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Theorem opnsubg 22204
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 2765 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
21subgss 17873 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
323ad2ant2 1164 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆 ⊆ (Base‘𝐺))
4 subgntr.h . . . . . 6 𝐽 = (TopOpen‘𝐺)
54, 1tgptopon 22179 . . . . 5 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
653ad2ant1 1163 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
7 toponuni 21012 . . . 4 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = 𝐽)
86, 7syl 17 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → (Base‘𝐺) = 𝐽)
93, 8sseqtrd 3803 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆 𝐽)
108difeq1d 3891 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → ((Base‘𝐺) ∖ 𝑆) = ( 𝐽𝑆))
11 df-ima 5292 . . . . . . . 8 ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) “ 𝑆) = ran ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ↾ 𝑆)
123adantr 472 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆 ⊆ (Base‘𝐺))
1312resmptd 5631 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ↾ 𝑆) = (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)))
1413rneqd 5523 . . . . . . . 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, 14syl5eq 2811 . . . . . . 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 1242 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝐺 ∈ TopGrp)
17 eldifi 3896 . . . . . . . . . 10 (𝑥 ∈ ((Base‘𝐺) ∖ 𝑆) → 𝑥 ∈ (Base‘𝐺))
1817adantl 473 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑥 ∈ (Base‘𝐺))
19 eqid 2765 . . . . . . . . . 10 (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) = (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦))
20 eqid 2765 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
2119, 1, 20, 4tgplacthmeo 22200 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
2216, 18, 21syl2anc 579 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
23 simpl3 1246 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆𝐽)
24 hmeoima 21862 . . . . . . . 8 (((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ∈ (𝐽Homeo𝐽) ∧ 𝑆𝐽) → ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) “ 𝑆) ∈ 𝐽)
2522, 23, 24syl2anc 579 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) “ 𝑆) ∈ 𝐽)
2615, 25eqeltrrd 2845 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) ∈ 𝐽)
27 tgpgrp 22175 . . . . . . . . 9 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2816, 27syl 17 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝐺 ∈ Grp)
29 eqid 2765 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
301, 20, 29grprid 17734 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥(+g𝐺)(0g𝐺)) = 𝑥)
3128, 18, 30syl2anc 579 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑥(+g𝐺)(0g𝐺)) = 𝑥)
32 simpl2 1244 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆 ∈ (SubGrp‘𝐺))
3329subg0cl 17880 . . . . . . . . 9 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑆)
3432, 33syl 17 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (0g𝐺) ∈ 𝑆)
35 ovex 6878 . . . . . . . 8 (𝑥(+g𝐺)(0g𝐺)) ∈ V
36 eqid 2765 . . . . . . . . 9 (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) = (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦))
37 oveq2 6854 . . . . . . . . 9 (𝑦 = (0g𝐺) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐺)(0g𝐺)))
3836, 37elrnmpt1s 5544 . . . . . . . 8 (((0g𝐺) ∈ 𝑆 ∧ (𝑥(+g𝐺)(0g𝐺)) ∈ V) → (𝑥(+g𝐺)(0g𝐺)) ∈ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)))
3934, 35, 38sylancl 580 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑥(+g𝐺)(0g𝐺)) ∈ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)))
4031, 39eqeltrrd 2845 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑥 ∈ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)))
4128adantr 472 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → 𝐺 ∈ Grp)
4218adantr 472 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → 𝑥 ∈ (Base‘𝐺))
4312sselda 3763 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → 𝑦 ∈ (Base‘𝐺))
441, 20grpcl 17711 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
4541, 42, 43, 44syl3anc 1490 . . . . . . . . 9 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
46 eldifn 3897 . . . . . . . . . . 11 (𝑥 ∈ ((Base‘𝐺) ∖ 𝑆) → ¬ 𝑥𝑆)
4746ad2antlr 718 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → ¬ 𝑥𝑆)
48 eqid 2765 . . . . . . . . . . . . . . 15 (-g𝐺) = (-g𝐺)
4948subgsubcl 17883 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑥(+g𝐺)𝑦) ∈ 𝑆𝑦𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) ∈ 𝑆)
50493com23 1156 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦𝑆 ∧ (𝑥(+g𝐺)𝑦) ∈ 𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) ∈ 𝑆)
51503expia 1150 . . . . . . . . . . . 12 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦) ∈ 𝑆 → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) ∈ 𝑆))
5232, 51sylan 575 . . . . . . . . . . 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 17787 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) = 𝑥)
5441, 42, 43, 53syl3anc 1490 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) = 𝑥)
5554eleq1d 2829 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → (((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) ∈ 𝑆𝑥𝑆))
5652, 55sylibd 230 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦) ∈ 𝑆𝑥𝑆))
5747, 56mtod 189 . . . . . . . . 9 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → ¬ (𝑥(+g𝐺)𝑦) ∈ 𝑆)
5845, 57eldifd 3745 . . . . . . . 8 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → (𝑥(+g𝐺)𝑦) ∈ ((Base‘𝐺) ∖ 𝑆))
5958fmpttd 6579 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)):𝑆⟶((Base‘𝐺) ∖ 𝑆))
6059frnd 6232 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) ⊆ ((Base‘𝐺) ∖ 𝑆))
61 eleq2 2833 . . . . . . . 8 (𝑢 = ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) → (𝑥𝑢𝑥 ∈ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦))))
62 sseq1 3788 . . . . . . . 8 (𝑢 = ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) → (𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆) ↔ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) ⊆ ((Base‘𝐺) ∖ 𝑆)))
6361, 62anbi12d 624 . . . . . . 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 3462 . . . . . 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 865 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)))
6665ralrimiva 3113 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → ∀𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)))
67 topontop 21011 . . . . . 6 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
686, 67syl 17 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝐽 ∈ Top)
69 eltop2 21073 . . . . 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 248 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → ((Base‘𝐺) ∖ 𝑆) ∈ 𝐽)
7210, 71eqeltrrd 2845 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → ( 𝐽𝑆) ∈ 𝐽)
73 eqid 2765 . . . 4 𝐽 = 𝐽
7473iscld 21125 . . 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 704 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆 ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  wrex 3056  Vcvv 3350  cdif 3731  wss 3734   cuni 4596  cmpt 4890  ran crn 5280  cres 5281  cima 5282  cfv 6070  (class class class)co 6846  Basecbs 16144  +gcplusg 16228  TopOpenctopn 16362  0gc0g 16380  Grpcgrp 17703  -gcsg 17705  SubGrpcsubg 17866  Topctop 20991  TopOnctopon 21008  Clsdccld 21114  Homeochmeo 21850  TopGrpctgp 22168
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 7151  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270
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-iun 4680  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 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-er 7951  df-map 8066  df-en 8165  df-dom 8166  df-sdom 8167  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-nn 11279  df-2 11339  df-ndx 16147  df-slot 16148  df-base 16150  df-sets 16151  df-ress 16152  df-plusg 16241  df-0g 16382  df-topgen 16384  df-plusf 17521  df-mgm 17522  df-sgrp 17564  df-mnd 17575  df-grp 17706  df-minusg 17707  df-sbg 17708  df-subg 17869  df-top 20992  df-topon 21009  df-topsp 21031  df-bases 21044  df-cld 21117  df-cn 21325  df-cnp 21326  df-tx 21659  df-hmeo 21852  df-tmd 22169  df-tgp 22170
This theorem is referenced by:  cldsubg  22207  tgpconncompss  22210
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