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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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