Mathbox for Mario Carneiro < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmlift3 Structured version   Visualization version   GIF version

Theorem cvmlift3 30564
 Description: A general version of cvmlift 30535. 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 ∈ SCon)
cvmlift3.l (𝜑𝐾 ∈ 𝑛-Locally PCon)
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 ∈ SCon)
5 cvmlift3.l . . 3 (𝜑𝐾 ∈ 𝑛-Locally PCon)
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 2621 . . . . . . . 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 1397 . . . . . . 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 3034 . . . . . 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 6522 . . . . 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 6102 . . . . . . . . . 10 (𝑐 = 𝑓 → (𝑐‘0) = (𝑓‘0))
1514eqeq1d 2612 . . . . . . . . 9 (𝑐 = 𝑓 → ((𝑐‘0) = 𝑂 ↔ (𝑓‘0) = 𝑂))
16 fveq1 6102 . . . . . . . . . 10 (𝑐 = 𝑓 → (𝑐‘1) = (𝑓‘1))
1716eqeq1d 2612 . . . . . . . . 9 (𝑐 = 𝑓 → ((𝑐‘1) = 𝑎 ↔ (𝑓‘1) = 𝑎))
18 coeq2 5202 . . . . . . . . . . . . . . 15 (𝑑 = 𝑔 → (𝐹𝑑) = (𝐹𝑔))
1918eqeq1d 2612 . . . . . . . . . . . . . 14 (𝑑 = 𝑔 → ((𝐹𝑑) = (𝐺𝑐) ↔ (𝐹𝑔) = (𝐺𝑐)))
20 fveq1 6102 . . . . . . . . . . . . . . 15 (𝑑 = 𝑔 → (𝑑‘0) = (𝑔‘0))
2120eqeq1d 2612 . . . . . . . . . . . . . 14 (𝑑 = 𝑔 → ((𝑑‘0) = 𝑃 ↔ (𝑔‘0) = 𝑃))
2219, 21anbi12d 743 . . . . . . . . . . . . 13 (𝑑 = 𝑔 → (((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺𝑐) ∧ (𝑔‘0) = 𝑃)))
2322cbvriotav 6522 . . . . . . . . . . . 12 (𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑐) ∧ (𝑔‘0) = 𝑃))
24 coeq2 5202 . . . . . . . . . . . . . . 15 (𝑐 = 𝑓 → (𝐺𝑐) = (𝐺𝑓))
2524eqeq2d 2620 . . . . . . . . . . . . . 14 (𝑐 = 𝑓 → ((𝐹𝑔) = (𝐺𝑐) ↔ (𝐹𝑔) = (𝐺𝑓)))
2625anbi1d 737 . . . . . . . . . . . . 13 (𝑐 = 𝑓 → (((𝐹𝑔) = (𝐺𝑐) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)))
2726riotabidv 6513 . . . . . . . . . . . 12 (𝑐 = 𝑓 → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑐) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)))
2823, 27syl5eq 2656 . . . . . . . . . . 11 (𝑐 = 𝑓 → (𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)))
2928fveq1d 6105 . . . . . . . . . 10 (𝑐 = 𝑓 → ((𝑑 ∈ (II Cn 𝐶)((𝐹𝑑) = (𝐺𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1))
3029eqeq1d 2612 . . . . . . . . 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 1391 . . . . . . . 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 3148 . . . . . . 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 2621 . . . . . . . . 9 (𝑎 = 𝑥 → ((𝑓‘1) = 𝑎 ↔ (𝑓‘1) = 𝑥))
34333anbi2d 1396 . . . . . . . 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 3034 . . . . . . 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 271 . . . . . 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 6513 . . . . 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 2656 . . . 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 4678 . . 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 2610 . . . 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 30494 . . 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 30563 . 2 (𝜑 → ∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
43 sconpcon 30463 . . . 4 (𝐾 ∈ SCon → 𝐾 ∈ PCon)
44 pconcon 30467 . . . 4 (𝐾 ∈ PCon → 𝐾 ∈ Con)
454, 43, 443syl 18 . . 3 (𝜑𝐾 ∈ Con)
46 pconcon 30467 . . . . . 6 (𝑥 ∈ PCon → 𝑥 ∈ Con)
4746ssriv 3572 . . . . 5 PCon ⊆ Con
48 nllyss 21093 . . . . 5 (PCon ⊆ Con → 𝑛-Locally PCon ⊆ 𝑛-Locally Con)
4947, 48ax-mp 5 . . . 4 𝑛-Locally PCon ⊆ 𝑛-Locally Con
5049, 5sseldi 3566 . . 3 (𝜑𝐾 ∈ 𝑛-Locally Con)
511, 2, 3, 45, 50, 6, 7, 8, 9cvmliftmo 30520 . 2 (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
52 reu5 3136 . 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 695 1 (𝜑 → ∃!𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  ∃*wrmo 2899  {crab 2900   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372   ↦ cmpt 4643  ◡ccnv 5037   ↾ cres 5040   “ cima 5041   ∘ ccom 5042  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  0cc0 9815  1c1 9816   ↾t crest 15904   Cn ccn 20838  Conccon 21024  𝑛-Locally cnlly 21078  Homeochmeo 21366  IIcii 22486  PConcpcon 30455  SConcscon 30456   CovMap ccvm 30491 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-ec 7631  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-cn 20841  df-cnp 20842  df-cmp 21000  df-con 21025  df-lly 21079  df-nlly 21080  df-tx 21175  df-hmeo 21368  df-xms 21935  df-ms 21936  df-tms 21937  df-ii 22488  df-htpy 22577  df-phtpy 22578  df-phtpc 22599  df-pco 22613  df-pcon 30457  df-scon 30458  df-cvm 30492 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator