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Theorem opnsubg 24032
Description: An open subgroup of a topological group is also closed. (Contributed by Mario Carneiro, 17-Sep-2015.)
Hypothesis
Ref Expression
subgntr.h 𝐽 = (TopOpen‘𝐺)
Assertion
Ref Expression
opnsubg ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆 ∈ (Clsd‘𝐽))

Proof of Theorem opnsubg
Dummy variables 𝑥 𝑢 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2728 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
21subgss 19089 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
323ad2ant2 1131 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆 ⊆ (Base‘𝐺))
4 subgntr.h . . . . . 6 𝐽 = (TopOpen‘𝐺)
54, 1tgptopon 24006 . . . . 5 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
653ad2ant1 1130 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
7 toponuni 22836 . . . 4 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = 𝐽)
86, 7syl 17 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → (Base‘𝐺) = 𝐽)
93, 8sseqtrd 4022 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆 𝐽)
108difeq1d 4121 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → ((Base‘𝐺) ∖ 𝑆) = ( 𝐽𝑆))
11 df-ima 5695 . . . . . . . 8 ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) “ 𝑆) = ran ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ↾ 𝑆)
123adantr 479 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆 ⊆ (Base‘𝐺))
1312resmptd 6049 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ↾ 𝑆) = (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)))
1413rneqd 5944 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ran ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ↾ 𝑆) = ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)))
1511, 14eqtrid 2780 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) “ 𝑆) = ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)))
16 simpl1 1188 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝐺 ∈ TopGrp)
17 eldifi 4127 . . . . . . . . . 10 (𝑥 ∈ ((Base‘𝐺) ∖ 𝑆) → 𝑥 ∈ (Base‘𝐺))
1817adantl 480 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑥 ∈ (Base‘𝐺))
19 eqid 2728 . . . . . . . . . 10 (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) = (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦))
20 eqid 2728 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
2119, 1, 20, 4tgplacthmeo 24027 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
2216, 18, 21syl2anc 582 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
23 simpl3 1190 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆𝐽)
24 hmeoima 23689 . . . . . . . 8 (((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ∈ (𝐽Homeo𝐽) ∧ 𝑆𝐽) → ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) “ 𝑆) ∈ 𝐽)
2522, 23, 24syl2anc 582 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) “ 𝑆) ∈ 𝐽)
2615, 25eqeltrrd 2830 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) ∈ 𝐽)
27 tgpgrp 24002 . . . . . . . . 9 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2816, 27syl 17 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝐺 ∈ Grp)
29 eqid 2728 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
301, 20, 29grprid 18932 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥(+g𝐺)(0g𝐺)) = 𝑥)
3128, 18, 30syl2anc 582 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑥(+g𝐺)(0g𝐺)) = 𝑥)
32 simpl2 1189 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆 ∈ (SubGrp‘𝐺))
3329subg0cl 19096 . . . . . . . . 9 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑆)
3432, 33syl 17 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (0g𝐺) ∈ 𝑆)
35 ovex 7459 . . . . . . . 8 (𝑥(+g𝐺)(0g𝐺)) ∈ V
36 eqid 2728 . . . . . . . . 9 (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) = (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦))
37 oveq2 7434 . . . . . . . . 9 (𝑦 = (0g𝐺) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐺)(0g𝐺)))
3836, 37elrnmpt1s 5963 . . . . . . . 8 (((0g𝐺) ∈ 𝑆 ∧ (𝑥(+g𝐺)(0g𝐺)) ∈ V) → (𝑥(+g𝐺)(0g𝐺)) ∈ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)))
3934, 35, 38sylancl 584 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑥(+g𝐺)(0g𝐺)) ∈ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)))
4031, 39eqeltrrd 2830 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑥 ∈ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)))
4128adantr 479 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → 𝐺 ∈ Grp)
4218adantr 479 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → 𝑥 ∈ (Base‘𝐺))
4312sselda 3982 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → 𝑦 ∈ (Base‘𝐺))
441, 20grpcl 18905 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
4541, 42, 43, 44syl3anc 1368 . . . . . . . . 9 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
46 eldifn 4128 . . . . . . . . . . 11 (𝑥 ∈ ((Base‘𝐺) ∖ 𝑆) → ¬ 𝑥𝑆)
4746ad2antlr 725 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → ¬ 𝑥𝑆)
48 eqid 2728 . . . . . . . . . . . . . . 15 (-g𝐺) = (-g𝐺)
4948subgsubcl 19099 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑥(+g𝐺)𝑦) ∈ 𝑆𝑦𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) ∈ 𝑆)
50493com23 1123 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦𝑆 ∧ (𝑥(+g𝐺)𝑦) ∈ 𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) ∈ 𝑆)
51503expia 1118 . . . . . . . . . . . 12 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦) ∈ 𝑆 → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) ∈ 𝑆))
5232, 51sylan 578 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦) ∈ 𝑆 → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) ∈ 𝑆))
531, 20, 48grppncan 18994 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) = 𝑥)
5441, 42, 43, 53syl3anc 1368 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) = 𝑥)
5554eleq1d 2814 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → (((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) ∈ 𝑆𝑥𝑆))
5652, 55sylibd 238 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦) ∈ 𝑆𝑥𝑆))
5747, 56mtod 197 . . . . . . . . 9 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → ¬ (𝑥(+g𝐺)𝑦) ∈ 𝑆)
5845, 57eldifd 3960 . . . . . . . 8 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → (𝑥(+g𝐺)𝑦) ∈ ((Base‘𝐺) ∖ 𝑆))
5958fmpttd 7130 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)):𝑆⟶((Base‘𝐺) ∖ 𝑆))
6059frnd 6735 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) ⊆ ((Base‘𝐺) ∖ 𝑆))
61 eleq2 2818 . . . . . . . 8 (𝑢 = ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) → (𝑥𝑢𝑥 ∈ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦))))
62 sseq1 4007 . . . . . . . 8 (𝑢 = ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) → (𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆) ↔ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) ⊆ ((Base‘𝐺) ∖ 𝑆)))
6361, 62anbi12d 630 . . . . . . 7 (𝑢 = ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) → ((𝑥𝑢𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)) ↔ (𝑥 ∈ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) ∧ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) ⊆ ((Base‘𝐺) ∖ 𝑆))))
6463rspcev 3611 . . . . . 6 ((ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) ∈ 𝐽 ∧ (𝑥 ∈ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) ∧ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) ⊆ ((Base‘𝐺) ∖ 𝑆))) → ∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)))
6526, 40, 60, 64syl12anc 835 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)))
6665ralrimiva 3143 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → ∀𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)))
67 topontop 22835 . . . . . 6 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
686, 67syl 17 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝐽 ∈ Top)
69 eltop2 22898 . . . . 5 (𝐽 ∈ Top → (((Base‘𝐺) ∖ 𝑆) ∈ 𝐽 ↔ ∀𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆))))
7068, 69syl 17 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → (((Base‘𝐺) ∖ 𝑆) ∈ 𝐽 ↔ ∀𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆))))
7166, 70mpbird 256 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → ((Base‘𝐺) ∖ 𝑆) ∈ 𝐽)
7210, 71eqeltrrd 2830 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → ( 𝐽𝑆) ∈ 𝐽)
73 eqid 2728 . . . 4 𝐽 = 𝐽
7473iscld 22951 . . 3 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 𝐽 ∧ ( 𝐽𝑆) ∈ 𝐽)))
7568, 74syl 17 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 𝐽 ∧ ( 𝐽𝑆) ∈ 𝐽)))
769, 72, 75mpbir2and 711 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆 ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3058  wrex 3067  Vcvv 3473  cdif 3946  wss 3949   cuni 4912  cmpt 5235  ran crn 5683  cres 5684  cima 5685  cfv 6553  (class class class)co 7426  Basecbs 17187  +gcplusg 17240  TopOpenctopn 17410  0gc0g 17428  Grpcgrp 18897  -gcsg 18899  SubGrpcsubg 19082  Topctop 22815  TopOnctopon 22832  Clsdccld 22940  Homeochmeo 23677  TopGrpctgp 23995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-er 8731  df-map 8853  df-en 8971  df-dom 8972  df-sdom 8973  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-0g 17430  df-topgen 17432  df-plusf 18606  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-grp 18900  df-minusg 18901  df-sbg 18902  df-subg 19085  df-top 22816  df-topon 22833  df-topsp 22855  df-bases 22869  df-cld 22943  df-cn 23151  df-cnp 23152  df-tx 23486  df-hmeo 23679  df-tmd 23996  df-tgp 23997
This theorem is referenced by:  cldsubg  24035  tgpconncompss  24038
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