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Theorem clssubg 22113
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 2760 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
31, 2tgptopon 22087 . . . . . 6 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
43adantr 472 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
5 topontop 20920 . . . . 5 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
64, 5syl 17 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐽 ∈ Top)
72subgss 17796 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
87adantl 473 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
9 toponuni 20921 . . . . . 6 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = 𝐽)
104, 9syl 17 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (Base‘𝐺) = 𝐽)
118, 10sseqtrd 3782 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 𝐽)
12 eqid 2760 . . . . 5 𝐽 = 𝐽
1312clsss3 21065 . . . 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 3783 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
1612sscls 21062 . . . 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 2760 . . . . . 6 (0g𝐺) = (0g𝐺)
1918subg0cl 17803 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑆)
2019adantl 473 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (0g𝐺) ∈ 𝑆)
21 ne0i 4064 . . . 4 ((0g𝐺) ∈ 𝑆𝑆 ≠ ∅)
2220, 21syl 17 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ≠ ∅)
23 ssn0 4119 . . 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 6816 . . . 4 (𝑥(-g𝐺)𝑦) = ((-g𝐺)‘⟨𝑥, 𝑦⟩)
26 opelxpi 5305 . . . . . . 7 ((𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆)) → ⟨𝑥, 𝑦⟩ ∈ (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)))
27 txcls 21609 . . . . . . . . . 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 1478 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) = (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)))
29 txtopon 21596 . . . . . . . . . . . . 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 20920 . . . . . . . . . . . 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 5643 . . . . . . . . . . . . 13 ((-g𝐺) “ 𝑆) ⊆ dom (-g𝐺)
34 tgpgrp 22083 . . . . . . . . . . . . . . . 16 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
3534adantr 472 . . . . . . . . . . . . . . 15 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
36 eqid 2760 . . . . . . . . . . . . . . . 16 (-g𝐺) = (-g𝐺)
372, 36grpsubf 17695 . . . . . . . . . . . . . . 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 6212 . . . . . . . . . . . . . 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 3794 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((-g𝐺) “ 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
42 toponuni 20921 . . . . . . . . . . . . 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 3782 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((-g𝐺) “ 𝑆) ⊆ (𝐽 ×t 𝐽))
4536subgsubcl 17806 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆𝑦𝑆) → (𝑥(-g𝐺)𝑦) ∈ 𝑆)
46453expb 1114 . . . . . . . . . . . . . . 15 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(-g𝐺)𝑦) ∈ 𝑆)
4746ralrimivva 3109 . . . . . . . . . . . . . 14 (𝑆 ∈ (SubGrp‘𝐺) → ∀𝑥𝑆𝑦𝑆 (𝑥(-g𝐺)𝑦) ∈ 𝑆)
48 fveq2 6352 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨𝑥, 𝑦⟩ → ((-g𝐺)‘𝑧) = ((-g𝐺)‘⟨𝑥, 𝑦⟩))
4948, 25syl6eqr 2812 . . . . . . . . . . . . . . . 16 (𝑧 = ⟨𝑥, 𝑦⟩ → ((-g𝐺)‘𝑧) = (𝑥(-g𝐺)𝑦))
5049eleq1d 2824 . . . . . . . . . . . . . . 15 (𝑧 = ⟨𝑥, 𝑦⟩ → (((-g𝐺)‘𝑧) ∈ 𝑆 ↔ (𝑥(-g𝐺)𝑦) ∈ 𝑆))
5150ralxp 5419 . . . . . . . . . . . . . 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 6209 . . . . . . . . . . . . . 14 ((-g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺) → Fun (-g𝐺))
5538, 54syl 17 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → Fun (-g𝐺))
56 xpss12 5281 . . . . . . . . . . . . . . 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 3783 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ dom (-g𝐺))
59 funimass5 6497 . . . . . . . . . . . . 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 2760 . . . . . . . . . . . 12 (𝐽 ×t 𝐽) = (𝐽 ×t 𝐽)
6362clsss 21060 . . . . . . . . . . 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 1477 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) ⊆ ((cls‘(𝐽 ×t 𝐽))‘((-g𝐺) “ 𝑆)))
651, 36tgpsubcn 22095 . . . . . . . . . . . 12 (𝐺 ∈ TopGrp → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
6665adantr 472 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
6712cncls2i 21276 . . . . . . . . . . 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 3754 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) ⊆ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
7028, 69eqsstr3d 3781 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)) ⊆ ((-g𝐺) “ ((cls‘𝐽)‘𝑆)))
7170sselda 3744 . . . . . . 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 6206 . . . . . . 7 ((-g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺) → (-g𝐺) Fn ((Base‘𝐺) × (Base‘𝐺)))
75 elpreima 6500 . . . . . . 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 2843 . . 3 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → (𝑥(-g𝐺)𝑦) ∈ ((cls‘𝐽)‘𝑆))
8079ralrimivva 3109 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑥 ∈ ((cls‘𝐽)‘𝑆)∀𝑦 ∈ ((cls‘𝐽)‘𝑆)(𝑥(-g𝐺)𝑦) ∈ ((cls‘𝐽)‘𝑆))
812, 36issubg4 17814 . . 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 1428 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 1632  wcel 2139  wne 2932  wral 3050  wss 3715  c0 4058  cop 4327   cuni 4588   × cxp 5264  ccnv 5265  dom cdm 5266  cima 5269  Fun wfun 6043   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6813  Basecbs 16059  TopOpenctopn 16284  0gc0g 16302  Grpcgrp 17623  -gcsg 17625  SubGrpcsubg 17789  Topctop 20900  TopOnctopon 20917  clsccl 21024   Cn ccn 21230   ×t ctx 21565  TopGrpctgp 22076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-0g 16304  df-topgen 16306  df-plusf 17442  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-grp 17626  df-minusg 17627  df-sbg 17628  df-subg 17792  df-top 20901  df-topon 20918  df-topsp 20939  df-bases 20952  df-cld 21025  df-ntr 21026  df-cls 21027  df-cn 21233  df-tx 21567  df-tmd 22077  df-tgp 22078
This theorem is referenced by:  clsnsg  22114  tgptsmscls  22154
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