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Theorem clssubg 22105
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 2752 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
31, 2tgptopon 22079 . . . . . 6 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
43adantr 472 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
5 topontop 20912 . . . . 5 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
64, 5syl 17 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐽 ∈ Top)
72subgss 17788 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
87adantl 473 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
9 toponuni 20913 . . . . . 6 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = 𝐽)
104, 9syl 17 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (Base‘𝐺) = 𝐽)
118, 10sseqtrd 3774 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 𝐽)
12 eqid 2752 . . . . 5 𝐽 = 𝐽
1312clsss3 21057 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
146, 11, 13syl2anc 696 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
1514, 10sseqtr4d 3775 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
1612sscls 21054 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
176, 11, 16syl2anc 696 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
18 eqid 2752 . . . . . 6 (0g𝐺) = (0g𝐺)
1918subg0cl 17795 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑆)
2019adantl 473 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (0g𝐺) ∈ 𝑆)
21 ne0i 4056 . . . 4 ((0g𝐺) ∈ 𝑆𝑆 ≠ ∅)
2220, 21syl 17 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ≠ ∅)
23 ssn0 4111 . . 3 ((𝑆 ⊆ ((cls‘𝐽)‘𝑆) ∧ 𝑆 ≠ ∅) → ((cls‘𝐽)‘𝑆) ≠ ∅)
2417, 22, 23syl2anc 696 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ≠ ∅)
25 df-ov 6808 . . . 4 (𝑥(-g𝐺)𝑦) = ((-g𝐺)‘⟨𝑥, 𝑦⟩)
26 opelxpi 5297 . . . . . . 7 ((𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆)) → ⟨𝑥, 𝑦⟩ ∈ (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)))
27 txcls 21601 . . . . . . . . . 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)))
284, 4, 8, 8, 27syl22anc 1474 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) = (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)))
29 txtopon 21588 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺))) → (𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))))
304, 4, 29syl2anc 696 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))))
31 topontop 20912 . . . . . . . . . . . 12 ((𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))) → (𝐽 ×t 𝐽) ∈ Top)
3230, 31syl 17 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐽 ×t 𝐽) ∈ Top)
33 cnvimass 5635 . . . . . . . . . . . . 13 ((-g𝐺) “ 𝑆) ⊆ dom (-g𝐺)
34 tgpgrp 22075 . . . . . . . . . . . . . . . 16 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
3534adantr 472 . . . . . . . . . . . . . . 15 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
36 eqid 2752 . . . . . . . . . . . . . . . 16 (-g𝐺) = (-g𝐺)
372, 36grpsubf 17687 . . . . . . . . . . . . . . 15 (𝐺 ∈ Grp → (-g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
3835, 37syl 17 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (-g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
39 fdm 6204 . . . . . . . . . . . . . 14 ((-g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺) → dom (-g𝐺) = ((Base‘𝐺) × (Base‘𝐺)))
4038, 39syl 17 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → dom (-g𝐺) = ((Base‘𝐺) × (Base‘𝐺)))
4133, 40syl5sseq 3786 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((-g𝐺) “ 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
42 toponuni 20913 . . . . . . . . . . . . 13 ((𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))) → ((Base‘𝐺) × (Base‘𝐺)) = (𝐽 ×t 𝐽))
4330, 42syl 17 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((Base‘𝐺) × (Base‘𝐺)) = (𝐽 ×t 𝐽))
4441, 43sseqtrd 3774 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((-g𝐺) “ 𝑆) ⊆ (𝐽 ×t 𝐽))
4536subgsubcl 17798 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆𝑦𝑆) → (𝑥(-g𝐺)𝑦) ∈ 𝑆)
46453expb 1113 . . . . . . . . . . . . . . 15 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(-g𝐺)𝑦) ∈ 𝑆)
4746ralrimivva 3101 . . . . . . . . . . . . . 14 (𝑆 ∈ (SubGrp‘𝐺) → ∀𝑥𝑆𝑦𝑆 (𝑥(-g𝐺)𝑦) ∈ 𝑆)
48 fveq2 6344 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨𝑥, 𝑦⟩ → ((-g𝐺)‘𝑧) = ((-g𝐺)‘⟨𝑥, 𝑦⟩))
4948, 25syl6eqr 2804 . . . . . . . . . . . . . . . 16 (𝑧 = ⟨𝑥, 𝑦⟩ → ((-g𝐺)‘𝑧) = (𝑥(-g𝐺)𝑦))
5049eleq1d 2816 . . . . . . . . . . . . . . 15 (𝑧 = ⟨𝑥, 𝑦⟩ → (((-g𝐺)‘𝑧) ∈ 𝑆 ↔ (𝑥(-g𝐺)𝑦) ∈ 𝑆))
5150ralxp 5411 . . . . . . . . . . . . . 14 (∀𝑧 ∈ (𝑆 × 𝑆)((-g𝐺)‘𝑧) ∈ 𝑆 ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(-g𝐺)𝑦) ∈ 𝑆)
5247, 51sylibr 224 . . . . . . . . . . . . 13 (𝑆 ∈ (SubGrp‘𝐺) → ∀𝑧 ∈ (𝑆 × 𝑆)((-g𝐺)‘𝑧) ∈ 𝑆)
5352adantl 473 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑧 ∈ (𝑆 × 𝑆)((-g𝐺)‘𝑧) ∈ 𝑆)
54 ffun 6201 . . . . . . . . . . . . . 14 ((-g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺) → Fun (-g𝐺))
5538, 54syl 17 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → Fun (-g𝐺))
56 xpss12 5273 . . . . . . . . . . . . . . 15 ((𝑆 ⊆ (Base‘𝐺) ∧ 𝑆 ⊆ (Base‘𝐺)) → (𝑆 × 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
578, 8, 56syl2anc 696 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
5857, 40sseqtr4d 3775 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ dom (-g𝐺))
59 funimass5 6489 . . . . . . . . . . . . 13 ((Fun (-g𝐺) ∧ (𝑆 × 𝑆) ⊆ dom (-g𝐺)) → ((𝑆 × 𝑆) ⊆ ((-g𝐺) “ 𝑆) ↔ ∀𝑧 ∈ (𝑆 × 𝑆)((-g𝐺)‘𝑧) ∈ 𝑆))
6055, 58, 59syl2anc 696 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((𝑆 × 𝑆) ⊆ ((-g𝐺) “ 𝑆) ↔ ∀𝑧 ∈ (𝑆 × 𝑆)((-g𝐺)‘𝑧) ∈ 𝑆))
6153, 60mpbird 247 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ ((-g𝐺) “ 𝑆))
62 eqid 2752 . . . . . . . . . . . 12 (𝐽 ×t 𝐽) = (𝐽 ×t 𝐽)
6362clsss 21052 . . . . . . . . . . 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)))
6432, 44, 61, 63syl3anc 1473 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) ⊆ ((cls‘(𝐽 ×t 𝐽))‘((-g𝐺) “ 𝑆)))
651, 36tgpsubcn 22087 . . . . . . . . . . . 12 (𝐺 ∈ TopGrp → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
6665adantr 472 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
6712cncls2i 21268 . . . . . . . . . . 11 (((-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) ∧ 𝑆 𝐽) → ((cls‘(𝐽 ×t 𝐽))‘((-g𝐺) “ 𝑆)) ⊆ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
6866, 11, 67syl2anc 696 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘((-g𝐺) “ 𝑆)) ⊆ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
6964, 68sstrd 3746 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) ⊆ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
7028, 69eqsstr3d 3773 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)) ⊆ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
7170sselda 3736 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ ⟨𝑥, 𝑦⟩ ∈ (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆))) → ⟨𝑥, 𝑦⟩ ∈ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
7226, 71sylan2 492 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → ⟨𝑥, 𝑦⟩ ∈ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
7334ad2antrr 764 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → 𝐺 ∈ Grp)
74 ffn 6198 . . . . . . 7 ((-g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺) → (-g𝐺) Fn ((Base‘𝐺) × (Base‘𝐺)))
75 elpreima 6492 . . . . . . 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))))
7673, 37, 74, 754syl 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))))
7772, 76mpbid 222 . . . . 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)))
7877simprd 482 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ ((cls‘𝐽)‘𝑆))
7925, 78syl5eqel 2835 . . 3 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → (𝑥(-g𝐺)𝑦) ∈ ((cls‘𝐽)‘𝑆))
8079ralrimivva 3101 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑥 ∈ ((cls‘𝐽)‘𝑆)∀𝑦 ∈ ((cls‘𝐽)‘𝑆)(𝑥(-g𝐺)𝑦) ∈ ((cls‘𝐽)‘𝑆))
812, 36issubg4 17806 . . 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))))
8235, 81syl 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))))
8315, 24, 80, 82mpbir3and 1425 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  wne 2924  wral 3042  wss 3707  c0 4050  cop 4319   cuni 4580   × cxp 5256  ccnv 5257  dom cdm 5258  cima 5261  Fun wfun 6035   Fn wfn 6036  wf 6037  cfv 6041  (class class class)co 6805  Basecbs 16051  TopOpenctopn 16276  0gc0g 16294  Grpcgrp 17615  -gcsg 17617  SubGrpcsubg 17781  Topctop 20892  TopOnctopon 20909  clsccl 21016   Cn ccn 21222   ×t ctx 21557  TopGrpctgp 22068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-iin 4667  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-er 7903  df-map 8017  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-ndx 16054  df-slot 16055  df-base 16057  df-sets 16058  df-ress 16059  df-plusg 16148  df-0g 16296  df-topgen 16298  df-plusf 17434  df-mgm 17435  df-sgrp 17477  df-mnd 17488  df-grp 17618  df-minusg 17619  df-sbg 17620  df-subg 17784  df-top 20893  df-topon 20910  df-topsp 20931  df-bases 20944  df-cld 21017  df-ntr 21018  df-cls 21019  df-cn 21225  df-tx 21559  df-tmd 22069  df-tgp 22070
This theorem is referenced by:  clsnsg  22106  tgptsmscls  22146
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