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Theorem clsnsg 22114
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 17827 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
2 subgntr.h . . . 4 𝐽 = (TopOpen‘𝐺)
32clssubg 22113 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
41, 3sylan2 492 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
5 df-ima 5279 . . . . . . 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 2760 . . . . . . . . . . . . . 14 (Base‘𝐺) = (Base‘𝐺)
72, 6tgptopon 22087 . . . . . . . . . . . . 13 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
87ad2antrr 764 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
9 topontop 20920 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
108, 9syl 17 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐽 ∈ Top)
111ad2antlr 765 . . . . . . . . . . . . 13 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 ∈ (SubGrp‘𝐺))
126subgss 17796 . . . . . . . . . . . . 13 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
1311, 12syl 17 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
14 toponuni 20921 . . . . . . . . . . . . 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 3782 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 𝐽)
17 eqid 2760 . . . . . . . . . . . 12 𝐽 = 𝐽
1817clsss3 21065 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
1910, 16, 18syl2anc 696 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
2019, 15sseqtr4d 3783 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
2120resmptd 5610 . . . . . . . 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 5508 . . . . . . 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 2806 . . . . . 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 2760 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
25 tgptmd 22084 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
2625ad2antrr 764 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐺 ∈ TopMnd)
27 simpr 479 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ (Base‘𝐺))
288, 8, 27cnmptc 21667 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
298cnmptid 21666 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ (𝐽 Cn 𝐽))
302, 24, 26, 8, 28, 29cnmpt1plusg 22092 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ∈ (𝐽 Cn 𝐽))
31 eqid 2760 . . . . . . . . . . 11 (-g𝐺) = (-g𝐺)
322, 31tgpsubcn 22095 . . . . . . . . . 10 (𝐺 ∈ TopGrp → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
3332ad2antrr 764 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
348, 30, 28, 33cnmpt12f 21671 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ∈ (𝐽 Cn 𝐽))
3517cnclsi 21278 . . . . . . . 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 696 . . . . . . 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 5279 . . . . . . . . . 10 ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ 𝑆) = ran ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ↾ 𝑆)
3813resmptd 5610 . . . . . . . . . . 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 5508 . . . . . . . . . 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 2806 . . . . . . . . 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 17828 . . . . . . . . . . . . 13 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ 𝑆)
42413expa 1112 . . . . . . . . . . . 12 (((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ 𝑆)
4342adantlll 756 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ 𝑆)
44 eqid 2760 . . . . . . . . . . 11 (𝑦𝑆 ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) = (𝑦𝑆 ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥))
4543, 44fmptd 6548 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦𝑆 ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)):𝑆𝑆)
46 frn 6214 . . . . . . . . . 10 ((𝑦𝑆 ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)):𝑆𝑆 → ran (𝑦𝑆 ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ⊆ 𝑆)
4745, 46syl 17 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ran (𝑦𝑆 ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ⊆ 𝑆)
4840, 47eqsstrd 3780 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ 𝑆) ⊆ 𝑆)
4917clsss 21060 . . . . . . . 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))
5010, 16, 48, 49syl3anc 1477 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ 𝑆)) ⊆ ((cls‘𝐽)‘𝑆))
5136, 50sstrd 3754 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐽)‘𝑆))
5223, 51eqsstr3d 3781 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ran (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ⊆ ((cls‘𝐽)‘𝑆))
53 ovex 6841 . . . . . . 7 ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ V
54 eqid 2760 . . . . . . 7 (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) = (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥))
5553, 54fnmpti 6183 . . . . . 6 (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) Fn ((cls‘𝐽)‘𝑆)
56 df-f 6053 . . . . . 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)))
5755, 56mpbiran 991 . . . . 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))
5852, 57sylibr 224 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)):((cls‘𝐽)‘𝑆)⟶((cls‘𝐽)‘𝑆))
5954fmpt 6544 . . . 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))
6058, 59sylibr 224 . . 3 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ∀𝑦 ∈ ((cls‘𝐽)‘𝑆)((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ ((cls‘𝐽)‘𝑆))
6160ralrimiva 3104 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ ((cls‘𝐽)‘𝑆)((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ ((cls‘𝐽)‘𝑆))
626, 24, 31isnsg3 17829 . 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)))
634, 61, 62sylanbrc 701 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (NrmSGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  wss 3715   cuni 4588  cmpt 4881  ran crn 5267  cres 5268  cima 5269   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6813  Basecbs 16059  +gcplusg 16143  TopOpenctopn 16284  -gcsg 17625  SubGrpcsubg 17789  NrmSGrpcnsg 17790  Topctop 20900  TopOnctopon 20917  clsccl 21024   Cn ccn 21230   ×t ctx 21565  TopMndctmd 22075  TopGrpctgp 22076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-0g 16304  df-topgen 16306  df-plusf 17442  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-grp 17626  df-minusg 17627  df-sbg 17628  df-subg 17792  df-nsg 17793  df-top 20901  df-topon 20918  df-topsp 20939  df-bases 20952  df-cld 21025  df-ntr 21026  df-cls 21027  df-cn 21233  df-cnp 21234  df-tx 21567  df-tmd 22077  df-tgp 22078
This theorem is referenced by: (None)
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