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Theorem opnsubg 21834
 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 2621 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
21subgss 17527 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
323ad2ant2 1081 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆 ⊆ (Base‘𝐺))
4 subgntr.h . . . . . 6 𝐽 = (TopOpen‘𝐺)
54, 1tgptopon 21809 . . . . 5 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
653ad2ant1 1080 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
7 toponuni 20651 . . . 4 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = 𝐽)
86, 7syl 17 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → (Base‘𝐺) = 𝐽)
93, 8sseqtrd 3625 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆 𝐽)
108difeq1d 3710 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → ((Base‘𝐺) ∖ 𝑆) = ( 𝐽𝑆))
11 df-ima 5092 . . . . . . . 8 ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) “ 𝑆) = ran ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ↾ 𝑆)
123adantr 481 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆 ⊆ (Base‘𝐺))
1312resmptd 5416 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ↾ 𝑆) = (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)))
1413rneqd 5318 . . . . . . . 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 2667 . . . . . . 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 1062 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝐺 ∈ TopGrp)
17 eldifi 3715 . . . . . . . . . 10 (𝑥 ∈ ((Base‘𝐺) ∖ 𝑆) → 𝑥 ∈ (Base‘𝐺))
1817adantl 482 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑥 ∈ (Base‘𝐺))
19 eqid 2621 . . . . . . . . . 10 (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) = (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦))
20 eqid 2621 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
2119, 1, 20, 4tgplacthmeo 21830 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
2216, 18, 21syl2anc 692 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
23 simpl3 1064 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆𝐽)
24 hmeoima 21491 . . . . . . . 8 (((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) ∈ (𝐽Homeo𝐽) ∧ 𝑆𝐽) → ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) “ 𝑆) ∈ 𝐽)
2522, 23, 24syl2anc 692 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g𝐺)𝑦)) “ 𝑆) ∈ 𝐽)
2615, 25eqeltrrd 2699 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) ∈ 𝐽)
27 tgpgrp 21805 . . . . . . . . 9 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2816, 27syl 17 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝐺 ∈ Grp)
29 eqid 2621 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
301, 20, 29grprid 17385 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥(+g𝐺)(0g𝐺)) = 𝑥)
3128, 18, 30syl2anc 692 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑥(+g𝐺)(0g𝐺)) = 𝑥)
32 simpl2 1063 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆 ∈ (SubGrp‘𝐺))
3329subg0cl 17534 . . . . . . . . 9 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑆)
3432, 33syl 17 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (0g𝐺) ∈ 𝑆)
35 ovex 6638 . . . . . . . 8 (𝑥(+g𝐺)(0g𝐺)) ∈ V
36 eqid 2621 . . . . . . . . 9 (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) = (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦))
37 oveq2 6618 . . . . . . . . 9 (𝑦 = (0g𝐺) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐺)(0g𝐺)))
3836, 37elrnmpt1s 5338 . . . . . . . 8 (((0g𝐺) ∈ 𝑆 ∧ (𝑥(+g𝐺)(0g𝐺)) ∈ V) → (𝑥(+g𝐺)(0g𝐺)) ∈ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)))
3934, 35, 38sylancl 693 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑥(+g𝐺)(0g𝐺)) ∈ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)))
4031, 39eqeltrrd 2699 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑥 ∈ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)))
4128adantr 481 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → 𝐺 ∈ Grp)
4218adantr 481 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → 𝑥 ∈ (Base‘𝐺))
4312sselda 3587 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → 𝑦 ∈ (Base‘𝐺))
441, 20grpcl 17362 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
4541, 42, 43, 44syl3anc 1323 . . . . . . . . 9 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
46 eldifn 3716 . . . . . . . . . . 11 (𝑥 ∈ ((Base‘𝐺) ∖ 𝑆) → ¬ 𝑥𝑆)
4746ad2antlr 762 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → ¬ 𝑥𝑆)
48 eqid 2621 . . . . . . . . . . . . . . 15 (-g𝐺) = (-g𝐺)
4948subgsubcl 17537 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑥(+g𝐺)𝑦) ∈ 𝑆𝑦𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) ∈ 𝑆)
50493com23 1268 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦𝑆 ∧ (𝑥(+g𝐺)𝑦) ∈ 𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) ∈ 𝑆)
51503expia 1264 . . . . . . . . . . . 12 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦) ∈ 𝑆 → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) ∈ 𝑆))
5232, 51sylan 488 . . . . . . . . . . 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 17438 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) = 𝑥)
5441, 42, 43, 53syl3anc 1323 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) = 𝑥)
5554eleq1d 2683 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → (((𝑥(+g𝐺)𝑦)(-g𝐺)𝑦) ∈ 𝑆𝑥𝑆))
5652, 55sylibd 229 . . . . . . . . . 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 3570 . . . . . . . 8 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦𝑆) → (𝑥(+g𝐺)𝑦) ∈ ((Base‘𝐺) ∖ 𝑆))
5958, 36fmptd 6346 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)):𝑆⟶((Base‘𝐺) ∖ 𝑆))
60 frn 6015 . . . . . . 7 ((𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)):𝑆⟶((Base‘𝐺) ∖ 𝑆) → ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) ⊆ ((Base‘𝐺) ∖ 𝑆))
6159, 60syl 17 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) ⊆ ((Base‘𝐺) ∖ 𝑆))
62 eleq2 2687 . . . . . . . 8 (𝑢 = ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) → (𝑥𝑢𝑥 ∈ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦))))
63 sseq1 3610 . . . . . . . 8 (𝑢 = ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) → (𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆) ↔ ran (𝑦𝑆 ↦ (𝑥(+g𝐺)𝑦)) ⊆ ((Base‘𝐺) ∖ 𝑆)))
6462, 63anbi12d 746 . . . . . . 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))))
6564rspcev 3298 . . . . . 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)))
6626, 40, 61, 65syl12anc 1321 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)))
6766ralrimiva 2961 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → ∀𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)))
68 topontop 20650 . . . . . 6 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
696, 68syl 17 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝐽 ∈ Top)
70 eltop2 20703 . . . . 5 (𝐽 ∈ Top → (((Base‘𝐺) ∖ 𝑆) ∈ 𝐽 ↔ ∀𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆))))
7169, 70syl 17 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → (((Base‘𝐺) ∖ 𝑆) ∈ 𝐽 ↔ ∀𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆))))
7267, 71mpbird 247 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → ((Base‘𝐺) ∖ 𝑆) ∈ 𝐽)
7310, 72eqeltrrd 2699 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → ( 𝐽𝑆) ∈ 𝐽)
74 eqid 2621 . . . 4 𝐽 = 𝐽
7574iscld 20754 . . 3 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 𝐽 ∧ ( 𝐽𝑆) ∈ 𝐽)))
7669, 75syl 17 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 𝐽 ∧ ( 𝐽𝑆) ∈ 𝐽)))
779, 73, 76mpbir2and 956 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆 ∈ (Clsd‘𝐽))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908  Vcvv 3189   ∖ cdif 3556   ⊆ wss 3559  ∪ cuni 4407   ↦ cmpt 4678  ran crn 5080   ↾ cres 5081   “ cima 5082  ⟶wf 5848  ‘cfv 5852  (class class class)co 6610  Basecbs 15792  +gcplusg 15873  TopOpenctopn 16014  0gc0g 16032  Grpcgrp 17354  -gcsg 17356  SubGrpcsubg 17520  Topctop 20630  TopOnctopon 20647  Clsdccld 20743  Homeochmeo 21479  TopGrpctgp 21798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-0g 16034  df-topgen 16036  df-plusf 17173  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-grp 17357  df-minusg 17358  df-sbg 17359  df-subg 17523  df-top 20631  df-topon 20648  df-topsp 20661  df-bases 20674  df-cld 20746  df-cn 20954  df-cnp 20955  df-tx 21288  df-hmeo 21481  df-tmd 21799  df-tgp 21800 This theorem is referenced by:  cldsubg  21837  tgpconncompss  21840
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