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Theorem cvmlift3 31018
 Description: A general version of cvmlift 30989. 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 2632 . . . . . . . 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 1402 . . . . . . 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 3045 . . . . . 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 6576 . . . . 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 6147 . . . . . . . . . 10 (𝑐 = 𝑓 → (𝑐‘0) = (𝑓‘0))
1514eqeq1d 2623 . . . . . . . . 9 (𝑐 = 𝑓 → ((𝑐‘0) = 𝑂 ↔ (𝑓‘0) = 𝑂))
16 fveq1 6147 . . . . . . . . . 10 (𝑐 = 𝑓 → (𝑐‘1) = (𝑓‘1))
1716eqeq1d 2623 . . . . . . . . 9 (𝑐 = 𝑓 → ((𝑐‘1) = 𝑎 ↔ (𝑓‘1) = 𝑎))
18 coeq2 5240 . . . . . . . . . . . . . . 15 (𝑑 = 𝑔 → (𝐹𝑑) = (𝐹𝑔))
1918eqeq1d 2623 . . . . . . . . . . . . . 14 (𝑑 = 𝑔 → ((𝐹𝑑) = (𝐺𝑐) ↔ (𝐹𝑔) = (𝐺𝑐)))
20 fveq1 6147 . . . . . . . . . . . . . . 15 (𝑑 = 𝑔 → (𝑑‘0) = (𝑔‘0))
2120eqeq1d 2623 . . . . . . . . . . . . . 14 (𝑑 = 𝑔 → ((𝑑‘0) = 𝑃 ↔ (𝑔‘0) = 𝑃))
2219, 21anbi12d 746 . . . . . . . . . . . . 13 (𝑑 = 𝑔 → (((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺𝑐) ∧ (𝑔‘0) = 𝑃)))
2322cbvriotav 6576 . . . . . . . . . . . 12 (𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑐) ∧ (𝑔‘0) = 𝑃))
24 coeq2 5240 . . . . . . . . . . . . . . 15 (𝑐 = 𝑓 → (𝐺𝑐) = (𝐺𝑓))
2524eqeq2d 2631 . . . . . . . . . . . . . 14 (𝑐 = 𝑓 → ((𝐹𝑔) = (𝐺𝑐) ↔ (𝐹𝑔) = (𝐺𝑓)))
2625anbi1d 740 . . . . . . . . . . . . 13 (𝑐 = 𝑓 → (((𝐹𝑔) = (𝐺𝑐) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)))
2726riotabidv 6567 . . . . . . . . . . . 12 (𝑐 = 𝑓 → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑐) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)))
2823, 27syl5eq 2667 . . . . . . . . . . 11 (𝑐 = 𝑓 → (𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)))
2928fveq1d 6150 . . . . . . . . . 10 (𝑐 = 𝑓 → ((𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1))
3029eqeq1d 2623 . . . . . . . . 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 1396 . . . . . . . 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 3160 . . . . . . 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 2632 . . . . . . . . 9 (𝑎 = 𝑥 → ((𝑓‘1) = 𝑎 ↔ (𝑓‘1) = 𝑥))
34333anbi2d 1401 . . . . . . . 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 3045 . . . . . . 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 6567 . . . . 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 2667 . . . 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 4710 . . 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 2621 . . . 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 30948 . . 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 31017 . 2 (𝜑 → ∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
43 sconnpconn 30917 . . . 4 (𝐾 ∈ SConn → 𝐾 ∈ PConn)
44 pconnconn 30921 . . . 4 (𝐾 ∈ PConn → 𝐾 ∈ Conn)
454, 43, 443syl 18 . . 3 (𝜑𝐾 ∈ Conn)
46 pconnconn 30921 . . . . . 6 (𝑥 ∈ PConn → 𝑥 ∈ Conn)
4746ssriv 3587 . . . . 5 PConn ⊆ Conn
48 nllyss 21193 . . . . 5 (PConn ⊆ Conn → 𝑛-Locally PConn ⊆ 𝑛-Locally Conn)
4947, 48ax-mp 5 . . . 4 𝑛-Locally PConn ⊆ 𝑛-Locally Conn
5049, 5sseldi 3581 . . 3 (𝜑𝐾 ∈ 𝑛-Locally Conn)
511, 2, 3, 45, 50, 6, 7, 8, 9cvmliftmo 30974 . 2 (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
52 reu5 3148 . 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 697 1 (𝜑 → ∃!𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908  ∃!wreu 2909  ∃*wrmo 2910  {crab 2911   ∖ cdif 3552   ∩ cin 3554   ⊆ wss 3555  ∅c0 3891  𝒫 cpw 4130  {csn 4148  ∪ cuni 4402   ↦ cmpt 4673  ◡ccnv 5073   ↾ cres 5076   “ cima 5077   ∘ ccom 5078  ‘cfv 5847  ℩crio 6564  (class class class)co 6604  0cc0 9880  1c1 9881   ↾t crest 16002   Cn ccn 20938  Conncconn 21124  𝑛-Locally cnlly 21178  Homeochmeo 21466  IIcii 22586  PConncpconn 30909  SConncsconn 30910   CovMap ccvm 30945 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-addf 9959  ax-mulf 9960 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-ec 7689  df-map 7804  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-fi 8261  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ioo 12121  df-ico 12123  df-icc 12124  df-fz 12269  df-fzo 12407  df-fl 12533  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-sum 14351  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-starv 15877  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-unif 15886  df-hom 15887  df-cco 15888  df-rest 16004  df-topn 16005  df-0g 16023  df-gsum 16024  df-topgen 16025  df-pt 16026  df-prds 16029  df-xrs 16083  df-qtop 16088  df-imas 16089  df-xps 16091  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-mulg 17462  df-cntz 17671  df-cmn 18116  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-cnfld 19666  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cld 20733  df-ntr 20734  df-cls 20735  df-nei 20812  df-cn 20941  df-cnp 20942  df-cmp 21100  df-conn 21125  df-lly 21179  df-nlly 21180  df-tx 21275  df-hmeo 21468  df-xms 22035  df-ms 22036  df-tms 22037  df-ii 22588  df-htpy 22677  df-phtpy 22678  df-phtpc 22699  df-pco 22713  df-pconn 30911  df-sconn 30912  df-cvm 30946 This theorem is referenced by: (None)
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