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Theorem cvmlift3 34938
Description: A general version of cvmlift 34909. If 𝐾 is simply connected and weakly locally path-connected, then there is a unique lift of functions on 𝐾 which commutes with the covering map. (Contributed by Mario Carneiro, 9-Jul-2015.)
Hypotheses
Ref Expression
cvmlift3.b 𝐵 = 𝐶
cvmlift3.y 𝑌 = 𝐾
cvmlift3.f (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
cvmlift3.k (𝜑𝐾 ∈ SConn)
cvmlift3.l (𝜑𝐾 ∈ 𝑛-Locally PConn)
cvmlift3.o (𝜑𝑂𝑌)
cvmlift3.g (𝜑𝐺 ∈ (𝐾 Cn 𝐽))
cvmlift3.p (𝜑𝑃𝐵)
cvmlift3.e (𝜑 → (𝐹𝑃) = (𝐺𝑂))
Assertion
Ref Expression
cvmlift3 (𝜑 → ∃!𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
Distinct variable groups:   𝑓,𝐽   𝑓,𝐹   𝐵,𝑓   𝑓,𝐺   𝐶,𝑓   𝜑,𝑓   𝑓,𝐾   𝑃,𝑓   𝑓,𝑂   𝑓,𝑌

Proof of Theorem cvmlift3
Dummy variables 𝑏 𝑐 𝑑 𝑘 𝑠 𝑧 𝑔 𝑎 𝑢 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvmlift3.b . . 3 𝐵 = 𝐶
2 cvmlift3.y . . 3 𝑌 = 𝐾
3 cvmlift3.f . . 3 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
4 cvmlift3.k . . 3 (𝜑𝐾 ∈ SConn)
5 cvmlift3.l . . 3 (𝜑𝐾 ∈ 𝑛-Locally PConn)
6 cvmlift3.o . . 3 (𝜑𝑂𝑌)
7 cvmlift3.g . . 3 (𝜑𝐺 ∈ (𝐾 Cn 𝐽))
8 cvmlift3.p . . 3 (𝜑𝑃𝐵)
9 cvmlift3.e . . 3 (𝜑 → (𝐹𝑃) = (𝐺𝑂))
10 eqeq2 2740 . . . . . . . 8 (𝑏 = 𝑧 → (((𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑏 ↔ ((𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧))
11103anbi3d 1439 . . . . . . 7 (𝑏 = 𝑧 → (((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑏) ↔ ((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧)))
1211rexbidv 3175 . . . . . 6 (𝑏 = 𝑧 → (∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑏) ↔ ∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧)))
1312cbvriotavw 7386 . . . . 5 (𝑏𝐵𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑏)) = (𝑧𝐵𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧))
14 fveq1 6896 . . . . . . . . . 10 (𝑐 = 𝑓 → (𝑐‘0) = (𝑓‘0))
1514eqeq1d 2730 . . . . . . . . 9 (𝑐 = 𝑓 → ((𝑐‘0) = 𝑂 ↔ (𝑓‘0) = 𝑂))
16 fveq1 6896 . . . . . . . . . 10 (𝑐 = 𝑓 → (𝑐‘1) = (𝑓‘1))
1716eqeq1d 2730 . . . . . . . . 9 (𝑐 = 𝑓 → ((𝑐‘1) = 𝑎 ↔ (𝑓‘1) = 𝑎))
18 coeq2 5861 . . . . . . . . . . . . . . 15 (𝑑 = 𝑔 → (𝐹𝑑) = (𝐹𝑔))
1918eqeq1d 2730 . . . . . . . . . . . . . 14 (𝑑 = 𝑔 → ((𝐹𝑑) = (𝐺𝑐) ↔ (𝐹𝑔) = (𝐺𝑐)))
20 fveq1 6896 . . . . . . . . . . . . . . 15 (𝑑 = 𝑔 → (𝑑‘0) = (𝑔‘0))
2120eqeq1d 2730 . . . . . . . . . . . . . 14 (𝑑 = 𝑔 → ((𝑑‘0) = 𝑃 ↔ (𝑔‘0) = 𝑃))
2219, 21anbi12d 631 . . . . . . . . . . . . 13 (𝑑 = 𝑔 → (((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺𝑐) ∧ (𝑔‘0) = 𝑃)))
2322cbvriotavw 7386 . . . . . . . . . . . 12 (𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑐) ∧ (𝑔‘0) = 𝑃))
24 coeq2 5861 . . . . . . . . . . . . . . 15 (𝑐 = 𝑓 → (𝐺𝑐) = (𝐺𝑓))
2524eqeq2d 2739 . . . . . . . . . . . . . 14 (𝑐 = 𝑓 → ((𝐹𝑔) = (𝐺𝑐) ↔ (𝐹𝑔) = (𝐺𝑓)))
2625anbi1d 630 . . . . . . . . . . . . 13 (𝑐 = 𝑓 → (((𝐹𝑔) = (𝐺𝑐) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)))
2726riotabidv 7378 . . . . . . . . . . . 12 (𝑐 = 𝑓 → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑐) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)))
2823, 27eqtrid 2780 . . . . . . . . . . 11 (𝑐 = 𝑓 → (𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)))
2928fveq1d 6899 . . . . . . . . . 10 (𝑐 = 𝑓 → ((𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1))
3029eqeq1d 2730 . . . . . . . . 9 (𝑐 = 𝑓 → (((𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧 ↔ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧))
3115, 17, 303anbi123d 1433 . . . . . . . 8 (𝑐 = 𝑓 → (((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧) ↔ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑎 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
3231cbvrexvw 3232 . . . . . . 7 (∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑎 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧))
33 eqeq2 2740 . . . . . . . . 9 (𝑎 = 𝑥 → ((𝑓‘1) = 𝑎 ↔ (𝑓‘1) = 𝑥))
34333anbi2d 1438 . . . . . . . 8 (𝑎 = 𝑥 → (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑎 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
3534rexbidv 3175 . . . . . . 7 (𝑎 = 𝑥 → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑎 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
3632, 35bitrid 283 . . . . . 6 (𝑎 = 𝑥 → (∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
3736riotabidv 7378 . . . . 5 (𝑎 = 𝑥 → (𝑧𝐵𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧)) = (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
3813, 37eqtrid 2780 . . . 4 (𝑎 = 𝑥 → (𝑏𝐵𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑏)) = (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
3938cbvmptv 5261 . . 3 (𝑎𝑌 ↦ (𝑏𝐵𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑏))) = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
40 eqid 2728 . . . 4 (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))}) = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})
4140cvmscbv 34868 . . 3 (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))}) = (𝑎𝐽 ↦ {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑏 = (𝐹𝑎) ∧ ∀𝑣𝑏 (∀𝑢 ∈ (𝑏 ∖ {𝑣})(𝑣𝑢) = ∅ ∧ (𝐹𝑣) ∈ ((𝐶t 𝑣)Homeo(𝐽t 𝑎))))})
421, 2, 3, 4, 5, 6, 7, 8, 9, 39, 41cvmlift3lem9 34937 . 2 (𝜑 → ∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
43 sconnpconn 34837 . . . 4 (𝐾 ∈ SConn → 𝐾 ∈ PConn)
44 pconnconn 34841 . . . 4 (𝐾 ∈ PConn → 𝐾 ∈ Conn)
454, 43, 443syl 18 . . 3 (𝜑𝐾 ∈ Conn)
46 pconnconn 34841 . . . . . 6 (𝑥 ∈ PConn → 𝑥 ∈ Conn)
4746ssriv 3984 . . . . 5 PConn ⊆ Conn
48 nllyss 23397 . . . . 5 (PConn ⊆ Conn → 𝑛-Locally PConn ⊆ 𝑛-Locally Conn)
4947, 48ax-mp 5 . . . 4 𝑛-Locally PConn ⊆ 𝑛-Locally Conn
5049, 5sselid 3978 . . 3 (𝜑𝐾 ∈ 𝑛-Locally Conn)
511, 2, 3, 45, 50, 6, 7, 8, 9cvmliftmo 34894 . 2 (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
52 reu5 3375 . 2 (∃!𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ↔ (∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃)))
5342, 51, 52sylanbrc 582 1 (𝜑 → ∃!𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  wral 3058  wrex 3067  ∃!wreu 3371  ∃*wrmo 3372  {crab 3429  cdif 3944  cin 3946  wss 3947  c0 4323  𝒫 cpw 4603  {csn 4629   cuni 4908  cmpt 5231  ccnv 5677  cres 5680  cima 5681  ccom 5682  cfv 6548  crio 7375  (class class class)co 7420  0cc0 11139  1c1 11140  t crest 17402   Cn ccn 23141  Conncconn 23328  𝑛-Locally cnlly 23382  Homeochmeo 23670  IIcii 24808  PConncpconn 34829  SConncsconn 34830   CovMap ccvm 34865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-inf2 9665  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216  ax-pre-sup 11217  ax-addf 11218
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7685  df-om 7871  df-1st 7993  df-2nd 7994  df-supp 8166  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-er 8725  df-ec 8727  df-map 8847  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9387  df-fi 9435  df-sup 9466  df-inf 9467  df-oi 9534  df-card 9963  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-div 11903  df-nn 12244  df-2 12306  df-3 12307  df-4 12308  df-5 12309  df-6 12310  df-7 12311  df-8 12312  df-9 12313  df-n0 12504  df-z 12590  df-dec 12709  df-uz 12854  df-q 12964  df-rp 13008  df-xneg 13125  df-xadd 13126  df-xmul 13127  df-ioo 13361  df-ico 13363  df-icc 13364  df-fz 13518  df-fzo 13661  df-fl 13790  df-seq 14000  df-exp 14060  df-hash 14323  df-cj 15079  df-re 15080  df-im 15081  df-sqrt 15215  df-abs 15216  df-clim 15465  df-sum 15666  df-struct 17116  df-sets 17133  df-slot 17151  df-ndx 17163  df-base 17181  df-ress 17210  df-plusg 17246  df-mulr 17247  df-starv 17248  df-sca 17249  df-vsca 17250  df-ip 17251  df-tset 17252  df-ple 17253  df-ds 17255  df-unif 17256  df-hom 17257  df-cco 17258  df-rest 17404  df-topn 17405  df-0g 17423  df-gsum 17424  df-topgen 17425  df-pt 17426  df-prds 17429  df-xrs 17484  df-qtop 17489  df-imas 17490  df-xps 17492  df-mre 17566  df-mrc 17567  df-acs 17569  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-submnd 18741  df-mulg 19024  df-cntz 19268  df-cmn 19737  df-psmet 21271  df-xmet 21272  df-met 21273  df-bl 21274  df-mopn 21275  df-cnfld 21280  df-top 22809  df-topon 22826  df-topsp 22848  df-bases 22862  df-cld 22936  df-ntr 22937  df-cls 22938  df-nei 23015  df-cn 23144  df-cnp 23145  df-cmp 23304  df-conn 23329  df-lly 23383  df-nlly 23384  df-tx 23479  df-hmeo 23672  df-xms 24239  df-ms 24240  df-tms 24241  df-ii 24810  df-cncf 24811  df-htpy 24909  df-phtpy 24910  df-phtpc 24931  df-pco 24945  df-pconn 34831  df-sconn 34832  df-cvm 34866
This theorem is referenced by: (None)
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