Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  subgntr Structured version   Visualization version   GIF version

Theorem subgntr 21720
 Description: A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 21722, the interior of a subgroup is either a subgroup, or empty. (Contributed by Mario Carneiro, 19-Sep-2015.)
Hypothesis
Ref Expression
subgntr.h 𝐽 = (TopOpen‘𝐺)
Assertion
Ref Expression
subgntr ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝑆𝐽)

Proof of Theorem subgntr
Dummy variables 𝑥 𝑢 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ima 5051 . . . . . 6 ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) “ ((int‘𝐽)‘𝑆)) = ran ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ↾ ((int‘𝐽)‘𝑆))
2 subgntr.h . . . . . . . . . . . 12 𝐽 = (TopOpen‘𝐺)
3 eqid 2610 . . . . . . . . . . . 12 (Base‘𝐺) = (Base‘𝐺)
42, 3tgptopon 21696 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
543ad2ant1 1075 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
65adantr 480 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
7 topontop 20541 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
85, 7syl 17 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝐽 ∈ Top)
98adantr 480 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐽 ∈ Top)
10 simpl2 1058 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
113subgss 17418 . . . . . . . . . . . 12 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
1210, 11syl 17 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝑆 ⊆ (Base‘𝐺))
13 toponuni 20542 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = 𝐽)
146, 13syl 17 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → (Base‘𝐺) = 𝐽)
1512, 14sseqtrd 3604 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝑆 𝐽)
16 eqid 2610 . . . . . . . . . . 11 𝐽 = 𝐽
1716ntropn 20663 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
189, 15, 17syl2anc 691 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
19 toponss 20544 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ ((int‘𝐽)‘𝑆) ∈ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
206, 18, 19syl2anc 691 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ((int‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
2120resmptd 5371 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ↾ ((int‘𝐽)‘𝑆)) = (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)))
2221rneqd 5274 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ran ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ↾ ((int‘𝐽)‘𝑆)) = ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)))
231, 22syl5eq 2656 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) “ ((int‘𝐽)‘𝑆)) = ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)))
24 simpl1 1057 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐺 ∈ TopGrp)
25 simpr 476 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝑥𝑆)
2616ntrss2 20671 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
279, 15, 26syl2anc 691 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
28 simpl3 1059 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐴 ∈ ((int‘𝐽)‘𝑆))
2927, 28sseldd 3569 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐴𝑆)
30 eqid 2610 . . . . . . . . . 10 (-g𝐺) = (-g𝐺)
3130subgsubcl 17428 . . . . . . . . 9 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆𝐴𝑆) → (𝑥(-g𝐺)𝐴) ∈ 𝑆)
3210, 25, 29, 31syl3anc 1318 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → (𝑥(-g𝐺)𝐴) ∈ 𝑆)
3312, 32sseldd 3569 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → (𝑥(-g𝐺)𝐴) ∈ (Base‘𝐺))
34 eqid 2610 . . . . . . . 8 (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) = (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦))
35 eqid 2610 . . . . . . . 8 (+g𝐺) = (+g𝐺)
3634, 3, 35, 2tgplacthmeo 21717 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ (𝑥(-g𝐺)𝐴) ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
3724, 33, 36syl2anc 691 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
38 hmeoima 21378 . . . . . 6 (((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ∈ (𝐽Homeo𝐽) ∧ ((int‘𝐽)‘𝑆) ∈ 𝐽) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) “ ((int‘𝐽)‘𝑆)) ∈ 𝐽)
3937, 18, 38syl2anc 691 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) “ ((int‘𝐽)‘𝑆)) ∈ 𝐽)
