝐿 ^ko 𝐾) ^ko 𝐽)))
821, 81syl5eqel 2889 . 2 (𝜑𝐹 ∈ ((𝐿 ^ko (𝐽 ×t 𝐾)) Cn ((𝐿 ^ko 𝐾) ^ko 𝐽)))
832, 3, 1, 35, 36, 8xkocnv 21831 . . . 4 (𝜑𝐹 = (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ↦ (𝑥𝑋, 𝑦𝑌 ↦ ((𝑔𝑥)‘𝑦))))
8413, 23op1std 7408 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
8584fveq2d 6412 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑔‘(1st𝑧)) = (𝑔𝑥))
8613, 23op2ndd 7409 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
8785, 86fveq12d 6415 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑔‘(1st𝑧))‘(2nd𝑧)) = ((𝑔𝑥)‘𝑦))
8887mpt2mpt 6982 . . . . 5 (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st𝑧))‘(2nd𝑧))) = (𝑥𝑋, 𝑦𝑌 ↦ ((𝑔𝑥)‘𝑦))
8988mpteq2i 4935 . . . 4 (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ↦ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st𝑧))‘(2nd𝑧)))) = (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ↦ (𝑥𝑋, 𝑦𝑌 ↦ ((𝑔𝑥)‘𝑦)))
9083, 89syl6eqr 2858 . . 3 (𝜑𝐹 = (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ↦ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st𝑧))‘(2nd𝑧)))))
91 nllytop 21490 . . . . . 6 (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ Top)
9235, 91syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
93 nllytop 21490 . . . . . . 7 (𝐾 ∈ 𝑛-Locally Comp → 𝐾 ∈ Top)
9436, 93syl 17 . . . . . 6 (𝜑𝐾 ∈ Top)
95 xkotop 21605 . . . . . 6 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ^ko 𝐾) ∈ Top)
9694, 8, 95syl2anc 575 . . . . 5 (𝜑 → (𝐿 ^ko 𝐾) ∈ Top)
97 eqid 2806 . . . . . 6 ((𝐿 ^ko 𝐾) ^ko 𝐽) = ((𝐿 ^ko 𝐾) ^ko 𝐽)
9897xkotopon 21617 . . . . 5 ((𝐽 ∈ Top ∧ (𝐿 ^ko 𝐾) ∈ Top) → ((𝐿 ^ko 𝐾) ^ko 𝐽) ∈ (TopOn‘(𝐽 Cn (𝐿 ^ko 𝐾))))
9992, 96, 98syl2anc 575 . . . 4 (𝜑 → ((𝐿 ^ko 𝐾) ^ko 𝐽) ∈ (TopOn‘(𝐽 Cn (𝐿 ^ko 𝐾))))
100 vex 3394 . . . . . . . . 9 𝑔 ∈ V
101100, 22op1std 7408 . . . . . . . 8 (𝑤 = ⟨𝑔, 𝑧⟩ → (1st𝑤) = 𝑔)
102100, 22op2ndd 7409 . . . . . . . . 9 (𝑤 = ⟨𝑔, 𝑧⟩ → (2nd𝑤) = 𝑧)
103102fveq2d 6412 . . . . . . . 8 (𝑤 = ⟨𝑔, 𝑧⟩ → (1st ‘(2nd𝑤)) = (1st𝑧))
104101, 103fveq12d 6415 . . . . . . 7 (𝑤 = ⟨𝑔, 𝑧⟩ → ((1st𝑤)‘(1st ‘(2nd𝑤))) = (𝑔‘(1st𝑧)))
105102fveq2d 6412 . . . . . . 7 (𝑤 = ⟨𝑔, 𝑧⟩ → (2nd ‘(2nd𝑤)) = (2nd𝑧))
106104, 105fveq12d 6415 . . . . . 6 (𝑤 = ⟨𝑔, 𝑧⟩ → (((1st𝑤)‘(1st ‘(2nd𝑤)))‘(2nd ‘(2nd𝑤))) = ((𝑔‘(1st𝑧))‘(2nd𝑧)))
107106mpt2mpt 6982 . . . . 5 (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (((1st𝑤)‘(1st ‘(2nd𝑤)))‘(2nd ‘(2nd𝑤)))) = (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st𝑧))‘(2nd𝑧)))
108 txtopon 21608 . . . . . . 7 ((((𝐿 ^ko 𝐾) ^ko 𝐽) ∈ (TopOn‘(𝐽 Cn (𝐿 ^ko 𝐾))) ∧ (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) → (((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) ∈ (TopOn‘((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))))
10999, 5, 108syl2anc 575 . . . . . 6 (𝜑 → (((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) ∈ (TopOn‘((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))))
1103adantr 468 . . . . . . . . . 10 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → 𝐾 ∈ (TopOn‘𝑌))
11134adantr 468 . . . . . . . . . 10 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → 𝐿 ∈ (TopOn‘ 𝐿))
1122adantr 468 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → 𝐽 ∈ (TopOn‘𝑋))
113 eqid 2806 . . . . . . . . . . . . . . 15 (𝐿 ^ko 𝐾) = (𝐿 ^ko 𝐾)
114113xkotopon 21617 . . . . . . . . . . . . . 14 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
11594, 8, 114syl2anc 575 . . . . . . . . . . . . 13 (𝜑 → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
116115adantr 468 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
117 xp1st 7430 . . . . . . . . . . . . 13 (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) → (1st𝑤) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
118117adantl 469 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → (1st𝑤) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
119 cnf2 21267 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ (1st𝑤) ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) → (1st𝑤):𝑋⟶(𝐾 Cn 𝐿))
120112, 116, 118, 119syl3anc 1483 . . . . . . . . . . 11 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → (1st𝑤):𝑋⟶(𝐾 Cn 𝐿))
121 xp2nd 7431 . . . . . . . . . . . . 13 (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) → (2nd𝑤) ∈ (𝑋 × 𝑌))
122121adantl 469 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → (2nd𝑤) ∈ (𝑋 × 𝑌))
123 xp1st 7430 . . . . . . . . . . . 12 ((2nd𝑤) ∈ (𝑋 × 𝑌) → (1st ‘(2nd𝑤)) ∈ 𝑋)
124122, 123syl 17 . . . . . . . . . . 11 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → (1st ‘(2nd𝑤)) ∈ 𝑋)
125120, 124ffvelrnd 6582 . . . . . . . . . 10 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → ((1st𝑤)‘(1st ‘(2nd𝑤))) ∈ (𝐾 Cn 𝐿))
126 cnf2 21267 . . . . . . . . . 10 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ ((1st𝑤)‘(1st ‘(2nd𝑤))) ∈ (𝐾 Cn 𝐿)) → ((1st𝑤)‘(1st ‘(2nd𝑤))):𝑌 𝐿)
127110, 111, 125, 126syl3anc 1483 . . . . . . . . 9 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → ((1st𝑤)‘(1st ‘(2nd𝑤))):𝑌 𝐿)
128127feqmptd 6470 . . . . . . . 8 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → ((1st𝑤)‘(1st ‘(2nd𝑤))) = (𝑦𝑌 ↦ (((1st𝑤)‘(1st ‘(2nd𝑤)))‘𝑦)))
129128mpteq2dva 4938 . . . . . . 7 (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ ((1st𝑤)‘(1st ‘(2nd𝑤)))) = (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑦𝑌 ↦ (((1st𝑤)‘(1st ‘(2nd𝑤)))‘𝑦))))
130101mpt2mpt 6982 . . . . . . . . . 10 (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1st𝑤)) = (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑔)
131120feqmptd 6470 . . . . . . . . . . 11 ((𝜑𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → (1st𝑤) = (𝑥𝑋 ↦ ((1st𝑤)‘𝑥)))
132131mpteq2dva 4938 . . . . . . . . . 10 (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1st𝑤)) = (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑥𝑋 ↦ ((1st𝑤)‘𝑥))))
133130, 132syl5eqr 2854 . . . . . . . . 9 (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑔) = (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑥𝑋 ↦ ((1st𝑤)‘𝑥))))
13499, 5cnmpt1st 21685 . . . . . . . . 9 (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑔) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn ((𝐿 ^ko 𝐾) ^ko 𝐽)))
135133, 134eqeltrrd 2886 . . . . . . . 8 (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑥𝑋 ↦ ((1st𝑤)‘𝑥))) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn ((𝐿 ^ko 𝐾) ^ko 𝐽)))
136103mpt2mpt 6982 . . . . . . . . 9 (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1st ‘(2nd𝑤))) = (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧))
13799, 5cnmpt2nd 21686 . . . . . . . . . 10 (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn (𝐽 ×t 𝐾)))
13853cbvmptv 4944 . . . . . . . . . . 11 (𝑡 ∈ (𝑋 × 𝑌) ↦ (1st𝑡)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧))
13984mpt2mpt 6982 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧)) = (𝑥𝑋, 𝑦𝑌𝑥)
1402, 3cnmpt1st 21685 . . . . . . . . . . . 12 (𝜑 → (𝑥𝑋, 𝑦𝑌𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
141139, 140syl5eqel 2889 . . . . . . . . . . 11 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
142138, 141syl5eqel 2889 . . . . . . . . . 10 (𝜑 → (𝑡 ∈ (𝑋 × 𝑌) ↦ (1st𝑡)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
14399, 5, 137, 5, 142, 53cnmpt21 21688 . . . . . . . . 9 (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧)) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐽))
144136, 143syl5eqel 2889 . . . . . . . 8 (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1st ‘(2nd𝑤))) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐽))
145 fveq2 6408 . . . . . . . 8 (𝑥 = (1st ‘(2nd𝑤)) → ((1st𝑤)‘𝑥) = ((1st𝑤)‘(1st ‘(2nd𝑤))))
146109, 2, 115, 35, 135, 144, 145cnmptk1p 21702 . . . . . . 7 (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ ((1st𝑤)‘(1st ‘(2nd𝑤)))) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn (𝐿 ^ko 𝐾)))
147129, 146eqeltrrd 2886 . . . . . 6 (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑦𝑌 ↦ (((1st𝑤)‘(1st ‘(2nd𝑤)))‘𝑦))) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn (𝐿 ^ko 𝐾)))
148105mpt2mpt 6982 . . . . . . 7 (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (2nd ‘(2nd𝑤))) = (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧))
14962cbvmptv 4944 . . . . . . . . 9 (𝑡 ∈ (𝑋 × 𝑌) ↦ (2nd𝑡)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧))
15086mpt2mpt 6982 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧)) = (𝑥𝑋, 𝑦𝑌𝑦)
1512, 3cnmpt2nd 21686 . . . . . . . . . 10 (𝜑 → (𝑥𝑋, 𝑦𝑌𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
152150, 151syl5eqel 2889 . . . . . . . . 9 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
153149, 152syl5eqel 2889 . . . . . . . 8 (𝜑 → (𝑡 ∈ (𝑋 × 𝑌) ↦ (2nd𝑡)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
15499, 5, 137, 5, 153, 62cnmpt21 21688 . . . . . . 7 (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧)) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐾))
155148, 154syl5eqel 2889 . . . . . 6 (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (2nd ‘(2nd𝑤))) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐾))
156 fveq2 6408 . . . . . 6 (𝑦 = (2nd ‘(2nd𝑤)) → (((1st𝑤)‘(1st ‘(2nd𝑤)))‘𝑦) = (((1st𝑤)‘(1st ‘(2nd𝑤)))‘(2nd ‘(2nd𝑤))))
157109, 3, 34, 36, 147, 155, 156cnmptk1p 21702 . . . . 5 (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (((1st𝑤)‘(1st ‘(2nd𝑤)))‘(2nd ‘(2nd𝑤)))) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐿))
158107, 157syl5eqelr 2890 . . . 4 (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st𝑧))‘(2nd𝑧))) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐿))
15999, 5, 158cnmpt2k 21705 . . 3 (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ↦ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st𝑧))‘(2nd𝑧)))) ∈ (((𝐿 ^ko 𝐾) ^ko 𝐽) Cn (𝐿 ^ko (𝐽 ×t 𝐾))))
16090, 159eqeltrd 2885 . 2 (𝜑𝐹 ∈ (((𝐿 ^ko 𝐾) ^ko 𝐽) Cn (𝐿 ^ko (𝐽 ×t 𝐾))))
161 ishmeo 21776 . 2 (𝐹 ∈ ((𝐿 ^ko (𝐽 ×t 𝐾))Homeo((𝐿 ^ko 𝐾) ^ko 𝐽)) ↔ (𝐹 ∈ ((𝐿 ^ko (𝐽 ×t 𝐾)) Cn ((𝐿 ^ko 𝐾) ^ko 𝐽)) ∧ 𝐹 ∈ (((𝐿 ^ko 𝐾) ^ko 𝐽) Cn (𝐿 ^ko (𝐽 ×t 𝐾)))))
16282, 160, 161sylanbrc 574 1 (𝜑𝐹 ∈ ((𝐿 ^ko (𝐽 ×t 𝐾))Homeo((𝐿 ^ko 𝐾) ^ko 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2156  cop 4376   cuni 4630  cmpt 4923   × cxp 5309  ccnv 5310  wf 6097  cfv 6101  (class class class)co 6874  cmpt2 6876  1st c1st 7396  2nd c2nd 7397  Topctop 20911  TopOnctopon 20928   Cn ccn 21242  Compccmp 21403  𝑛-Locally cnlly 21482   ×t ctx 21577   ^ko cxko 21578  Homeochmeo 21770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-iin 4715  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-1st 7398  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-2o 7797  df-oadd 7800  df-er 7979  df-map 8094  df-ixp 8146  df-en 8193  df-dom 8194  df-sdom 8195  df-fin 8196  df-fi 8556  df-rest 16288  df-topgen 16309  df-pt 16310  df-top 20912  df-topon 20929  df-bases 20964  df-ntr 21038  df-nei 21116  df-cn 21245  df-cnp 21246  df-cmp 21404  df-nlly 21484  df-tx 21579  df-xko 21580  df-hmeo 21772
This theorem is referenced by: (None)
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