MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clsnsg Structured version   Visualization version   GIF version

Theorem clsnsg 22647
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 18250 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
2 subgntr.h . . . 4 𝐽 = (TopOpen‘𝐺)
32clssubg 22646 . . 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 5562 . . . . . . 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 2821 . . . . . . . . . . . . . 14 (Base‘𝐺) = (Base‘𝐺)
72, 6tgptopon 22620 . . . . . . . . . . . . 13 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
87ad2antrr 722 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
9 topontop 21451 . . . . . . . . . . . 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 18220 . . . . . . . . . . . . 13 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
1311, 12syl 17 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
14 toponuni 21452 . . . . . . . . . . . . 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 4006 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 𝐽)
17 eqid 2821 . . . . . . . . . . . 12 𝐽 = 𝐽
1817clsss3 21597 . . . . . . . . . . 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, 15sseqtrrd 4007 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
2120resmptd 5902 . . . . . . . 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 5802 . . . . . . 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 2868 . . . . . 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 2821 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
25 tgptmd 22617 . . . . . . . . . . 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 22200 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
298cnmptid 22199 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ (𝐽 Cn 𝐽))
302, 24, 26, 8, 28, 29cnmpt1plusg 22625 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ∈ (𝐽 Cn 𝐽))
31 eqid 2821 . . . . . . . . . . 11 (-g𝐺) = (-g𝐺)
322, 31tgpsubcn 22628 . . . . . . . . . 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 22204 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ∈ (𝐽 Cn 𝐽))
3517cnclsi 21810 . . . . . . . 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 5562 . . . . . . . . . 10 ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ 𝑆) = ran ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ↾ 𝑆)
3813resmptd 5902 . . . . . . . . . . 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 5802 . . . . . . . . . 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 2868 . . . . . . . . 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 18251 . . . . . . . . . . . 12 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ 𝑆)
4241ad4ant234 1167 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ 𝑆)
4342fmpttd 6872 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦𝑆 ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)):𝑆𝑆)
4443frnd 6515 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ran (𝑦𝑆 ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ⊆ 𝑆)
4540, 44eqsstrd 4004 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ 𝑆) ⊆ 𝑆)
4617clsss 21592 . . . . . . . 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 1363 . . . . . . 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 3976 . . . . . 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 4005 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ran (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ⊆ ((cls‘𝐽)‘𝑆))
50 ovex 7178 . . . . . . 7 ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ V
51 eqid 2821 . . . . . . 7 (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) = (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥))
5250, 51fnmpti 6485 . . . . . 6 (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) Fn ((cls‘𝐽)‘𝑆)
53 df-f 6353 . . . . . 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 6867 . . . 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 3182 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ ((cls‘𝐽)‘𝑆)((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ ((cls‘𝐽)‘𝑆))
596, 24, 31isnsg3 18252 . 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 1528  wcel 2105  wral 3138  wss 3935   cuni 4832  cmpt 5138  ran crn 5550  cres 5551  cima 5552   Fn wfn 6344  wf 6345  cfv 6349  (class class class)co 7145  Basecbs 16473  +gcplusg 16555  TopOpenctopn 16685  -gcsg 18045  SubGrpcsubg 18213  NrmSGrpcnsg 18214  Topctop 21431  TopOnctopon 21448  clsccl 21556   Cn ccn 21762   ×t ctx 22098  TopMndctmd 22608  TopGrpctgp 22609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-iin 4915  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-ress 16481  df-plusg 16568  df-0g 16705  df-topgen 16707  df-plusf 17841  df-mgm 17842  df-sgrp 17891  df-mnd 17902  df-grp 18046  df-minusg 18047  df-sbg 18048  df-subg 18216  df-nsg 18217  df-top 21432  df-topon 21449  df-topsp 21471  df-bases 21484  df-cld 21557  df-ntr 21558  df-cls 21559  df-cn 21765  df-cnp 21766  df-tx 22100  df-tmd 22610  df-tgp 22611
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator