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Theorem qustgphaus 21736
 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 2610 . . . . . . . 8 (0g𝐺) = (0g𝐺)
31, 2qus0 17475 . . . . . . 7 (𝑌 ∈ (NrmSGrp‘𝐺) → [(0g𝐺)](𝐺 ~QG 𝑌) = (0g𝐻))
433ad2ant2 1076 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g𝐺)](𝐺 ~QG 𝑌) = (0g𝐻))
5 tgpgrp 21692 . . . . . . . . 9 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
653ad2ant1 1075 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐺 ∈ Grp)
7 eqid 2610 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
87, 2grpidcl 17273 . . . . . . . 8 (𝐺 ∈ Grp → (0g𝐺) ∈ (Base‘𝐺))
96, 8syl 17 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (0g𝐺) ∈ (Base‘𝐺))
10 ovex 6577 . . . . . . . 8 (𝐺 ~QG 𝑌) ∈ V
1110ecelqsi 7690 . . . . . . 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 2689 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (0g𝐻) ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
1413snssd 4281 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {(0g𝐻)} ⊆ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
15 eqid 2610 . . . . . . 7 (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌))
1615mptpreima 5545 . . . . . 6 ((𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g𝐻)}) = {𝑥 ∈ (Base‘𝐺) ∣ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)}}
17 nsgsubg 17449 . . . . . . . . . . 11 (𝑌 ∈ (NrmSGrp‘𝐺) → 𝑌 ∈ (SubGrp‘𝐺))
18173ad2ant2 1076 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ∈ (SubGrp‘𝐺))
19 eqid 2610 . . . . . . . . . . 11 (𝐺 ~QG 𝑌) = (𝐺 ~QG 𝑌)
207, 19, 2eqgid 17469 . . . . . . . . . 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 17418 . . . . . . . . . 10 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ (Base‘𝐺))
2318, 22syl 17 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ⊆ (Base‘𝐺))
2421, 23eqsstrd 3602 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g𝐺)](𝐺 ~QG 𝑌) ⊆ (Base‘𝐺))
25 sseqin2 3779 . . . . . . . 8 ([(0g𝐺)](𝐺 ~QG 𝑌) ⊆ (Base‘𝐺) ↔ ((Base‘𝐺) ∩ [(0g𝐺)](𝐺 ~QG 𝑌)) = [(0g𝐺)](𝐺 ~QG 𝑌))
2624, 25sylib 207 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ((Base‘𝐺) ∩ [(0g𝐺)](𝐺 ~QG 𝑌)) = [(0g𝐺)](𝐺 ~QG 𝑌))
277, 19eqger 17467 . . . . . . . . . . . . 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 7678 . . . . . . . . . . 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 2612 . . . . . . . . . 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 267 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((0g𝐺)(𝐺 ~QG 𝑌)𝑥 ↔ (0g𝐻) = [𝑥](𝐺 ~QG 𝑌)))
34 vex 3176 . . . . . . . . . 10 𝑥 ∈ V
35 fvex 6113 . . . . . . . . . 10 (0g𝐺) ∈ V
3634, 35elec 7673 . . . . . . . . 9 (𝑥 ∈ [(0g𝐺)](𝐺 ~QG 𝑌) ↔ (0g𝐺)(𝐺 ~QG 𝑌)𝑥)
37 fvex 6113 . . . . . . . . . . 11 (0g𝐻) ∈ V
3837elsn2 4158 . . . . . . . . . 10 ([𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)} ↔ [𝑥](𝐺 ~QG 𝑌) = (0g𝐻))
39 eqcom 2617 . . . . . . . . . 10 ([𝑥](𝐺 ~QG 𝑌) = (0g𝐻) ↔ (0g𝐻) = [𝑥](𝐺 ~QG 𝑌))
4038, 39bitri 263 . . . . . . . . 9 ([𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)} ↔ (0g𝐻) = [𝑥](𝐺 ~QG 𝑌))
4133, 36, 403bitr4g 302 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥 ∈ [(0g𝐺)](𝐺 ~QG 𝑌) ↔ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)}))
4241rabbi2dva 3783 . . . . . . 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 2652 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {𝑥 ∈ (Base‘𝐺) ∣ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)}} = 𝑌)
4416, 43syl5eq 2656 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ((𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g𝐻)}) = 𝑌)
45 simp3 1056 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ∈ (Clsd‘𝐽))
4644, 45eqeltrd 2688 . . . 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 21696 . . . . . 6 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
49483ad2ant1 1075 . . . . 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 2611 . . . . . 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 1054 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐺 ∈ TopGrp)
5450, 51, 15, 52, 53quslem 16026 . . . . 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 21326 . . . . 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 691 . . . 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 959 . . 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 16025 . . . . 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 21333 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐾 = (𝐽 qTop (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌))))
6160fveq2d 6107 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (Clsd‘𝐾) = (Clsd‘(𝐽 qTop (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)))))
6257, 61eleqtrrd 2691 . 2 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {(0g𝐻)} ∈ (Clsd‘𝐾))
631qustgp 21735 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺)) → 𝐻 ∈ TopGrp)
64633adant3 1074 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐻 ∈ TopGrp)
65 eqid 2610 . . . 4 (0g𝐻) = (0g𝐻)
6665, 59tgphaus 21730 . . 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 246 1 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐾 ∈ Haus)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  {csn 4125   class class class wbr 4583   ↦ cmpt 4643  ◡ccnv 5037   “ cima 5041  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549   Er wer 7626  [cec 7627   / cqs 7628  Basecbs 15695  TopOpenctopn 15905  0gc0g 15923   qTop cqtop 15986   /s cqus 15988  Grpcgrp 17245  SubGrpcsubg 17411  NrmSGrpcnsg 17412   ~QG cqg 17413  TopOnctopon 20518  Clsdccld 20630  Hauscha 20922  TopGrpctgp 21685 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-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 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  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-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-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-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  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-fz 12198  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-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-rest 15906  df-topn 15907  df-0g 15925  df-topgen 15927  df-qtop 15990  df-imas 15991  df-qus 15992  df-plusf 17064  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-nsg 17415  df-eqg 17416  df-oppg 17599  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-cn 20841  df-cnp 20842  df-t1 20928  df-haus 20929  df-tx 21175  df-hmeo 21368  df-tmd 21686  df-tgp 21687 This theorem is referenced by: (None)
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