4023, 39eqeltrrd 2689 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ∈ 𝐽)
41 tgpgrp 21692 . . . . . . 7 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
4224, 41syl 17 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐺 ∈ Grp)
43113ad2ant2 1076 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ (Base‘𝐺))
4443sselda 3568 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝑥 ∈ (Base‘𝐺))
4520, 28sseldd 3569 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐴 ∈ (Base‘𝐺))
463, 35, 30grpnpcan 17330 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝐴 ∈ (Base‘𝐺)) → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝐴) = 𝑥)
4742, 44, 45, 46syl3anc 1318 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝐴) = 𝑥)
48 ovex 6577 . . . . . 6 ((𝑥(-g𝐺)𝐴)(+g𝐺)𝐴) ∈ V
49 eqid 2610 . . . . . . 7 (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) = (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦))
50 oveq2 6557 . . . . . . 7 (𝑦 = 𝐴 → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦) = ((𝑥(-g𝐺)𝐴)(+g𝐺)𝐴))
5149, 50elrnmpt1s 5294 . . . . . 6 ((𝐴 ∈ ((int‘𝐽)‘𝑆) ∧ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝐴) ∈ V) → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝐴) ∈ ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)))
5228, 48, 51sylancl 693 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝐴) ∈ ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)))
5347, 52eqeltrrd 2689 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝑥 ∈ ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)))
5410adantr 480 . . . . . . 7 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ∈ (SubGrp‘𝐺))
5532adantr 480 . . . . . . 7 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → (𝑥(-g𝐺)𝐴) ∈ 𝑆)
5627sselda 3568 . . . . . . 7 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → 𝑦𝑆)
5735subgcl 17427 . . . . . . 7 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑥(-g𝐺)𝐴) ∈ 𝑆𝑦𝑆) → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦) ∈ 𝑆)
5854, 55, 56, 57syl3anc 1318 . . . . . 6 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦) ∈ 𝑆)
5958, 49fmptd 6292 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)):((int‘𝐽)‘𝑆)⟶𝑆)
60 frn 5966 . . . . 5 ((𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)):((int‘𝐽)‘𝑆)⟶𝑆 → ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ⊆ 𝑆)
6159, 60syl 17 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ⊆ 𝑆)
62 eleq2 2677 . . . . . 6 (𝑢 = ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) → (𝑥𝑢𝑥 ∈ ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦))))
63 sseq1 3589 . . . . . 6 (𝑢 = ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) → (𝑢𝑆 ↔ ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ⊆ 𝑆))
6462, 63anbi12d 743 . . . . 5 (𝑢 = ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) → ((𝑥𝑢𝑢𝑆) ↔ (𝑥 ∈ ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ∧ ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ⊆ 𝑆)))
6564rspcev 3282 . . . 4 ((ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ∈ 𝐽 ∧ (𝑥 ∈ ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ∧ ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ⊆ 𝑆)) → ∃𝑢𝐽 (𝑥𝑢𝑢𝑆))
6640, 53, 61, 65syl12anc 1316 . . 3 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ∃𝑢𝐽 (𝑥𝑢𝑢𝑆))
6766ralrimiva 2949 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → ∀𝑥𝑆𝑢𝐽 (𝑥𝑢𝑢𝑆))
68 eltop2 20590 . . 3 (𝐽 ∈ Top → (𝑆𝐽 ↔ ∀𝑥𝑆𝑢𝐽 (𝑥𝑢𝑢𝑆)))
698, 68syl 17 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → (𝑆𝐽 ↔ ∀𝑥𝑆𝑢𝐽 (𝑥𝑢𝑢𝑆)))
7067, 69mpbird 246 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝑆𝐽)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∪ cuni 4372   ↦ cmpt 4643  ran crn 5039   ↾ cres 5040   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  TopOpenctopn 15905  Grpcgrp 17245  -gcsg 17247  SubGrpcsubg 17411  Topctop 20517  TopOnctopon 20518  intcnt 20631  Homeochmeo 21366  TopGrpctgp 21685 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-topgen 15927  df-plusf 17064  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-ntr 20634  df-cn 20841  df-cnp 20842  df-tx 21175  df-hmeo 21368  df-tmd 21686  df-tgp 21687 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator