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Theorem clssubg 24117
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 2737 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
31, 2tgptopon 24090 . . . . . 6 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
43adantr 480 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
5 topontop 22919 . . . . 5 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
64, 5syl 17 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐽 ∈ Top)
72subgss 19145 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
87adantl 481 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
9 toponuni 22920 . . . . . 6 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = 𝐽)
104, 9syl 17 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (Base‘𝐺) = 𝐽)
118, 10sseqtrd 4020 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 𝐽)
12 eqid 2737 . . . . 5 𝐽 = 𝐽
1312clsss3 23067 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
146, 11, 13syl2anc 584 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
1514, 10sseqtrrd 4021 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
1612sscls 23064 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
176, 11, 16syl2anc 584 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
18 eqid 2737 . . . . . 6 (0g𝐺) = (0g𝐺)
1918subg0cl 19152 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑆)
2019adantl 481 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (0g𝐺) ∈ 𝑆)
2120ne0d 4342 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ≠ ∅)
22 ssn0 4404 . . 3 ((𝑆 ⊆ ((cls‘𝐽)‘𝑆) ∧ 𝑆 ≠ ∅) → ((cls‘𝐽)‘𝑆) ≠ ∅)
2317, 21, 22syl2anc 584 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ≠ ∅)
24 df-ov 7434 . . . 4 (𝑥(-g𝐺)𝑦) = ((-g𝐺)‘⟨𝑥, 𝑦⟩)
25 opelxpi 5722 . . . . . . 7 ((𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆)) → ⟨𝑥, 𝑦⟩ ∈ (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)))
26 txcls 23612 . . . . . . . . . 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 839 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) = (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)))
28 txtopon 23599 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺))) → (𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))))
294, 4, 28syl2anc 584 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))))
30 topontop 22919 . . . . . . . . . . . 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 6100 . . . . . . . . . . . . 13 ((-g𝐺) “ 𝑆) ⊆ dom (-g𝐺)
33 tgpgrp 24086 . . . . . . . . . . . . . . 15 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
3433adantr 480 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
35 eqid 2737 . . . . . . . . . . . . . . 15 (-g𝐺) = (-g𝐺)
362, 35grpsubf 19037 . . . . . . . . . . . . . 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 6755 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((-g𝐺) “ 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
39 toponuni 22920 . . . . . . . . . . . . 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 4020 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((-g𝐺) “ 𝑆) ⊆ (𝐽 ×t 𝐽))
4235subgsubcl 19155 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆𝑦𝑆) → (𝑥(-g𝐺)𝑦) ∈ 𝑆)
43423expb 1121 . . . . . . . . . . . . . . 15 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(-g𝐺)𝑦) ∈ 𝑆)
4443ralrimivva 3202 . . . . . . . . . . . . . 14 (𝑆 ∈ (SubGrp‘𝐺) → ∀𝑥𝑆𝑦𝑆 (𝑥(-g𝐺)𝑦) ∈ 𝑆)
45 fveq2 6906 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨𝑥, 𝑦⟩ → ((-g𝐺)‘𝑧) = ((-g𝐺)‘⟨𝑥, 𝑦⟩))
4645, 24eqtr4di 2795 . . . . . . . . . . . . . . . 16 (𝑧 = ⟨𝑥, 𝑦⟩ → ((-g𝐺)‘𝑧) = (𝑥(-g𝐺)𝑦))
4746eleq1d 2826 . . . . . . . . . . . . . . 15 (𝑧 = ⟨𝑥, 𝑦⟩ → (((-g𝐺)‘𝑧) ∈ 𝑆 ↔ (𝑥(-g𝐺)𝑦) ∈ 𝑆))
4847ralxp 5852 . . . . . . . . . . . . . 14 (∀𝑧 ∈ (𝑆 × 𝑆)((-g𝐺)‘𝑧) ∈ 𝑆 ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(-g𝐺)𝑦) ∈ 𝑆)
