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Theorem clsnsg 22405
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 18069 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
2 subgntr.h . . . 4 𝐽 = (TopOpen‘𝐺)
32clssubg 22404 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
41, 3sylan2 592 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
5 df-ima 5463 . . . . . . 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 2797 . . . . . . . . . . . . . 14 (Base‘𝐺) = (Base‘𝐺)
72, 6tgptopon 22378 . . . . . . . . . . . . 13 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
87ad2antrr 722 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
9 topontop 21209 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
108, 9syl 17 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐽 ∈ Top)
111ad2antlr 723 . . . . . . . . . . . . 13 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 ∈ (SubGrp‘𝐺))
126subgss 18038 . . . . . . . . . . . . 13 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
1311, 12syl 17 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
14 toponuni 21210 . . . . . . . . . . . . 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 3934 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 𝐽)
17 eqid 2797 . . . . . . . . . . . 12 𝐽 = 𝐽
1817clsss3 21355 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
1910, 16, 18syl2anc 584 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
2019, 15sseqtr4d 3935 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
2120resmptd 5796 . . . . . . . 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 5697 . . . . . . 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 2797 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
25 tgptmd 22375 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
2625ad2antrr 722 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐺 ∈ TopMnd)
27 simpr 485 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ (Base‘𝐺))
288, 8, 27cnmptc 21958 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
298cnmptid 21957 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ (𝐽 Cn 𝐽))
302, 24, 26, 8, 28, 29cnmpt1plusg 22383 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ∈ (𝐽 Cn 𝐽))
31 eqid 2797 . . . . . . . . . . 11 (-g𝐺) = (-g𝐺)
322, 31tgpsubcn 22386 . . . . . . . . . 10 (𝐺 ∈ TopGrp → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
3332ad2antrr 722 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
348, 30, 28, 33cnmpt12f 21962 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ∈ (𝐽 Cn 𝐽))
3517cnclsi 21568 . . . . . . . 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 584 . . . . . . 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 5463 . . . . . . . . . 10 ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ 𝑆) = ran ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ↾ 𝑆)
3813resmptd 5796 . . . . . . . . . . 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 5697 . . . . . . . . . 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 18070 . . . . . . . . . . . 12 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ 𝑆)
4241ad4ant234 1168 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ 𝑆)
4342fmpttd 6749 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦𝑆 ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)):𝑆𝑆)
4443frnd 6396 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ran (𝑦𝑆 ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ⊆ 𝑆)
4540, 44eqsstrd 3932 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ 𝑆) ⊆ 𝑆)
4617clsss 21350 . . . . . . . 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 1364 . . . . . . 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 3905 . . . . . 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 3933 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ran (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ⊆ ((cls‘𝐽)‘𝑆))
50 ovex 7055 . . . . . . 7 ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ V
51 eqid 2797 . . . . . . 7 (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) = (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥))
5250, 51fnmpti 6366 . . . . . 6 (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) Fn ((cls‘𝐽)‘𝑆)
53 df-f 6236 . . . . . 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 705 . . . . 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 235 . . . 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 6744 . . . 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 235 . . 3 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ∀𝑦 ∈ ((cls‘𝐽)‘𝑆)((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ ((cls‘𝐽)‘𝑆))
5857ralrimiva 3151 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ ((cls‘𝐽)‘𝑆)((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ ((cls‘𝐽)‘𝑆))
596, 24, 31isnsg3 18071 . 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 583 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (NrmSGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1525  wcel 2083  wral 3107  wss 3865   cuni 4751  cmpt 5047  ran crn 5451  cres 5452  cima 5453   Fn wfn 6227  wf 6228  cfv 6232  (class class class)co 7023  Basecbs 16316  +gcplusg 16398  TopOpenctopn 16528  -gcsg 17867  SubGrpcsubg 18031  NrmSGrpcnsg 18032  Topctop 21189  TopOnctopon 21206  clsccl 21314   Cn ccn 21520   ×t ctx 21856  TopMndctmd 22366  TopGrpctgp 22367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-iin 4834  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-er 8146  df-map 8265  df-en 8365  df-dom 8366  df-sdom 8367  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-nn 11493  df-2 11554  df-ndx 16319  df-slot 16320  df-base 16322  df-sets 16323  df-ress 16324  df-plusg 16411  df-0g 16548  df-topgen 16550  df-plusf 17684  df-mgm 17685  df-sgrp 17727  df-mnd 17738  df-grp 17868  df-minusg 17869  df-sbg 17870  df-subg 18034  df-nsg 18035  df-top 21190  df-topon 21207  df-topsp 21229  df-bases 21242  df-cld 21315  df-ntr 21316  df-cls 21317  df-cn 21523  df-cnp 21524  df-tx 21858  df-tmd 22368  df-tgp 22369
This theorem is referenced by: (None)
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