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Theorem qustgphaus 24152
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 2740 . . . . . . . 8 (0g𝐺) = (0g𝐺)
31, 2qus0 19229 . . . . . . 7 (𝑌 ∈ (NrmSGrp‘𝐺) → [(0g𝐺)](𝐺 ~QG 𝑌) = (0g𝐻))
433ad2ant2 1134 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g𝐺)](𝐺 ~QG 𝑌) = (0g𝐻))
5 tgpgrp 24107 . . . . . . . . 9 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
653ad2ant1 1133 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐺 ∈ Grp)
7 eqid 2740 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
87, 2grpidcl 19005 . . . . . . . 8 (𝐺 ∈ Grp → (0g𝐺) ∈ (Base‘𝐺))
96, 8syl 17 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (0g𝐺) ∈ (Base‘𝐺))
10 ovex 7481 . . . . . . . 8 (𝐺 ~QG 𝑌) ∈ V
1110ecelqsi 8831 . . . . . . 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 2845 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (0g𝐻) ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
1413snssd 4834 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {(0g𝐻)} ⊆ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
15 eqid 2740 . . . . . . 7 (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌))
1615mptpreima 6269 . . . . . 6 ((𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g𝐻)}) = {𝑥 ∈ (Base‘𝐺) ∣ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)}}
17 nsgsubg 19198 . . . . . . . . . . 11 (𝑌 ∈ (NrmSGrp‘𝐺) → 𝑌 ∈ (SubGrp‘𝐺))
18173ad2ant2 1134 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ∈ (SubGrp‘𝐺))
19 eqid 2740 . . . . . . . . . . 11 (𝐺 ~QG 𝑌) = (𝐺 ~QG 𝑌)
207, 19, 2eqgid 19220 . . . . . . . . . 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 19167 . . . . . . . . . 10 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ (Base‘𝐺))
2318, 22syl 17 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ⊆ (Base‘𝐺))
2421, 23eqsstrd 4047 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g𝐺)](𝐺 ~QG 𝑌) ⊆ (Base‘𝐺))
25 sseqin2 4244 . . . . . . . 8 ([(0g𝐺)](𝐺 ~QG 𝑌) ⊆ (Base‘𝐺) ↔ ((Base‘𝐺) ∩ [(0g𝐺)](𝐺 ~QG 𝑌)) = [(0g𝐺)](𝐺 ~QG 𝑌))
2624, 25sylib 218 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ((Base‘𝐺) ∩ [(0g𝐺)](𝐺 ~QG 𝑌)) = [(0g𝐺)](𝐺 ~QG 𝑌))
277, 19eqger 19218 . . . . . . . . . . . . 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 8814 . . . . . . . . . . 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 2742 . . . . . . . . . 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 3492 . . . . . . . . . 10 𝑥 ∈ V
35 fvex 6933 . . . . . . . . . 10 (0g𝐺) ∈ V
3634, 35elec 8809 . . . . . . . . 9 (𝑥 ∈ [(0g𝐺)](𝐺 ~QG 𝑌) ↔ (0g𝐺)(𝐺 ~QG 𝑌)𝑥)
37 fvex 6933 . . . . . . . . . . 11 (0g𝐻) ∈ V
3837elsn2 4687 . . . . . . . . . 10 ([𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)} ↔ [𝑥](𝐺 ~QG 𝑌) = (0g𝐻))
39 eqcom 2747 . . . . . . . . . 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 4247 . . . . . . 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 2788 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {𝑥 ∈ (Base‘𝐺) ∣ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)}} = 𝑌)
4416, 43eqtrid 2792 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ((𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g𝐻)}) = 𝑌)
45 simp3 1138 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ∈ (Clsd‘𝐽))
4644, 45eqeltrd 2844 . . . 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 24111 . . . . . 6 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
49483ad2ant1 1133 . . . . 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 2741 . . . . . 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 1136 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐺 ∈ TopGrp)
5450, 51, 15, 52, 53quslem 17603 . . . . 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 23742 . . . . 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 583 . . . 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 17602 . . . . 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 23749 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐾 = (𝐽 qTop (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌))))
6160fveq2d 6924 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (Clsd‘𝐾) = (Clsd‘(𝐽 qTop (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)))))
6257, 61eleqtrrd 2847 . 2 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {(0g𝐻)} ∈ (Clsd‘𝐾))
631qustgp 24151 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺)) → 𝐻 ∈ TopGrp)
64633adant3 1132 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐻 ∈ TopGrp)
65 eqid 2740 . . . 4 (0g𝐻) = (0g𝐻)
6665, 59tgphaus 24146 . . 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 206  wa 395  w3a 1087   = wceq 1537  wcel 2108  {crab 3443  Vcvv 3488  cin 3975  wss 3976  {csn 4648   class class class wbr 5166  cmpt 5249  ccnv 5699  cima 5703  ontowfo 6571  cfv 6573  (class class class)co 7448   Er wer 8760  [cec 8761   / cqs 8762  Basecbs 17258  TopOpenctopn 17481  0gc0g 17499   qTop cqtop 17563   /s cqus 17565  Grpcgrp 18973  SubGrpcsubg 19160  NrmSGrpcnsg 19161   ~QG cqg 19162  TopOnctopon 22937  Clsdccld 23045  Hauscha 23337  TopGrpctgp 24100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-tpos 8267  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-ec 8765  df-qs 8769  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-rest 17482  df-topn 17483  df-0g 17501  df-topgen 17503  df-qtop 17567  df-imas 17568  df-qus 17569  df-plusf 18677  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-minusg 18977  df-sbg 18978  df-subg 19163  df-nsg 19164  df-eqg 19165  df-oppg 19386  df-top 22921  df-topon 22938  df-topsp 22960  df-bases 22974  df-cld 23048  df-cn 23256  df-cnp 23257  df-t1 23343  df-haus 23344  df-tx 23591  df-hmeo 23784  df-tmd 24101  df-tgp 24102
This theorem is referenced by: (None)
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