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Theorem cvmlift3 32582
Description: A general version of cvmlift 32553. 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 2833 . . . . . . . 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 3283 . . . . . 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 7098 . . . . 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 6642 . . . . . . . . . 10 (𝑐 = 𝑓 → (𝑐‘0) = (𝑓‘0))
1514eqeq1d 2823 . . . . . . . . 9 (𝑐 = 𝑓 → ((𝑐‘0) = 𝑂 ↔ (𝑓‘0) = 𝑂))
16 fveq1 6642 . . . . . . . . . 10 (𝑐 = 𝑓 → (𝑐‘1) = (𝑓‘1))
1716eqeq1d 2823 . . . . . . . . 9 (𝑐 = 𝑓 → ((𝑐‘1) = 𝑎 ↔ (𝑓‘1) = 𝑎))
18 coeq2 5702 . . . . . . . . . . . . . . 15 (𝑑 = 𝑔 → (𝐹𝑑) = (𝐹𝑔))
1918eqeq1d 2823 . . . . . . . . . . . . . 14 (𝑑 = 𝑔 → ((𝐹𝑑) = (𝐺𝑐) ↔ (𝐹𝑔) = (𝐺𝑐)))
20 fveq1 6642 . . . . . . . . . . . . . . 15 (𝑑 = 𝑔 → (𝑑‘0) = (𝑔‘0))
2120eqeq1d 2823 . . . . . . . . . . . . . 14 (𝑑 = 𝑔 → ((𝑑‘0) = 𝑃 ↔ (𝑔‘0) = 𝑃))
2219, 21anbi12d 633 . . . . . . . . . . . . 13 (𝑑 = 𝑔 → (((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺𝑐) ∧ (𝑔‘0) = 𝑃)))
2322cbvriotavw 7098 . . . . . . . . . . . 12 (𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑐) ∧ (𝑔‘0) = 𝑃))
24 coeq2 5702 . . . . . . . . . . . . . . 15 (𝑐 = 𝑓 → (𝐺𝑐) = (𝐺𝑓))
2524eqeq2d 2832 . . . . . . . . . . . . . 14 (𝑐 = 𝑓 → ((𝐹𝑔) = (𝐺𝑐) ↔ (𝐹𝑔) = (𝐺𝑓)))
2625anbi1d 632 . . . . . . . . . . . . 13 (𝑐 = 𝑓 → (((𝐹𝑔) = (𝐺𝑐) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)))
2726riotabidv 7090 . . . . . . . . . . . 12 (𝑐 = 𝑓 → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑐) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)))
2823, 27syl5eq 2868 . . . . . . . . . . 11 (𝑐 = 𝑓 → (𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)))
2928fveq1d 6645 . . . . . . . . . 10 (𝑐 = 𝑓 → ((𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1))
3029eqeq1d 2823 . . . . . . . . 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 3427 . . . . . . 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 2833 . . . . . . . . 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 3283 . . . . . . 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 286 . . . . . 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 7090 . . . . 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 2868 . . . 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 5142 . . 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 2821 . . . 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 32512 . . 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 32581 . 2 (𝜑 → ∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
43 sconnpconn 32481 . . . 4 (𝐾 ∈ SConn → 𝐾 ∈ PConn)
44 pconnconn 32485 . . . 4 (𝐾 ∈ PConn → 𝐾 ∈ Conn)
454, 43, 443syl 18 . . 3 (𝜑𝐾 ∈ Conn)
46 pconnconn 32485 . . . . . 6 (𝑥 ∈ PConn → 𝑥 ∈ Conn)
4746ssriv 3947 . . . . 5 PConn ⊆ Conn
48 nllyss 22063 . . . . 5 (PConn ⊆ Conn → 𝑛-Locally PConn ⊆ 𝑛-Locally Conn)
4947, 48ax-mp 5 . . . 4 𝑛-Locally PConn ⊆ 𝑛-Locally Conn
5049, 5sseldi 3941 . . 3 (𝜑𝐾 ∈ 𝑛-Locally Conn)
511, 2, 3, 45, 50, 6, 7, 8, 9cvmliftmo 32538 . 2 (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
52 reu5 3407 . 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 586 1 (𝜑 → ∃!𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3126  wrex 3127  ∃!wreu 3128  ∃*wrmo 3129  {crab 3130  cdif 3907  cin 3909  wss 3910  c0 4266  𝒫 cpw 4512  {csn 4540   cuni 4811  cmpt 5119  ccnv 5527  cres 5530  cima 5531  ccom 5532  cfv 6328  crio 7087  (class class class)co 7130  0cc0 10514  1c1 10515  t crest 16672   Cn ccn 21807  Conncconn 21994  𝑛-Locally cnlly 22048  Homeochmeo 22336  IIcii 23458  PConncpconn 32473  SConncsconn 32474   CovMap ccvm 32509
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 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-inf2 9080  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591  ax-pre-sup 10592  ax-addf 10593  ax-mulf 10594
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-iin 4895  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-of 7384  df-om 7556  df-1st 7664  df-2nd 7665  df-supp 7806  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-2o 8078  df-oadd 8081  df-er 8264  df-ec 8266  df-map 8383  df-ixp 8437  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-fsupp 8810  df-fi 8851  df-sup 8882  df-inf 8883  df-oi 8950  df-card 9344  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-div 11275  df-nn 11616  df-2 11678  df-3 11679  df-4 11680  df-5 11681  df-6 11682  df-7 11683  df-8 11684  df-9 11685  df-n0 11876  df-z 11960  df-dec 12077  df-uz 12222  df-q 12327  df-rp 12368  df-xneg 12485  df-xadd 12486  df-xmul 12487  df-ioo 12720  df-ico 12722  df-icc 12723  df-fz 12876  df-fzo 13017  df-fl 13145  df-seq 13353  df-exp 13414  df-hash 13675  df-cj 14437  df-re 14438  df-im 14439  df-sqrt 14573  df-abs 14574  df-clim 14824  df-sum 15022  df-struct 16463  df-ndx 16464  df-slot 16465  df-base 16467  df-sets 16468  df-ress 16469  df-plusg 16556  df-mulr 16557  df-starv 16558  df-sca 16559  df-vsca 16560  df-ip 16561  df-tset 16562  df-ple 16563  df-ds 16565  df-unif 16566  df-hom 16567  df-cco 16568  df-rest 16674  df-topn 16675  df-0g 16693  df-gsum 16694  df-topgen 16695  df-pt 16696  df-prds 16699  df-xrs 16753  df-qtop 16758  df-imas 16759  df-xps 16761  df-mre 16835  df-mrc 16836  df-acs 16838  df-mgm 17830  df-sgrp 17879  df-mnd 17890  df-submnd 17935  df-mulg 18203  df-cntz 18425  df-cmn 18886  df-psmet 20512  df-xmet 20513  df-met 20514  df-bl 20515  df-mopn 20516  df-cnfld 20521  df-top 21477  df-topon 21494  df-topsp 21516  df-bases 21529  df-cld 21602  df-ntr 21603  df-cls 21604  df-nei 21681  df-cn 21810  df-cnp 21811  df-cmp 21970  df-conn 21995  df-lly 22049  df-nlly 22050  df-tx 22145  df-hmeo 22338  df-xms 22905  df-ms 22906  df-tms 22907  df-ii 23460  df-htpy 23553  df-phtpy 23554  df-phtpc 23575  df-pco 23588  df-pconn 32475  df-sconn 32476  df-cvm 32510
This theorem is referenced by: (None)
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