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Theorem qustgphaus 23402
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 2738 . . . . . . . 8 (0g𝐺) = (0g𝐺)
31, 2qus0 18923 . . . . . . 7 (𝑌 ∈ (NrmSGrp‘𝐺) → [(0g𝐺)](𝐺 ~QG 𝑌) = (0g𝐻))
433ad2ant2 1135 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g𝐺)](𝐺 ~QG 𝑌) = (0g𝐻))
5 tgpgrp 23357 . . . . . . . . 9 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
653ad2ant1 1134 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐺 ∈ Grp)
7 eqid 2738 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
87, 2grpidcl 18714 . . . . . . . 8 (𝐺 ∈ Grp → (0g𝐺) ∈ (Base‘𝐺))
96, 8syl 17 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (0g𝐺) ∈ (Base‘𝐺))
10 ovex 7383 . . . . . . . 8 (𝐺 ~QG 𝑌) ∈ V
1110ecelqsi 8646 . . . . . . 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 2840 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (0g𝐻) ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
1413snssd 4768 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {(0g𝐻)} ⊆ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
15 eqid 2738 . . . . . . 7 (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌))
1615mptpreima 6187 . . . . . 6 ((𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g𝐻)}) = {𝑥 ∈ (Base‘𝐺) ∣ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)}}
17 nsgsubg 18895 . . . . . . . . . . 11 (𝑌 ∈ (NrmSGrp‘𝐺) → 𝑌 ∈ (SubGrp‘𝐺))
18173ad2ant2 1135 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ∈ (SubGrp‘𝐺))
19 eqid 2738 . . . . . . . . . . 11 (𝐺 ~QG 𝑌) = (𝐺 ~QG 𝑌)
207, 19, 2eqgid 18917 . . . . . . . . . 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 18864 . . . . . . . . . 10 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ (Base‘𝐺))
2318, 22syl 17 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ⊆ (Base‘𝐺))
2421, 23eqsstrd 3981 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g𝐺)](𝐺 ~QG 𝑌) ⊆ (Base‘𝐺))
25 sseqin2 4174 . . . . . . . 8 ([(0g𝐺)](𝐺 ~QG 𝑌) ⊆ (Base‘𝐺) ↔ ((Base‘𝐺) ∩ [(0g𝐺)](𝐺 ~QG 𝑌)) = [(0g𝐺)](𝐺 ~QG 𝑌))
2624, 25sylib 217 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ((Base‘𝐺) ∩ [(0g𝐺)](𝐺 ~QG 𝑌)) = [(0g𝐺)](𝐺 ~QG 𝑌))
277, 19eqger 18915 . . . . . . . . . . . . 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 8631 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ((0g𝐺)(𝐺 ~QG 𝑌)𝑥 ↔ [(0g𝐺)](𝐺 ~QG 𝑌) = [𝑥](𝐺 ~QG 𝑌)))
3029adantr 482 . . . . . . . . . 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 482 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → [(0g𝐺)](𝐺 ~QG 𝑌) = (0g𝐻))
3231eqeq1d 2740 . . . . . . . . . 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 279 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((0g𝐺)(𝐺 ~QG 𝑌)𝑥 ↔ (0g𝐻) = [𝑥](𝐺 ~QG 𝑌)))
34 vex 3448 . . . . . . . . . 10 𝑥 ∈ V
35 fvex 6851 . . . . . . . . . 10 (0g𝐺) ∈ V
3634, 35elec 8626 . . . . . . . . 9 (𝑥 ∈ [(0g𝐺)](𝐺 ~QG 𝑌) ↔ (0g𝐺)(𝐺 ~QG 𝑌)𝑥)
37 fvex 6851 . . . . . . . . . . 11 (0g𝐻) ∈ V
3837elsn2 4624 . . . . . . . . . 10 ([𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)} ↔ [𝑥](𝐺 ~QG 𝑌) = (0g𝐻))
39 eqcom 2745 . . . . . . . . . 10 ([𝑥](𝐺 ~QG 𝑌) = (0g𝐻) ↔ (0g𝐻) = [𝑥](𝐺 ~QG 𝑌))
4038, 39bitri 275 . . . . . . . . 9 ([𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)} ↔ (0g𝐻) = [𝑥](𝐺 ~QG 𝑌))
4133, 36, 403bitr4g 314 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥 ∈ [(0g𝐺)](𝐺 ~QG 𝑌) ↔ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)}))
4241rabbi2dva 4176 . . . . . . 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 2786 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {𝑥 ∈ (Base‘𝐺) ∣ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)}} = 𝑌)
4416, 43eqtrid 2790 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ((𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g𝐻)}) = 𝑌)
45 simp3 1139 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ∈ (Clsd‘𝐽))
4644, 45eqeltrd 2839 . . . 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 23361 . . . . . 6 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
49483ad2ant1 1134 . . . . 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 2739 . . . . . 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 1137 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐺 ∈ TopGrp)
5450, 51, 15, 52, 53quslem 17361 . . . . 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 22992 . . . . 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 585 . . . 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 712 . . 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 17360 . . . . 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 22999 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐾 = (𝐽 qTop (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌))))
6160fveq2d 6842 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (Clsd‘𝐾) = (Clsd‘(𝐽 qTop (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)))))
6257, 61eleqtrrd 2842 . 2 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {(0g𝐻)} ∈ (Clsd‘𝐾))
631qustgp 23401 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺)) → 𝐻 ∈ TopGrp)
64633adant3 1133 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐻 ∈ TopGrp)
65 eqid 2738 . . . 4 (0g𝐻) = (0g𝐻)
6665, 59tgphaus 23396 . . 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 257 1 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐾 ∈ Haus)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  {crab 3406  Vcvv 3444  cin 3908  wss 3909  {csn 4585   class class class wbr 5104  cmpt 5187  ccnv 5630  cima 5634  ontowfo 6490  cfv 6492  (class class class)co 7350   Er wer 8579  [cec 8580   / cqs 8581  Basecbs 17019  TopOpenctopn 17239  0gc0g 17257   qTop cqtop 17321   /s cqus 17323  Grpcgrp 18684  SubGrpcsubg 18857  NrmSGrpcnsg 18858   ~QG cqg 18859  TopOnctopon 22187  Clsdccld 22295  Hauscha 22587  TopGrpctgp 23350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663  ax-cnex 11041  ax-resscn 11042  ax-1cn 11043  ax-icn 11044  ax-addcl 11045  ax-addrcl 11046  ax-mulcl 11047  ax-mulrcl 11048  ax-mulcom 11049  ax-addass 11050  ax-mulass 11051  ax-distr 11052  ax-i2m1 11053  ax-1ne0 11054  ax-1rid 11055  ax-rnegex 11056  ax-rrecex 11057  ax-cnre 11058  ax-pre-lttri 11059  ax-pre-lttrn 11060  ax-pre-ltadd 11061  ax-pre-mulgt0 11062
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6250  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7306  df-ov 7353  df-oprab 7354  df-mpo 7355  df-om 7794  df-1st 7912  df-2nd 7913  df-tpos 8125  df-frecs 8180  df-wrecs 8211  df-recs 8285  df-rdg 8324  df-1o 8380  df-er 8582  df-ec 8584  df-qs 8588  df-map 8701  df-en 8818  df-dom 8819  df-sdom 8820  df-fin 8821  df-sup 9312  df-inf 9313  df-pnf 11125  df-mnf 11126  df-xr 11127  df-ltxr 11128  df-le 11129  df-sub 11321  df-neg 11322  df-nn 12088  df-2 12150  df-3 12151  df-4 12152  df-5 12153  df-6 12154  df-7 12155  df-8 12156  df-9 12157  df-n0 12348  df-z 12434  df-dec 12553  df-uz 12698  df-fz 13355  df-struct 16955  df-sets 16972  df-slot 16990  df-ndx 17002  df-base 17020  df-ress 17049  df-plusg 17082  df-mulr 17083  df-sca 17085  df-vsca 17086  df-ip 17087  df-tset 17088  df-ple 17089  df-ds 17091  df-rest 17240  df-topn 17241  df-0g 17259  df-topgen 17261  df-qtop 17325  df-imas 17326  df-qus 17327  df-plusf 18432  df-mgm 18433  df-sgrp 18482  df-mnd 18493  df-grp 18687  df-minusg 18688  df-sbg 18689  df-subg 18860  df-nsg 18861  df-eqg 18862  df-oppg 19059  df-top 22171  df-topon 22188  df-topsp 22210  df-bases 22224  df-cld 22298  df-cn 22506  df-cnp 22507  df-t1 22593  df-haus 22594  df-tx 22841  df-hmeo 23034  df-tmd 23351  df-tgp 23352
This theorem is referenced by: (None)
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