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Theorem cvmlift3 31436
Description: A general version of cvmlift 31407. 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 2662 . . . . . . . 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 1445 . . . . . . 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 3081 . . . . . 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)))
1312cbvriotav 6662 . . . . 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 6228 . . . . . . . . . 10 (𝑐 = 𝑓 → (𝑐‘0) = (𝑓‘0))
1514eqeq1d 2653 . . . . . . . . 9 (𝑐 = 𝑓 → ((𝑐‘0) = 𝑂 ↔ (𝑓‘0) = 𝑂))
16 fveq1 6228 . . . . . . . . . 10 (𝑐 = 𝑓 → (𝑐‘1) = (𝑓‘1))
1716eqeq1d 2653 . . . . . . . . 9 (𝑐 = 𝑓 → ((𝑐‘1) = 𝑎 ↔ (𝑓‘1) = 𝑎))
18 coeq2 5313 . . . . . . . . . . . . . . 15 (𝑑 = 𝑔 → (𝐹𝑑) = (𝐹𝑔))
1918eqeq1d 2653 . . . . . . . . . . . . . 14 (𝑑 = 𝑔 → ((𝐹𝑑) = (𝐺𝑐) ↔ (𝐹𝑔) = (𝐺𝑐)))
20 fveq1 6228 . . . . . . . . . . . . . . 15 (𝑑 = 𝑔 → (𝑑‘0) = (𝑔‘0))
2120eqeq1d 2653 . . . . . . . . . . . . . 14 (𝑑 = 𝑔 → ((𝑑‘0) = 𝑃 ↔ (𝑔‘0) = 𝑃))
2219, 21anbi12d 747 . . . . . . . . . . . . 13 (𝑑 = 𝑔 → (((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺𝑐) ∧ (𝑔‘0) = 𝑃)))
2322cbvriotav 6662 . . . . . . . . . . . 12 (𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑐) ∧ (𝑔‘0) = 𝑃))
24 coeq2 5313 . . . . . . . . . . . . . . 15 (𝑐 = 𝑓 → (𝐺𝑐) = (𝐺𝑓))
2524eqeq2d 2661 . . . . . . . . . . . . . 14 (𝑐 = 𝑓 → ((𝐹𝑔) = (𝐺𝑐) ↔ (𝐹𝑔) = (𝐺𝑓)))
2625anbi1d 741 . . . . . . . . . . . . 13 (𝑐 = 𝑓 → (((𝐹𝑔) = (𝐺𝑐) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)))
2726riotabidv 6653 . . . . . . . . . . . 12 (𝑐 = 𝑓 → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑐) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)))
2823, 27syl5eq 2697 . . . . . . . . . . 11 (𝑐 = 𝑓 → (𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)))
2928fveq1d 6231 . . . . . . . . . 10 (𝑐 = 𝑓 → ((𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1))
3029eqeq1d 2653 . . . . . . . . 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 1439 . . . . . . . 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)))
3231cbvrexv 3202 . . . . . . 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 2662 . . . . . . . . 9 (𝑎 = 𝑥 → ((𝑓‘1) = 𝑎 ↔ (𝑓‘1) = 𝑥))
34333anbi2d 1444 . . . . . . . 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 3081 . . . . . . 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, 35syl5bb 272 . . . . . 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 6653 . . . . 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, 37syl5eq 2697 . . . 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 4783 . . 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 2651 . . . 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 31366 . . 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 31435 . 2 (𝜑 → ∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
43 sconnpconn 31335 . . . 4 (𝐾 ∈ SConn → 𝐾 ∈ PConn)
44 pconnconn 31339 . . . 4 (𝐾 ∈ PConn → 𝐾 ∈ Conn)
454, 43, 443syl 18 . . 3 (𝜑𝐾 ∈ Conn)
46 pconnconn 31339 . . . . . 6 (𝑥 ∈ PConn → 𝑥 ∈ Conn)
4746ssriv 3640 . . . . 5 PConn ⊆ Conn
48 nllyss 21331 . . . . 5 (PConn ⊆ Conn → 𝑛-Locally PConn ⊆ 𝑛-Locally Conn)
4947, 48ax-mp 5 . . . 4 𝑛-Locally PConn ⊆ 𝑛-Locally Conn
5049, 5sseldi 3634 . . 3 (𝜑𝐾 ∈ 𝑛-Locally Conn)
511, 2, 3, 45, 50, 6, 7, 8, 9cvmliftmo 31392 . 2 (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
52 reu5 3189 . 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 699 1 (𝜑 → ∃!𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  ∃!wreu 2943  ∃*wrmo 2944  {crab 2945  cdif 3604  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210   cuni 4468  cmpt 4762  ccnv 5142  cres 5145  cima 5146  ccom 5147  cfv 5926  crio 6650  (class class class)co 6690  0cc0 9974  1c1 9975  t crest 16128   Cn ccn 21076  Conncconn 21262  𝑛-Locally cnlly 21316  Homeochmeo 21604  IIcii 22725  PConncpconn 31327  SConncsconn 31328   CovMap ccvm 31363
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  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  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-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  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-se 5103  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-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-ec 7789  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-cn 21079  df-cnp 21080  df-cmp 21238  df-conn 21263  df-lly 21317  df-nlly 21318  df-tx 21413  df-hmeo 21606  df-xms 22172  df-ms 22173  df-tms 22174  df-ii 22727  df-htpy 22816  df-phtpy 22817  df-phtpc 22838  df-pco 22851  df-pconn 31329  df-sconn 31330  df-cvm 31364
This theorem is referenced by: (None)
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