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Theorem clsnsg 22722
 Description: The closure of a normal subgroup is a normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
Hypothesis
Ref Expression
subgntr.h 𝐽 = (TopOpen‘𝐺)
Assertion
Ref Expression
clsnsg ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (NrmSGrp‘𝐺))

Proof of Theorem clsnsg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nsgsubg 18305 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
2 subgntr.h . . . 4 𝐽 = (TopOpen‘𝐺)
32clssubg 22721 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
41, 3sylan2 595 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
5 df-ima 5532 . . . . . . 7 ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ ((cls‘𝐽)‘𝑆)) = ran ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ↾ ((cls‘𝐽)‘𝑆))
6 eqid 2798 . . . . . . . . . . . . . 14 (Base‘𝐺) = (Base‘𝐺)
72, 6tgptopon 22694 . . . . . . . . . . . . 13 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
87ad2antrr 725 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
9 topontop 21525 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
108, 9syl 17 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐽 ∈ Top)
111ad2antlr 726 . . . . . . . . . . . . 13 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 ∈ (SubGrp‘𝐺))
126subgss 18275 . . . . . . . . . . . . 13 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
1311, 12syl 17 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
14 toponuni 21526 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = 𝐽)
158, 14syl 17 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (Base‘𝐺) = 𝐽)
1613, 15sseqtrd 3955 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 𝐽)
17 eqid 2798 . . . . . . . . . . . 12 𝐽 = 𝐽
1817clsss3 21671 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
1910, 16, 18syl2anc 587 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
2019, 15sseqtrrd 3956 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
2120resmptd 5875 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ↾ ((cls‘𝐽)‘𝑆)) = (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)))
2221rneqd 5772 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ran ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ↾ ((cls‘𝐽)‘𝑆)) = ran (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)))
235, 22syl5eq 2845 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ ((cls‘𝐽)‘𝑆)) = ran (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)))
24 eqid 2798 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
25 tgptmd 22691 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
2625ad2antrr 725 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐺 ∈ TopMnd)
27 simpr 488 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ (Base‘𝐺))
288, 8, 27cnmptc 22274 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
298cnmptid 22273 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ (𝐽 Cn 𝐽))
302, 24, 26, 8, 28, 29cnmpt1plusg 22699 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ∈ (𝐽 Cn 𝐽))
31 eqid 2798 . . . . . . . . . . 11 (-g𝐺) = (-g𝐺)
322, 31tgpsubcn 22702 . . . . . . . . . 10 (𝐺 ∈ TopGrp → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
3332ad2antrr 725 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
348, 30, 28, 33cnmpt12f 22278 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ∈ (𝐽 Cn 𝐽))
3517cnclsi 21884 . . . . . . . 8 (((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ∈ (𝐽 Cn 𝐽) ∧ 𝑆 𝐽) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐽)‘((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ 𝑆)))
3634, 16, 35syl2anc 587 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐽)‘((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ 𝑆)))
37 df-ima 5532 . . . . . . . . . 10 ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ 𝑆) = ran ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ↾ 𝑆)
3813resmptd 5875 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ↾ 𝑆) = (𝑦𝑆 ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)))
3938rneqd 5772 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ran ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ↾ 𝑆) = ran (𝑦𝑆 ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)))
4037, 39syl5eq 2845 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ 𝑆) = ran (𝑦𝑆 ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)))
416, 24, 31nsgconj 18306 . . . . . . . . . . . 12 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ 𝑆)
4241ad4ant234 1172 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ 𝑆)
4342fmpttd 6856 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦𝑆 ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)):𝑆𝑆)
4443frnd 6494 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ran (𝑦𝑆 ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ⊆ 𝑆)
4540, 44eqsstrd 3953 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ 𝑆) ⊆ 𝑆)
4617clsss 21666 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 𝐽 ∧ ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ 𝑆) ⊆ 𝑆) → ((cls‘𝐽)‘((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ 𝑆)) ⊆ ((cls‘𝐽)‘𝑆))
4710, 16, 45, 46syl3anc 1368 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ 𝑆)) ⊆ ((cls‘𝐽)‘𝑆))
4836, 47sstrd 3925 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐽)‘𝑆))
4923, 48eqsstrrd 3954 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ran (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ⊆ ((cls‘𝐽)‘𝑆))
50 ovex 7168 . . . . . . 7 ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ V
51 eqid 2798 . . . . . . 7 (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) = (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥))
5250, 51fnmpti 6463 . . . . . 6 (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) Fn ((cls‘𝐽)‘𝑆)
53 df-f 6328 . . . . . 6 ((𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)):((cls‘𝐽)‘𝑆)⟶((cls‘𝐽)‘𝑆) ↔ ((𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) Fn ((cls‘𝐽)‘𝑆) ∧ ran (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ⊆ ((cls‘𝐽)‘𝑆)))
5452, 53mpbiran 708 . . . . 5 ((𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)):((cls‘𝐽)‘𝑆)⟶((cls‘𝐽)‘𝑆) ↔ ran (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ⊆ ((cls‘𝐽)‘𝑆))
5549, 54sylibr 237 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)):((cls‘𝐽)‘𝑆)⟶((cls‘𝐽)‘𝑆))
5651fmpt 6851 . . . 4 (∀𝑦 ∈ ((cls‘𝐽)‘𝑆)((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ ((cls‘𝐽)‘𝑆) ↔ (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)):((cls‘𝐽)‘𝑆)⟶((cls‘𝐽)‘𝑆))
5755, 56sylibr 237 . . 3 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ∀𝑦 ∈ ((cls‘𝐽)‘𝑆)((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ ((cls‘𝐽)‘𝑆))
5857ralrimiva 3149 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ ((cls‘𝐽)‘𝑆)((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ ((cls‘𝐽)‘𝑆))
596, 24, 31isnsg3 18307 . 2 (((cls‘𝐽)‘𝑆) ∈ (NrmSGrp‘𝐺) ↔ (((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ ((cls‘𝐽)‘𝑆)((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ ((cls‘𝐽)‘𝑆)))
604, 58, 59sylanbrc 586 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (NrmSGrp‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ⊆ wss 3881  ∪ cuni 4800   ↦ cmpt 5110  ran crn 5520   ↾ cres 5521   “ cima 5522   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  Basecbs 16477  +gcplusg 16559  TopOpenctopn 16689  -gcsg 18099  SubGrpcsubg 18268  NrmSGrpcnsg 18269  Topctop 21505  TopOnctopon 21522  clsccl 21630   Cn ccn 21836   ×t ctx 22172  TopMndctmd 22682  TopGrpctgp 22683 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-cnex 10584  ax-resscn 10585  ax-1cn 10586  ax-icn 10587  ax-addcl 10588  ax-addrcl 10589  ax-mulcl 10590  ax-mulrcl 10591  ax-mulcom 10592  ax-addass 10593  ax-mulass 10594  ax-distr 10595  ax-i2m1 10596  ax-1ne0 10597  ax-1rid 10598  ax-rnegex 10599  ax-rrecex 10600  ax-cnre 10601  ax-pre-lttri 10602  ax-pre-lttrn 10603  ax-pre-ltadd 10604  ax-pre-mulgt0 10605 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-1st 7673  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-er 8274  df-map 8393  df-en 8495  df-dom 8496  df-sdom 8497  df-pnf 10668  df-mnf 10669  df-xr 10670  df-ltxr 10671  df-le 10672  df-sub 10863  df-neg 10864  df-nn 11628  df-2 11690  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-plusg 16572  df-0g 16709  df-topgen 16711  df-plusf 17845  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-grp 18100  df-minusg 18101  df-sbg 18102  df-subg 18271  df-nsg 18272  df-top 21506  df-topon 21523  df-topsp 21545  df-bases 21558  df-cld 21631  df-ntr 21632  df-cls 21633  df-cn 21839  df-cnp 21840  df-tx 22174  df-tmd 22684  df-tgp 22685 This theorem is referenced by: (None)
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