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Theorem clsnsg 21960
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 17673 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
2 subgntr.h . . . 4 𝐽 = (TopOpen‘𝐺)
32clssubg 21959 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
41, 3sylan2 490 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
5 df-ima 5156 . . . . . . 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 2651 . . . . . . . . . . . . . 14 (Base‘𝐺) = (Base‘𝐺)
72, 6tgptopon 21933 . . . . . . . . . . . . 13 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
87ad2antrr 762 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
9 topontop 20766 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
108, 9syl 17 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐽 ∈ Top)
111ad2antlr 763 . . . . . . . . . . . . 13 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 ∈ (SubGrp‘𝐺))
126subgss 17642 . . . . . . . . . . . . 13 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
1311, 12syl 17 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
14 toponuni 20767 . . . . . . . . . . . . 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 3674 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑆 𝐽)
17 eqid 2651 . . . . . . . . . . . 12 𝐽 = 𝐽
1817clsss3 20911 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
1910, 16, 18syl2anc 694 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
2019, 15sseqtr4d 3675 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
2120resmptd 5487 . . . . . . . 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 5385 . . . . . . 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 2697 . . . . . 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 2651 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
25 tgptmd 21930 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
2625ad2antrr 762 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐺 ∈ TopMnd)
27 simpr 476 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ (Base‘𝐺))
288, 8, 27cnmptc 21513 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
298cnmptid 21512 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ (𝐽 Cn 𝐽))
302, 24, 26, 8, 28, 29cnmpt1plusg 21938 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ∈ (𝐽 Cn 𝐽))
31 eqid 2651 . . . . . . . . . . 11 (-g𝐺) = (-g𝐺)
322, 31tgpsubcn 21941 . . . . . . . . . 10 (𝐺 ∈ TopGrp → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
3332ad2antrr 762 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
348, 30, 28, 33cnmpt12f 21517 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ∈ (𝐽 Cn 𝐽))
3517cnclsi 21124 . . . . . . . 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 694 . . . . . . 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 5156 . . . . . . . . . 10 ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ 𝑆) = ran ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ↾ 𝑆)
3813resmptd 5487 . . . . . . . . . . 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 5385 . . . . . . . . . 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 2697 . . . . . . . . 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 17674 . . . . . . . . . . . . 13 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ 𝑆)
42413expa 1284 . . . . . . . . . . . 12 (((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ 𝑆)
4342adantlll 754 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ 𝑆)
44 eqid 2651 . . . . . . . . . . 11 (𝑦𝑆 ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) = (𝑦𝑆 ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥))
4543, 44fmptd 6425 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦𝑆 ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)):𝑆𝑆)
46 frn 6091 . . . . . . . . . 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 3672 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) “ 𝑆) ⊆ 𝑆)
4917clsss 20906 . . . . . . . 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 1366 . . . . . . 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 3646 . . . . . 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 3673 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → ran (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) ⊆ ((cls‘𝐽)‘𝑆))
53 ovex 6718 . . . . . . 7 ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ V
54 eqid 2651 . . . . . . 7 (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) = (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥))
5553, 54fnmpti 6060 . . . . . 6 (𝑦 ∈ ((cls‘𝐽)‘𝑆) ↦ ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥)) Fn ((cls‘𝐽)‘𝑆)
56 df-f 5930 . . . . . 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 973 . . . . 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 6421 . . . 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 2995 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ ((cls‘𝐽)‘𝑆)((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ ((cls‘𝐽)‘𝑆))
626, 24, 31isnsg3 17675 . 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 699 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (NrmSGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  wss 3607   cuni 4468  cmpt 4762  ran crn 5144  cres 5145  cima 5146   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  TopOpenctopn 16129  -gcsg 17471  SubGrpcsubg 17635  NrmSGrpcnsg 17636  Topctop 20746  TopOnctopon 20763  clsccl 20870   Cn ccn 21076   ×t ctx 21411  TopMndctmd 21921  TopGrpctgp 21922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-0g 16149  df-topgen 16151  df-plusf 17288  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-sbg 17474  df-subg 17638  df-nsg 17639  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-cn 21079  df-cnp 21080  df-tx 21413  df-tmd 21923  df-tgp 21924
This theorem is referenced by: (None)
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