MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  qustgphaus Structured version   Visualization version   GIF version

Theorem qustgphaus 21973
Description: The quotient of a topological group by a closed normal subgroup is a Hausdorff topological group. In particular, the quotient by the closure of the identity is a Hausdorff topological group, isomorphic to both the Kolmogorov quotient and the Hausdorff quotient operations on topological spaces (because T0 and Hausdorff coincide for topological groups). (Contributed by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
qustgp.h 𝐻 = (𝐺 /s (𝐺 ~QG 𝑌))
qustgphaus.j 𝐽 = (TopOpen‘𝐺)
qustgphaus.k 𝐾 = (TopOpen‘𝐻)
Assertion
Ref Expression
qustgphaus ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐾 ∈ Haus)

Proof of Theorem qustgphaus
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 qustgp.h . . . . . . . 8 𝐻 = (𝐺 /s (𝐺 ~QG 𝑌))
2 eqid 2651 . . . . . . . 8 (0g𝐺) = (0g𝐺)
31, 2qus0 17699 . . . . . . 7 (𝑌 ∈ (NrmSGrp‘𝐺) → [(0g𝐺)](𝐺 ~QG 𝑌) = (0g𝐻))
433ad2ant2 1103 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g𝐺)](𝐺 ~QG 𝑌) = (0g𝐻))
5 tgpgrp 21929 . . . . . . . . 9 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
653ad2ant1 1102 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐺 ∈ Grp)
7 eqid 2651 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
87, 2grpidcl 17497 . . . . . . . 8 (𝐺 ∈ Grp → (0g𝐺) ∈ (Base‘𝐺))
96, 8syl 17 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (0g𝐺) ∈ (Base‘𝐺))
10 ovex 6718 . . . . . . . 8 (𝐺 ~QG 𝑌) ∈ V
1110ecelqsi 7846 . . . . . . 7 ((0g𝐺) ∈ (Base‘𝐺) → [(0g𝐺)](𝐺 ~QG 𝑌) ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
129, 11syl 17 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g𝐺)](𝐺 ~QG 𝑌) ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
134, 12eqeltrrd 2731 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (0g𝐻) ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
1413snssd 4372 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {(0g𝐻)} ⊆ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
15 eqid 2651 . . . . . . 7 (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌))
1615mptpreima 5666 . . . . . 6 ((𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g𝐻)}) = {𝑥 ∈ (Base‘𝐺) ∣ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)}}
17 nsgsubg 17673 . . . . . . . . . . 11 (𝑌 ∈ (NrmSGrp‘𝐺) → 𝑌 ∈ (SubGrp‘𝐺))
18173ad2ant2 1103 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ∈ (SubGrp‘𝐺))
19 eqid 2651 . . . . . . . . . . 11 (𝐺 ~QG 𝑌) = (𝐺 ~QG 𝑌)
207, 19, 2eqgid 17693 . . . . . . . . . 10 (𝑌 ∈ (SubGrp‘𝐺) → [(0g𝐺)](𝐺 ~QG 𝑌) = 𝑌)
2118, 20syl 17 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g𝐺)](𝐺 ~QG 𝑌) = 𝑌)
227subgss 17642 . . . . . . . . . 10 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ (Base‘𝐺))
2318, 22syl 17 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ⊆ (Base‘𝐺))
2421, 23eqsstrd 3672 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g𝐺)](𝐺 ~QG 𝑌) ⊆ (Base‘𝐺))
25 sseqin2 3850 . . . . . . . 8 ([(0g𝐺)](𝐺 ~QG 𝑌) ⊆ (Base‘𝐺) ↔ ((Base‘𝐺) ∩ [(0g𝐺)](𝐺 ~QG 𝑌)) = [(0g𝐺)](𝐺 ~QG 𝑌))
2624, 25sylib 208 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ((Base‘𝐺) ∩ [(0g𝐺)](𝐺 ~QG 𝑌)) = [(0g𝐺)](𝐺 ~QG 𝑌))
277, 19eqger 17691 . . . . . . . . . . . . 13 (𝑌 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝑌) Er (Base‘𝐺))
2818, 27syl 17 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝐺 ~QG 𝑌) Er (Base‘𝐺))
2928, 9erth 7834 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ((0g𝐺)(𝐺 ~QG 𝑌)𝑥 ↔ [(0g𝐺)](𝐺 ~QG 𝑌) = [𝑥](𝐺 ~QG 𝑌)))
3029adantr 480 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((0g𝐺)(𝐺 ~QG 𝑌)𝑥 ↔ [(0g𝐺)](𝐺 ~QG 𝑌) = [𝑥](𝐺 ~QG 𝑌)))
314adantr 480 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → [(0g𝐺)](𝐺 ~QG 𝑌) = (0g𝐻))
3231eqeq1d 2653 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → ([(0g𝐺)](𝐺 ~QG 𝑌) = [𝑥](𝐺 ~QG 𝑌) ↔ (0g𝐻) = [𝑥](𝐺 ~QG 𝑌)))
3330, 32bitrd 268 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((0g𝐺)(𝐺 ~QG 𝑌)𝑥 ↔ (0g𝐻) = [𝑥](𝐺 ~QG 𝑌)))
34 vex 3234 . . . . . . . . . 10 𝑥 ∈ V
35 fvex 6239 . . . . . . . . . 10 (0g𝐺) ∈ V
3634, 35elec 7829 . . . . . . . . 9 (𝑥 ∈ [(0g𝐺)](𝐺 ~QG 𝑌) ↔ (0g𝐺)(𝐺 ~QG 𝑌)𝑥)
37 fvex 6239 . . . . . . . . . . 11 (0g𝐻) ∈ V
3837elsn2 4244 . . . . . . . . . 10 ([𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)} ↔ [𝑥](𝐺 ~QG 𝑌) = (0g𝐻))
39 eqcom 2658 . . . . . . . . . 10 ([𝑥](𝐺 ~QG 𝑌) = (0g𝐻) ↔ (0g𝐻) = [𝑥](𝐺 ~QG 𝑌))
4038, 39bitri 264 . . . . . . . . 9 ([𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)} ↔ (0g𝐻) = [𝑥](𝐺 ~QG 𝑌))
4133, 36, 403bitr4g 303 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥 ∈ [(0g𝐺)](𝐺 ~QG 𝑌) ↔ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)}))
4241rabbi2dva 3854 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ((Base‘𝐺) ∩ [(0g𝐺)](𝐺 ~QG 𝑌)) = {𝑥 ∈ (Base‘𝐺) ∣ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)}})
4326, 42, 213eqtr3d 2693 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {𝑥 ∈ (Base‘𝐺) ∣ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)}} = 𝑌)
4416, 43syl5eq 2697 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ((𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g𝐻)}) = 𝑌)
45 simp3 1083 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ∈ (Clsd‘𝐽))
4644, 45eqeltrd 2730 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ((𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g𝐻)}) ∈ (Clsd‘𝐽))
47 qustgphaus.j . . . . . . 7 𝐽 = (TopOpen‘𝐺)
4847, 7tgptopon 21933 . . . . . 6 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
49483ad2ant1 1102 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
501a1i 11 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐻 = (𝐺 /s (𝐺 ~QG 𝑌)))
51 eqidd 2652 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (Base‘𝐺) = (Base‘𝐺))
5210a1i 11 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝐺 ~QG 𝑌) ∈ V)
53 simp1 1081 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐺 ∈ TopGrp)
5450, 51, 15, 52, 53quslem 16250 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)):(Base‘𝐺)–onto→((Base‘𝐺) / (𝐺 ~QG 𝑌)))
55 qtopcld 21564 . . . . 5 ((𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)):(Base‘𝐺)–onto→((Base‘𝐺) / (𝐺 ~QG 𝑌))) → ({(0g𝐻)} ∈ (Clsd‘(𝐽 qTop (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)))) ↔ ({(0g𝐻)} ⊆ ((Base‘𝐺) / (𝐺 ~QG 𝑌)) ∧ ((𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g𝐻)}) ∈ (Clsd‘𝐽))))
5649, 54, 55syl2anc 694 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ({(0g𝐻)} ∈ (Clsd‘(𝐽 qTop (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)))) ↔ ({(0g𝐻)} ⊆ ((Base‘𝐺) / (𝐺 ~QG 𝑌)) ∧ ((𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g𝐻)}) ∈ (Clsd‘𝐽))))
5714, 46, 56mpbir2and 977 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {(0g𝐻)} ∈ (Clsd‘(𝐽 qTop (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)))))
5850, 51, 15, 52, 53qusval 16249 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐻 = ((𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “s 𝐺))
59 qustgphaus.k . . . . 5 𝐾 = (TopOpen‘𝐻)
6058, 51, 54, 53, 47, 59imastopn 21571 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐾 = (𝐽 qTop (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌))))
6160fveq2d 6233 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (Clsd‘𝐾) = (Clsd‘(𝐽 qTop (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)))))
6257, 61eleqtrrd 2733 . 2 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {(0g𝐻)} ∈ (Clsd‘𝐾))
631qustgp 21972 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺)) → 𝐻 ∈ TopGrp)
64633adant3 1101 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐻 ∈ TopGrp)
65 eqid 2651 . . . 4 (0g𝐻) = (0g𝐻)
6665, 59tgphaus 21967 . . 3 (𝐻 ∈ TopGrp → (𝐾 ∈ Haus ↔ {(0g𝐻)} ∈ (Clsd‘𝐾)))
6764, 66syl 17 . 2 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝐾 ∈ Haus ↔ {(0g𝐻)} ∈ (Clsd‘𝐾)))
6862, 67mpbird 247 1 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐾 ∈ Haus)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  {crab 2945  Vcvv 3231  cin 3606  wss 3607  {csn 4210   class class class wbr 4685  cmpt 4762  ccnv 5142  cima 5146  ontowfo 5924  cfv 5926  (class class class)co 6690   Er wer 7784  [cec 7785   / cqs 7786  Basecbs 15904  TopOpenctopn 16129  0gc0g 16147   qTop cqtop 16210   /s cqus 16212  Grpcgrp 17469  SubGrpcsubg 17635  NrmSGrpcnsg 17636   ~QG cqg 17637  TopOnctopon 20763  Clsdccld 20868  Hauscha 21160  TopGrpctgp 21922
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-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
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  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-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-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-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-tpos 7397  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-ec 7789  df-qs 7793  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  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-fz 12365  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-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-rest 16130  df-topn 16131  df-0g 16149  df-topgen 16151  df-qtop 16214  df-imas 16215  df-qus 16216  df-plusf 17288  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-sbg 17474  df-subg 17638  df-nsg 17639  df-eqg 17640  df-oppg 17822  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-cn 21079  df-cnp 21080  df-t1 21166  df-haus 21167  df-tx 21413  df-hmeo 21606  df-tmd 21923  df-tgp 21924
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator