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Theorem clssubg 22436
Description: The closure of a subgroup in a topological group is a subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
Hypothesis
Ref Expression
subgntr.h 𝐽 = (TopOpen‘𝐺)
Assertion
Ref Expression
clssubg ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))

Proof of Theorem clssubg
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subgntr.h . . . . . . 7 𝐽 = (TopOpen‘𝐺)
2 eqid 2773 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
31, 2tgptopon 22410 . . . . . 6 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
43adantr 473 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
5 topontop 21241 . . . . 5 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
64, 5syl 17 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐽 ∈ Top)
72subgss 18077 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
87adantl 474 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
9 toponuni 21242 . . . . . 6 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = 𝐽)
104, 9syl 17 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (Base‘𝐺) = 𝐽)
118, 10sseqtrd 3892 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 𝐽)
12 eqid 2773 . . . . 5 𝐽 = 𝐽
1312clsss3 21387 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
146, 11, 13syl2anc 576 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
1514, 10sseqtr4d 3893 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
1612sscls 21384 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
176, 11, 16syl2anc 576 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
18 eqid 2773 . . . . . 6 (0g𝐺) = (0g𝐺)
1918subg0cl 18084 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑆)
2019adantl 474 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (0g𝐺) ∈ 𝑆)
2120ne0d 4182 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ≠ ∅)
22 ssn0 4235 . . 3 ((𝑆 ⊆ ((cls‘𝐽)‘𝑆) ∧ 𝑆 ≠ ∅) → ((cls‘𝐽)‘𝑆) ≠ ∅)
2317, 21, 22syl2anc 576 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ≠ ∅)
24 df-ov 6978 . . . 4 (𝑥(-g𝐺)𝑦) = ((-g𝐺)‘⟨𝑥, 𝑦⟩)
25 opelxpi 5441 . . . . . . 7 ((𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆)) → ⟨𝑥, 𝑦⟩ ∈ (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)))
26 txcls 21932 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺))) ∧ (𝑆 ⊆ (Base‘𝐺) ∧ 𝑆 ⊆ (Base‘𝐺))) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) = (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)))
274, 4, 8, 8, 26syl22anc 827 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) = (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)))
28 txtopon 21919 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺))) → (𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))))
294, 4, 28syl2anc 576 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))))
30 topontop 21241 . . . . . . . . . . . 12 ((𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))) → (𝐽 ×t 𝐽) ∈ Top)
3129, 30syl 17 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐽 ×t 𝐽) ∈ Top)
32 cnvimass 5787 . . . . . . . . . . . . 13 ((-g𝐺) “ 𝑆) ⊆ dom (-g𝐺)
33 tgpgrp 22406 . . . . . . . . . . . . . . 15 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
3433adantr 473 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
35 eqid 2773 . . . . . . . . . . . . . . 15 (-g𝐺) = (-g𝐺)
362, 35grpsubf 17978 . . . . . . . . . . . . . 14 (𝐺 ∈ Grp → (-g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
3734, 36syl 17 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (-g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
3832, 37fssdm 6358 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((-g𝐺) “ 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
39 toponuni 21242 . . . . . . . . . . . . 13 ((𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))) → ((Base‘𝐺) × (Base‘𝐺)) = (𝐽 ×t 𝐽))
4029, 39syl 17 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((Base‘𝐺) × (Base‘𝐺)) = (𝐽 ×t 𝐽))
4138, 40sseqtrd 3892 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((-g𝐺) “ 𝑆) ⊆ (𝐽 ×t 𝐽))
4235subgsubcl 18087 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆𝑦𝑆) → (𝑥(-g𝐺)𝑦) ∈ 𝑆)
43423expb 1101 . . . . . . . . . . . . . . 15 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(-g𝐺)𝑦) ∈ 𝑆)
4443ralrimivva 3136 . . . . . . . . . . . . . 14 (𝑆 ∈ (SubGrp‘𝐺) → ∀𝑥𝑆𝑦𝑆 (𝑥(-g𝐺)𝑦) ∈ 𝑆)
45 fveq2 6497 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨𝑥, 𝑦⟩ → ((-g𝐺)‘𝑧) = ((-g𝐺)‘⟨𝑥, 𝑦⟩))
4645, 24syl6eqr 2827 . . . . . . . . . . . . . . . 16 (𝑧 = ⟨𝑥, 𝑦⟩ → ((-g𝐺)‘𝑧) = (𝑥(-g𝐺)𝑦))
4746eleq1d 2845 . . . . . . . . . . . . . . 15 (𝑧 = ⟨𝑥, 𝑦⟩ → (((-g𝐺)‘𝑧) ∈ 𝑆 ↔ (𝑥(-g𝐺)𝑦) ∈ 𝑆))
4847ralxp 5559 . . . . . . . . . . . . . 14 (∀𝑧 ∈ (𝑆 × 𝑆)((-g𝐺)‘𝑧) ∈ 𝑆 ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(-g𝐺)𝑦) ∈ 𝑆)
4944, 48sylibr 226 . . . . . . . . . . . . 13 (𝑆 ∈ (SubGrp‘𝐺) → ∀𝑧 ∈ (𝑆 × 𝑆)((-g𝐺)‘𝑧) ∈ 𝑆)
5049adantl 474 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑧 ∈ (𝑆 × 𝑆)((-g𝐺)‘𝑧) ∈ 𝑆)
5137ffund 6346 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → Fun (-g𝐺))
52 xpss12 5419 . . . . . . . . . . . . . . 15 ((𝑆 ⊆ (Base‘𝐺) ∧ 𝑆 ⊆ (Base‘𝐺)) → (𝑆 × 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
538, 8, 52syl2anc 576 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
5437fdmd 6351 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → dom (-g𝐺) = ((Base‘𝐺) × (Base‘𝐺)))
5553, 54sseqtr4d 3893 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ dom (-g𝐺))
56 funimass5 6649 . . . . . . . . . . . . 13 ((Fun (-g𝐺) ∧ (𝑆 × 𝑆) ⊆ dom (-g𝐺)) → ((𝑆 × 𝑆) ⊆ ((-g𝐺) “ 𝑆) ↔ ∀𝑧 ∈ (𝑆 × 𝑆)((-g𝐺)‘𝑧) ∈ 𝑆))
5751, 55, 56syl2anc 576 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((𝑆 × 𝑆) ⊆ ((-g𝐺) “ 𝑆) ↔ ∀𝑧 ∈ (𝑆 × 𝑆)((-g𝐺)‘𝑧) ∈ 𝑆))
5850, 57mpbird 249 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ ((-g𝐺) “ 𝑆))
59 eqid 2773 . . . . . . . . . . . 12 (𝐽 ×t 𝐽) = (𝐽 ×t 𝐽)
6059clsss 21382 . . . . . . . . . . 11 (((𝐽 ×t 𝐽) ∈ Top ∧ ((-g𝐺) “ 𝑆) ⊆ (𝐽 ×t 𝐽) ∧ (𝑆 × 𝑆) ⊆ ((-g𝐺) “ 𝑆)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) ⊆ ((cls‘(𝐽 ×t 𝐽))‘((-g𝐺) “ 𝑆)))
6131, 41, 58, 60syl3anc 1352 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) ⊆ ((cls‘(𝐽 ×t 𝐽))‘((-g𝐺) “ 𝑆)))
621, 35tgpsubcn 22418 . . . . . . . . . . . 12 (𝐺 ∈ TopGrp → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
6362adantr 473 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
6412cncls2i 21598 . . . . . . . . . . 11 (((-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) ∧ 𝑆 𝐽) → ((cls‘(𝐽 ×t 𝐽))‘((-g𝐺) “ 𝑆)) ⊆ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
6563, 11, 64syl2anc 576 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘((-g𝐺) “ 𝑆)) ⊆ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
6661, 65sstrd 3863 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) ⊆ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
6727, 66eqsstr3d 3891 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)) ⊆ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
6867sselda 3853 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ ⟨𝑥, 𝑦⟩ ∈ (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆))) → ⟨𝑥, 𝑦⟩ ∈ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
6925, 68sylan2 584 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → ⟨𝑥, 𝑦⟩ ∈ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
7033ad2antrr 714 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → 𝐺 ∈ Grp)
71 ffn 6342 . . . . . . 7 ((-g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺) → (-g𝐺) Fn ((Base‘𝐺) × (Base‘𝐺)))
72 elpreima 6652 . . . . . . 7 ((-g𝐺) Fn ((Base‘𝐺) × (Base‘𝐺)) → (⟨𝑥, 𝑦⟩ ∈ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)) ↔ (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ∧ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ ((cls‘𝐽)‘𝑆))))
7370, 36, 71, 724syl 19 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → (⟨𝑥, 𝑦⟩ ∈ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)) ↔ (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ∧ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ ((cls‘𝐽)‘𝑆))))
7469, 73mpbid 224 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ∧ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ ((cls‘𝐽)‘𝑆)))
7574simprd 488 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ ((cls‘𝐽)‘𝑆))
7624, 75syl5eqel 2865 . . 3 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → (𝑥(-g𝐺)𝑦) ∈ ((cls‘𝐽)‘𝑆))
7776ralrimivva 3136 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑥 ∈ ((cls‘𝐽)‘𝑆)∀𝑦 ∈ ((cls‘𝐽)‘𝑆)(𝑥(-g𝐺)𝑦) ∈ ((cls‘𝐽)‘𝑆))
782, 35issubg4 18095 . . 3 (𝐺 ∈ Grp → (((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺) ↔ (((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺) ∧ ((cls‘𝐽)‘𝑆) ≠ ∅ ∧ ∀𝑥 ∈ ((cls‘𝐽)‘𝑆)∀𝑦 ∈ ((cls‘𝐽)‘𝑆)(𝑥(-g𝐺)𝑦) ∈ ((cls‘𝐽)‘𝑆))))
7934, 78syl 17 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺) ↔ (((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺) ∧ ((cls‘𝐽)‘𝑆) ≠ ∅ ∧ ∀𝑥 ∈ ((cls‘𝐽)‘𝑆)∀𝑦 ∈ ((cls‘𝐽)‘𝑆)(𝑥(-g𝐺)𝑦) ∈ ((cls‘𝐽)‘𝑆))))
8015, 23, 77, 79mpbir3and 1323 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1069   = wceq 1508  wcel 2051  wne 2962  wral 3083  wss 3824  c0 4173  cop 4442   cuni 4709   × cxp 5402  ccnv 5403  dom cdm 5404  cima 5407  Fun wfun 6180   Fn wfn 6181  wf 6182  cfv 6186  (class class class)co 6975  Basecbs 16338  TopOpenctopn 16550  0gc0g 16568  Grpcgrp 17904  -gcsg 17906  SubGrpcsubg 18070  Topctop 21221  TopOnctopon 21238  clsccl 21346   Cn ccn 21552   ×t ctx 21888  TopGrpctgp 22399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278  ax-cnex 10390  ax-resscn 10391  ax-1cn 10392  ax-icn 10393  ax-addcl 10394  ax-addrcl 10395  ax-mulcl 10396  ax-mulrcl 10397  ax-mulcom 10398  ax-addass 10399  ax-mulass 10400  ax-distr 10401  ax-i2m1 10402  ax-1ne0 10403  ax-1rid 10404  ax-rnegex 10405  ax-rrecex 10406  ax-cnre 10407  ax-pre-lttri 10408  ax-pre-lttrn 10409  ax-pre-ltadd 10410  ax-pre-mulgt0 10411
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-nel 3069  df-ral 3088  df-rex 3089  df-reu 3090  df-rmo 3091  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-pss 3840  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-int 4747  df-iun 4791  df-iin 4792  df-br 4927  df-opab 4989  df-mpt 5006  df-tr 5028  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-riota 6936  df-ov 6978  df-oprab 6979  df-mpo 6980  df-om 7396  df-1st 7500  df-2nd 7501  df-wrecs 7749  df-recs 7811  df-rdg 7849  df-er 8088  df-map 8207  df-en 8306  df-dom 8307  df-sdom 8308  df-pnf 10475  df-mnf 10476  df-xr 10477  df-ltxr 10478  df-le 10479  df-sub 10671  df-neg 10672  df-nn 11439  df-2 11502  df-ndx 16341  df-slot 16342  df-base 16344  df-sets 16345  df-ress 16346  df-plusg 16433  df-0g 16570  df-topgen 16572  df-plusf 17722  df-mgm 17723  df-sgrp 17765  df-mnd 17776  df-grp 17907  df-minusg 17908  df-sbg 17909  df-subg 18073  df-top 21222  df-topon 21239  df-topsp 21261  df-bases 21274  df-cld 21347  df-ntr 21348  df-cls 21349  df-cn 21555  df-tx 21890  df-tmd 22400  df-tgp 22401
This theorem is referenced by:  clsnsg  22437  tgptsmscls  22477
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