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Theorem qustgphaus 22723
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 2819 . . . . . . . 8 (0g𝐺) = (0g𝐺)
31, 2qus0 18330 . . . . . . 7 (𝑌 ∈ (NrmSGrp‘𝐺) → [(0g𝐺)](𝐺 ~QG 𝑌) = (0g𝐻))
433ad2ant2 1128 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g𝐺)](𝐺 ~QG 𝑌) = (0g𝐻))
5 tgpgrp 22678 . . . . . . . . 9 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
653ad2ant1 1127 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐺 ∈ Grp)
7 eqid 2819 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
87, 2grpidcl 18123 . . . . . . . 8 (𝐺 ∈ Grp → (0g𝐺) ∈ (Base‘𝐺))
96, 8syl 17 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (0g𝐺) ∈ (Base‘𝐺))
10 ovex 7181 . . . . . . . 8 (𝐺 ~QG 𝑌) ∈ V
1110ecelqsi 8345 . . . . . . 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 2912 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (0g𝐻) ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
1413snssd 4734 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {(0g𝐻)} ⊆ ((Base‘𝐺) / (𝐺 ~QG 𝑌)))
15 eqid 2819 . . . . . . 7 (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌))
1615mptpreima 6085 . . . . . 6 ((𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g𝐻)}) = {𝑥 ∈ (Base‘𝐺) ∣ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)}}
17 nsgsubg 18302 . . . . . . . . . . 11 (𝑌 ∈ (NrmSGrp‘𝐺) → 𝑌 ∈ (SubGrp‘𝐺))
18173ad2ant2 1128 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ∈ (SubGrp‘𝐺))
19 eqid 2819 . . . . . . . . . . 11 (𝐺 ~QG 𝑌) = (𝐺 ~QG 𝑌)
207, 19, 2eqgid 18324 . . . . . . . . . 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 18272 . . . . . . . . . 10 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ (Base‘𝐺))
2318, 22syl 17 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ⊆ (Base‘𝐺))
2421, 23eqsstrd 4003 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → [(0g𝐺)](𝐺 ~QG 𝑌) ⊆ (Base‘𝐺))
25 sseqin2 4190 . . . . . . . 8 ([(0g𝐺)](𝐺 ~QG 𝑌) ⊆ (Base‘𝐺) ↔ ((Base‘𝐺) ∩ [(0g𝐺)](𝐺 ~QG 𝑌)) = [(0g𝐺)](𝐺 ~QG 𝑌))
2624, 25sylib 220 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ((Base‘𝐺) ∩ [(0g𝐺)](𝐺 ~QG 𝑌)) = [(0g𝐺)](𝐺 ~QG 𝑌))
277, 19eqger 18322 . . . . . . . . . . . . 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 8330 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ((0g𝐺)(𝐺 ~QG 𝑌)𝑥 ↔ [(0g𝐺)](𝐺 ~QG 𝑌) = [𝑥](𝐺 ~QG 𝑌)))
3029adantr 483 . . . . . . . . . 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 483 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → [(0g𝐺)](𝐺 ~QG 𝑌) = (0g𝐻))
3231eqeq1d 2821 . . . . . . . . . 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 281 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((0g𝐺)(𝐺 ~QG 𝑌)𝑥 ↔ (0g𝐻) = [𝑥](𝐺 ~QG 𝑌)))
34 vex 3496 . . . . . . . . . 10 𝑥 ∈ V
35 fvex 6676 . . . . . . . . . 10 (0g𝐺) ∈ V
3634, 35elec 8325 . . . . . . . . 9 (𝑥 ∈ [(0g𝐺)](𝐺 ~QG 𝑌) ↔ (0g𝐺)(𝐺 ~QG 𝑌)𝑥)
37 fvex 6676 . . . . . . . . . . 11 (0g𝐻) ∈ V
3837elsn2 4596 . . . . . . . . . 10 ([𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)} ↔ [𝑥](𝐺 ~QG 𝑌) = (0g𝐻))
39 eqcom 2826 . . . . . . . . . 10 ([𝑥](𝐺 ~QG 𝑌) = (0g𝐻) ↔ (0g𝐻) = [𝑥](𝐺 ~QG 𝑌))
4038, 39bitri 277 . . . . . . . . 9 ([𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)} ↔ (0g𝐻) = [𝑥](𝐺 ~QG 𝑌))
4133, 36, 403bitr4g 316 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥 ∈ [(0g𝐺)](𝐺 ~QG 𝑌) ↔ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)}))
4241rabbi2dva 4192 . . . . . . 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 2862 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {𝑥 ∈ (Base‘𝐺) ∣ [𝑥](𝐺 ~QG 𝑌) ∈ {(0g𝐻)}} = 𝑌)
4416, 43syl5eq 2866 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ((𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)) “ {(0g𝐻)}) = 𝑌)
45 simp3 1132 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ∈ (Clsd‘𝐽))
4644, 45eqeltrd 2911 . . . 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 22682 . . . . . 6 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
49483ad2ant1 1127 . . . . 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 2820 . . . . . 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 1130 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐺 ∈ TopGrp)
5450, 51, 15, 52, 53quslem 16808 . . . . 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 22313 . . . . 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 586 . . . 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 711 . . 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 16807 . . . . 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 22320 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐾 = (𝐽 qTop (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌))))
6160fveq2d 6667 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (Clsd‘𝐾) = (Clsd‘(𝐽 qTop (𝑥 ∈ (Base‘𝐺) ↦ [𝑥](𝐺 ~QG 𝑌)))))
6257, 61eleqtrrd 2914 . 2 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → {(0g𝐻)} ∈ (Clsd‘𝐾))
631qustgp 22722 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺)) → 𝐻 ∈ TopGrp)
64633adant3 1126 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐻 ∈ TopGrp)
65 eqid 2819 . . . 4 (0g𝐻) = (0g𝐻)
6665, 59tgphaus 22717 . . 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 259 1 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐾 ∈ Haus)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1081   = wceq 1530  wcel 2107  {crab 3140  Vcvv 3493  cin 3933  wss 3934  {csn 4559   class class class wbr 5057  cmpt 5137  ccnv 5547  cima 5551  ontowfo 6346  cfv 6348  (class class class)co 7148   Er wer 8278  [cec 8279   / cqs 8280  Basecbs 16475  TopOpenctopn 16687  0gc0g 16705   qTop cqtop 16768   /s cqus 16770  Grpcgrp 18095  SubGrpcsubg 18265  NrmSGrpcnsg 18266   ~QG cqg 18267  TopOnctopon 21510  Clsdccld 21616  Hauscha 21908  TopGrpctgp 22671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-tpos 7884  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-ec 8283  df-qs 8287  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-inf 8899  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12885  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-rest 16688  df-topn 16689  df-0g 16707  df-topgen 16709  df-qtop 16772  df-imas 16773  df-qus 16774  df-plusf 17843  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099  df-sbg 18100  df-subg 18268  df-nsg 18269  df-eqg 18270  df-oppg 18466  df-top 21494  df-topon 21511  df-topsp 21533  df-bases 21546  df-cld 21619  df-cn 21827  df-cnp 21828  df-t1 21914  df-haus 21915  df-tx 22162  df-hmeo 22355  df-tmd 22672  df-tgp 22673
This theorem is referenced by: (None)
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