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Theorem opnsubg 22708
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 2819 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
21subgss 18272 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
323ad2ant2 1129 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆 ⊆ (Base‘𝐺))
4 subgntr.h . . . . . 6 𝐽 = (TopOpen‘𝐺)
54, 1tgptopon 22682 . . . . 5 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
653ad2ant1 1128 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
7 toponuni 21514 . . . 4 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = 𝐽)
86, 7syl 17 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → (Base‘𝐺) = 𝐽)
93, 8sseqtrd 4005 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆 𝐽)
108difeq1d 4096 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → ((Base‘𝐺) ∖ 𝑆) = ( 𝐽𝑆))
11 df-ima 5561 . . . . . . . 8 ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) “ 𝑆) = ran ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ↾ 𝑆)
123adantr 483 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆 ⊆ (Base‘𝐺))
1312resmptd 5901 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ↾ 𝑆) = (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)))
1413rneqd 5801 . . . . . . . 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 2866 . . . . . . 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 1186 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝐺 ∈ TopGrp)
17 eldifi 4101 . . . . . . . . . 10 (𝑥 ∈ ((Base‘𝐺) ∖ 𝑆) → 𝑥 ∈ (Base‘𝐺))
1817adantl 484 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑥 ∈ (Base‘𝐺))
19 eqid 2819 . . . . . . . . . 10 (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) = (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦))
20 eqid 2819 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
2119, 1, 20, 4tgplacthmeo 22703 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
2216, 18, 21syl2anc 586 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
23 simpl3 1188 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆𝐽)
24 hmeoima 22365 . . . . . . . 8 (((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ∈ (𝐽Homeo𝐽) ∧ 𝑆𝐽) → ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) “ 𝑆) ∈ 𝐽)
2522, 23, 24syl2anc 586 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) “ 𝑆) ∈ 𝐽)
2615, 25eqeltrrd 2912 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) ∈ 𝐽)
27 tgpgrp 22678 . . . . . . . . 9 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2816, 27syl 17 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝐺 ∈ Grp)
29 eqid 2819 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
301, 20, 29grprid 18126 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥(+g𝐺)(0g𝐺)) = 𝑥)
3128, 18, 30syl2anc 586 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑥(+g𝐺)(0g𝐺)) = 𝑥)
32 simpl2 1187 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆 ∈ (SubGrp‘𝐺))
3329subg0cl 18279 . . . . . . . . 9 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑆)
3432, 33syl 17 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (0g𝐺) ∈ 𝑆)
35 ovex 7181 . . . . . . . 8 (𝑥(+g𝐺)(0g𝐺)) ∈ V
36 eqid 2819 . . . . . . . . 9 (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) = (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦))
37 oveq2 7156 . . . . . . . . 9 (𝑦 = (0g𝐺) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐺)(0g𝐺)))
3836, 37elrnmpt1s 5822 . . . . . . . 8 (((0g𝐺) ∈ 𝑆 ∧ (𝑥(+g𝐺)(0g𝐺)) ∈ V) → (𝑥(+g𝐺)(0g𝐺)) ∈ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)))
3934, 35, 38sylancl 588 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑥(+g𝐺)(0g𝐺)) ∈ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)))
4031, 39eqeltrrd 2912 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑥 ∈ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)))
4128adantr 483 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → 𝐺 ∈ Grp)
4218adantr 483 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → 𝑥 ∈ (Base‘𝐺))
4312sselda 3965 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → 𝑦 ∈ (Base‘𝐺))
441, 20grpcl 18103 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
4541, 42, 43, 44syl3anc 1366 . . . . . . . . 9 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
46 eldifn 4102 . . . . . . . . . . 11 (𝑥 ∈ ((Base‘𝐺) ∖ 𝑆) → ¬ 𝑥𝑆)
4746ad2antlr 725 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → ¬ 𝑥𝑆)
48 eqid 2819 . . . . . . . . . . . . . . 15 (-g𝐺) = (-g𝐺)
4948subgsubcl 18282 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑥(+g𝐺)𝑦) ∈ 𝑆𝑦𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) ∈ 𝑆)
50493com23 1121 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦𝑆 ∧ (𝑥(+g𝐺)𝑦) ∈ 𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) ∈ 𝑆)
51503expia 1116 . . . . . . . . . . . 12 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦) ∈ 𝑆 → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) ∈ 𝑆))
5232, 51sylan 582 . . . . . . . . . . 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 18182 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) = 𝑥)
5441, 42, 43, 53syl3anc 1366 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) = 𝑥)
5554eleq1d 2895 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → (((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) ∈ 𝑆𝑥𝑆))
5652, 55sylibd 241 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦) ∈ 𝑆𝑥𝑆))
5747, 56mtod 200 . . . . . . . . 9 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → ¬ (𝑥(+g𝐺)𝑦) ∈ 𝑆)
5845, 57eldifd 3945 . . . . . . . 8 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → (𝑥(+g𝐺)𝑦) ∈ ((Base‘𝐺) ∖ 𝑆))
5958fmpttd 6872 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)):𝑆⟶((Base‘𝐺) ∖ 𝑆))
6059frnd 6514 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) ⊆ ((Base‘𝐺) ∖ 𝑆))
61 eleq2 2899 . . . . . . . 8 (𝑢 = ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) → (𝑥𝑢𝑥 ∈ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦))))
62 sseq1 3990 . . . . . . . 8 (𝑢 = ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) → (𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆) ↔ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) ⊆ ((Base‘𝐺) ∖ 𝑆)))
6361, 62anbi12d 632 . . . . . . 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 3621 . . . . . 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 834 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)))
6665ralrimiva 3180 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → ∀𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)))
67 topontop 21513 . . . . . 6 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
686, 67syl 17 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝐽 ∈ Top)
69 eltop2 21575 . . . . 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 259 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → ((Base‘𝐺) ∖ 𝑆) ∈ 𝐽)
7210, 71eqeltrrd 2912 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → ( 𝐽𝑆) ∈ 𝐽)
73 eqid 2819 . . . 4 𝐽 = 𝐽
7473iscld 21627 . . 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 208  wa 398  w3a 1082   = wceq 1531  wcel 2108  wral 3136  wrex 3137  Vcvv 3493  cdif 3931  wss 3934   cuni 4830  cmpt 5137  ran crn 5549  cres 5550  cima 5551  cfv 6348  (class class class)co 7148  Basecbs 16475  +gcplusg 16557  TopOpenctopn 16687  0gc0g 16705  Grpcgrp 18095  -gcsg 18097  SubGrpcsubg 18265  Topctop 21493  TopOnctopon 21510  Clsdccld 21616  Homeochmeo 22353  TopGrpctgp 22671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-0g 16707  df-topgen 16709  df-plusf 17843  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099  df-sbg 18100  df-subg 18268  df-top 21494  df-topon 21511  df-topsp 21533  df-bases 21546  df-cld 21619  df-cn 21827  df-cnp 21828  df-tx 22162  df-hmeo 22355  df-tmd 22672  df-tgp 22673
This theorem is referenced by:  cldsubg  22711  tgpconncompss  22714
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