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Theorem clssubg 22194
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 2765 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
31, 2tgptopon 22168 . . . . . 6 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
43adantr 472 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
5 topontop 21000 . . . . 5 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
64, 5syl 17 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐽 ∈ Top)
72subgss 17862 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
87adantl 473 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
9 toponuni 21001 . . . . . 6 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = 𝐽)
104, 9syl 17 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (Base‘𝐺) = 𝐽)
118, 10sseqtrd 3803 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 𝐽)
12 eqid 2765 . . . . 5 𝐽 = 𝐽
1312clsss3 21146 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
146, 11, 13syl2anc 579 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
1514, 10sseqtr4d 3804 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
1612sscls 21143 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
176, 11, 16syl2anc 579 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
18 eqid 2765 . . . . . 6 (0g𝐺) = (0g𝐺)
1918subg0cl 17869 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑆)
2019adantl 473 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (0g𝐺) ∈ 𝑆)
2120ne0d 4088 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ≠ ∅)
22 ssn0 4140 . . 3 ((𝑆 ⊆ ((cls‘𝐽)‘𝑆) ∧ 𝑆 ≠ ∅) → ((cls‘𝐽)‘𝑆) ≠ ∅)
2317, 21, 22syl2anc 579 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ≠ ∅)
24 df-ov 6847 . . . 4 (𝑥(-g𝐺)𝑦) = ((-g𝐺)‘⟨𝑥, 𝑦⟩)
25 opelxpi 5316 . . . . . . 7 ((𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆)) → ⟨𝑥, 𝑦⟩ ∈ (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)))
26 txcls 21690 . . . . . . . . . 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 867 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) = (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)))
28 txtopon 21677 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺))) → (𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))))
294, 4, 28syl2anc 579 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))))
30 topontop 21000 . . . . . . . . . . . 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 5669 . . . . . . . . . . . . 13 ((-g𝐺) “ 𝑆) ⊆ dom (-g𝐺)
33 tgpgrp 22164 . . . . . . . . . . . . . . 15 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
3433adantr 472 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
35 eqid 2765 . . . . . . . . . . . . . . 15 (-g𝐺) = (-g𝐺)
362, 35grpsubf 17764 . . . . . . . . . . . . . 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 6241 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((-g𝐺) “ 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
39 toponuni 21001 . . . . . . . . . . . . 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 3803 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((-g𝐺) “ 𝑆) ⊆ (𝐽 ×t 𝐽))
4235subgsubcl 17872 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆𝑦𝑆) → (𝑥(-g𝐺)𝑦) ∈ 𝑆)
43423expb 1149 . . . . . . . . . . . . . . 15 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(-g𝐺)𝑦) ∈ 𝑆)
4443ralrimivva 3118 . . . . . . . . . . . . . 14 (𝑆 ∈ (SubGrp‘𝐺) → ∀𝑥𝑆𝑦𝑆 (𝑥(-g𝐺)𝑦) ∈ 𝑆)
45 fveq2 6377 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨𝑥, 𝑦⟩ → ((-g𝐺)‘𝑧) = ((-g𝐺)‘⟨𝑥, 𝑦⟩))
4645, 24syl6eqr 2817 . . . . . . . . . . . . . . . 16 (𝑧 = ⟨𝑥, 𝑦⟩ → ((-g𝐺)‘𝑧) = (𝑥(-g𝐺)𝑦))
4746eleq1d 2829 . . . . . . . . . . . . . . 15 (𝑧 = ⟨𝑥, 𝑦⟩ → (((-g𝐺)‘𝑧) ∈ 𝑆 ↔ (𝑥(-g𝐺)𝑦) ∈ 𝑆))
4847ralxp 5434 . . . . . . . . . . . . . 14 (∀𝑧 ∈ (𝑆 × 𝑆)((-g𝐺)‘𝑧) ∈ 𝑆 ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(-g𝐺)𝑦) ∈ 𝑆)
4944, 48sylibr 225 . . . . . . . . . . . . 13 (𝑆 ∈ (SubGrp‘𝐺) → ∀𝑧 ∈ (𝑆 × 𝑆)((-g𝐺)‘𝑧) ∈ 𝑆)
5049adantl 473 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑧 ∈ (𝑆 × 𝑆)((-g𝐺)‘𝑧) ∈ 𝑆)
5137ffund 6229 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → Fun (-g𝐺))
52 xpss12 5294 . . . . . . . . . . . . . . 15 ((𝑆 ⊆ (Base‘𝐺) ∧ 𝑆 ⊆ (Base‘𝐺)) → (𝑆 × 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
538, 8, 52syl2anc 579 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
5437fdmd 6234 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → dom (-g𝐺) = ((Base‘𝐺) × (Base‘𝐺)))
5553, 54sseqtr4d 3804 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ dom (-g𝐺))
56 funimass5 6526 . . . . . . . . . . . . 13 ((Fun (-g𝐺) ∧ (𝑆 × 𝑆) ⊆ dom (-g𝐺)) → ((𝑆 × 𝑆) ⊆ ((-g𝐺) “ 𝑆) ↔ ∀𝑧 ∈ (𝑆 × 𝑆)((-g𝐺)‘𝑧) ∈ 𝑆))
5751, 55, 56syl2anc 579 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((𝑆 × 𝑆) ⊆ ((-g𝐺) “ 𝑆) ↔ ∀𝑧 ∈ (𝑆 × 𝑆)((-g𝐺)‘𝑧) ∈ 𝑆))
5850, 57mpbird 248 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ ((-g𝐺) “ 𝑆))
59 eqid 2765 . . . . . . . . . . . 12 (𝐽 ×t 𝐽) = (𝐽 ×t 𝐽)
6059clsss 21141 . . . . . . . . . . 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 1490 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) ⊆ ((cls‘(𝐽 ×t 𝐽))‘((-g𝐺) “ 𝑆)))
621, 35tgpsubcn 22176 . . . . . . . . . . . 12 (𝐺 ∈ TopGrp → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
6362adantr 472 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
6412cncls2i 21357 . . . . . . . . . . 11 (((-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) ∧ 𝑆 𝐽) → ((cls‘(𝐽 ×t 𝐽))‘((-g𝐺) “ 𝑆)) ⊆ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
6563, 11, 64syl2anc 579 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘((-g𝐺) “ 𝑆)) ⊆ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
6661, 65sstrd 3773 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) ⊆ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
6727, 66eqsstr3d 3802 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)) ⊆ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
6867sselda 3763 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ ⟨𝑥, 𝑦⟩ ∈ (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆))) → ⟨𝑥, 𝑦⟩ ∈ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
6925, 68sylan2 586 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → ⟨𝑥, 𝑦⟩ ∈ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
7033ad2antrr 717 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → 𝐺 ∈ Grp)
71 ffn 6225 . . . . . . 7 ((-g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺) → (-g𝐺) Fn ((Base‘𝐺) × (Base‘𝐺)))
72 elpreima 6529 . . . . . . 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 223 . . . . 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 489 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ ((cls‘𝐽)‘𝑆))
7624, 75syl5eqel 2848 . . 3 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → (𝑥(-g𝐺)𝑦) ∈ ((cls‘𝐽)‘𝑆))
7776ralrimivva 3118 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑥 ∈ ((cls‘𝐽)‘𝑆)∀𝑦 ∈ ((cls‘𝐽)‘𝑆)(𝑥(-g𝐺)𝑦) ∈ ((cls‘𝐽)‘𝑆))
782, 35issubg4 17880 . . 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 1442 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  wss 3734  c0 4081  cop 4342   cuni 4596   × cxp 5277  ccnv 5278  dom cdm 5279  cima 5282  Fun wfun 6064   Fn wfn 6065  wf 6066  cfv 6070  (class class class)co 6844  Basecbs 16133  TopOpenctopn 16351  0gc0g 16369  Grpcgrp 17692  -gcsg 17694  SubGrpcsubg 17855  Topctop 20980  TopOnctopon 20997  clsccl 21105   Cn ccn 21311   ×t ctx 21646  TopGrpctgp 22157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-iin 4681  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-er 7949  df-map 8064  df-en 8163  df-dom 8164  df-sdom 8165  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-nn 11277  df-2 11337  df-ndx 16136  df-slot 16137  df-base 16139  df-sets 16140  df-ress 16141  df-plusg 16230  df-0g 16371  df-topgen 16373  df-plusf 17510  df-mgm 17511  df-sgrp 17553  df-mnd 17564  df-grp 17695  df-minusg 17696  df-sbg 17697  df-subg 17858  df-top 20981  df-topon 20998  df-topsp 21020  df-bases 21033  df-cld 21106  df-ntr 21107  df-cls 21108  df-cn 21314  df-tx 21648  df-tmd 22158  df-tgp 22159
This theorem is referenced by:  clsnsg  22195  tgptsmscls  22235
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