4944, 48sylibr 234 . . . . . . . . . . . . 13 (𝑆 ∈ (SubGrp‘𝐺) → ∀𝑧 ∈ (𝑆 × 𝑆)((-g𝐺)‘𝑧) ∈ 𝑆)
5049adantl 481 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑧 ∈ (𝑆 × 𝑆)((-g𝐺)‘𝑧) ∈ 𝑆)
5137ffund 6740 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → Fun (-g𝐺))
52 xpss12 5700 . . . . . . . . . . . . . . 15 ((𝑆 ⊆ (Base‘𝐺) ∧ 𝑆 ⊆ (Base‘𝐺)) → (𝑆 × 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
538, 8, 52syl2anc 584 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
5437fdmd 6746 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → dom (-g𝐺) = ((Base‘𝐺) × (Base‘𝐺)))
5553, 54sseqtrrd 4021 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ dom (-g𝐺))
56 funimass5 7075 . . . . . . . . . . . . 13 ((Fun (-g𝐺) ∧ (𝑆 × 𝑆) ⊆ dom (-g𝐺)) → ((𝑆 × 𝑆) ⊆ ((-g𝐺) “ 𝑆) ↔ ∀𝑧 ∈ (𝑆 × 𝑆)((-g𝐺)‘𝑧) ∈ 𝑆))
5751, 55, 56syl2anc 584 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((𝑆 × 𝑆) ⊆ ((-g𝐺) “ 𝑆) ↔ ∀𝑧 ∈ (𝑆 × 𝑆)((-g𝐺)‘𝑧) ∈ 𝑆))
5850, 57mpbird 257 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ ((-g𝐺) “ 𝑆))
59 eqid 2737 . . . . . . . . . . . 12 (𝐽 ×t 𝐽) = (𝐽 ×t 𝐽)
6059clsss 23062 . . . . . . . . . . 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 1373 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) ⊆ ((cls‘(𝐽 ×t 𝐽))‘((-g𝐺) “ 𝑆)))
621, 35tgpsubcn 24098 . . . . . . . . . . . 12 (𝐺 ∈ TopGrp → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
6362adantr 480 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
6412cncls2i 23278 . . . . . . . . . . 11 (((-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) ∧ 𝑆 𝐽) → ((cls‘(𝐽 ×t 𝐽))‘((-g𝐺) “ 𝑆)) ⊆ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
6563, 11, 64syl2anc 584 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘((-g𝐺) “ 𝑆)) ⊆ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
6661, 65sstrd 3994 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) ⊆ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
6727, 66eqsstrrd 4019 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)) ⊆ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
6867sselda 3983 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ ⟨𝑥, 𝑦⟩ ∈ (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆))) → ⟨𝑥, 𝑦⟩ ∈ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
6925, 68sylan2 593 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → ⟨𝑥, 𝑦⟩ ∈ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
7033ad2antrr 726 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → 𝐺 ∈ Grp)
71 ffn 6736 . . . . . . 7 ((-g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺) → (-g𝐺) Fn ((Base‘𝐺) × (Base‘𝐺)))
72 elpreima 7078 . . . . . . 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 232 . . . . 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 495 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ ((cls‘𝐽)‘𝑆))
7624, 75eqeltrid 2845 . . 3 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → (𝑥(-g𝐺)𝑦) ∈ ((cls‘𝐽)‘𝑆))
7776ralrimivva 3202 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑥 ∈ ((cls‘𝐽)‘𝑆)∀𝑦 ∈ ((cls‘𝐽)‘𝑆)(𝑥(-g𝐺)𝑦) ∈ ((cls‘𝐽)‘𝑆))
782, 35issubg4 19163 . . 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 1343 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wss 3951  c0 4333  cop 4632   cuni 4907   × cxp 5683  ccnv 5684  dom cdm 5685  cima 5688  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  TopOpenctopn 17466  0gc0g 17484  Grpcgrp 18951  -gcsg 18953  SubGrpcsubg 19138  Topctop 22899  TopOnctopon 22916  clsccl 23026   Cn ccn 23232   ×t ctx 23568  TopGrpctgp 24079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-topgen 17488  df-plusf 18652  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-cn 23235  df-tx 23570  df-tmd 24080  df-tgp 24081
This theorem is referenced by:  clsnsg  24118  tgptsmscls  24158
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