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

Theorem subgntr 23458
Description: A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 23460, 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 5646 . . . . . 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 2736 . . . . . . . . . . . 12 (Base‘𝐺) = (Base‘𝐺)
42, 3tgptopon 23433 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
543ad2ant1 1133 . . . . . . . . . 10 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
65adantr 481 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
7 topontop 22262 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
85, 7syl 17 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝐽 ∈ Top)
98adantr 481 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐽 ∈ Top)
10 simpl2 1192 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
113subgss 18929 . . . . . . . . . . . 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 22263 . . . . . . . . . . . 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 3984 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝑆 𝐽)
16 eqid 2736 . . . . . . . . . . 11 𝐽 = 𝐽
1716ntropn 22400 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
189, 15, 17syl2anc 584 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
19 toponss 22276 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ ((int‘𝐽)‘𝑆) ∈ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
206, 18, 19syl2anc 584 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ((int‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
2120resmptd 5994 . . . . . . 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 5893 . . . . . 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, 22eqtrid 2788 . . . . 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 1191 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐺 ∈ TopGrp)
25 simpr 485 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝑥𝑆)
2616ntrss2 22408 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
279, 15, 26syl2anc 584 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
28 simpl3 1193 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐴 ∈ ((int‘𝐽)‘𝑆))
2927, 28sseldd 3945 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐴𝑆)
30 eqid 2736 . . . . . . . . . 10 (-g𝐺) = (-g𝐺)
3130subgsubcl 18939 . . . . . . . . 9 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆𝐴𝑆) → (𝑥(-g𝐺)𝐴) ∈ 𝑆)
3210, 25, 29, 31syl3anc 1371 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → (𝑥(-g𝐺)𝐴) ∈ 𝑆)
3312, 32sseldd 3945 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → (𝑥(-g𝐺)𝐴) ∈ (Base‘𝐺))
34 eqid 2736 . . . . . . . 8 (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) = (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦))
35 eqid 2736 . . . . . . . 8 (+g𝐺) = (+g𝐺)
3634, 3, 35, 2tgplacthmeo 23454 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ (𝑥(-g𝐺)𝐴) ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
3724, 33, 36syl2anc 584 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
38 hmeoima 23116 . . . . . 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 584 . . . . 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 2839 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ∈ 𝐽)
41 tgpgrp 23429 . . . . . . 7 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
4224, 41syl 17 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐺 ∈ Grp)
43113ad2ant2 1134 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ (Base‘𝐺))
4443sselda 3944 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝑥 ∈ (Base‘𝐺))
4520, 28sseldd 3945 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐴 ∈ (Base‘𝐺))
463, 35, 30grpnpcan 18839 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝐴 ∈ (Base‘𝐺)) → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝐴) = 𝑥)
4742, 44, 45, 46syl3anc 1371 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝐴) = 𝑥)
48 ovex 7390 . . . . . 6 ((𝑥(-g𝐺)𝐴)(+g𝐺)𝐴) ∈ V
49 eqid 2736 . . . . . . 7 (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) = (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦))
50 oveq2 7365 . . . . . . 7 (𝑦 = 𝐴 → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦) = ((𝑥(-g𝐺)𝐴)(+g𝐺)𝐴))
5149, 50elrnmpt1s 5912 . . . . . 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 586 . . . . 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 2839 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝑥 ∈ ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)))
5410adantr 481 . . . . . . 7 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ∈ (SubGrp‘𝐺))
5532adantr 481 . . . . . . 7 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → (𝑥(-g𝐺)𝐴) ∈ 𝑆)
5627sselda 3944 . . . . . . 7 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → 𝑦𝑆)
5735subgcl 18938 . . . . . . 7 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑥(-g𝐺)𝐴) ∈ 𝑆𝑦𝑆) → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦) ∈ 𝑆)
5854, 55, 56, 57syl3anc 1371 . . . . . 6 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦) ∈ 𝑆)
5958fmpttd 7063 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)):((int‘𝐽)‘𝑆)⟶𝑆)
6059frnd 6676 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ⊆ 𝑆)
61 eleq2 2826 . . . . . 6 (𝑢 = ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) → (𝑥𝑢𝑥 ∈ ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦))))
62 sseq1 3969 . . . . . 6 (𝑢 = ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) → (𝑢𝑆 ↔ ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ⊆ 𝑆))
6361, 62anbi12d 631 . . . . 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)))
6463rspcev 3581 . . . 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))
6540, 53, 60, 64syl12anc 835 . . 3 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ∃𝑢𝐽 (𝑥𝑢𝑢𝑆))
6665ralrimiva 3143 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → ∀𝑥𝑆𝑢𝐽 (𝑥𝑢𝑢𝑆))
67 eltop2 22325 . . 3 (𝐽 ∈ Top → (𝑆𝐽 ↔ ∀𝑥𝑆𝑢𝐽 (𝑥𝑢𝑢𝑆)))
688, 67syl 17 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → (𝑆𝐽 ↔ ∀𝑥𝑆𝑢𝐽 (𝑥𝑢𝑢𝑆)))
6966, 68mpbird 256 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝑆𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  wss 3910   cuni 4865  cmpt 5188  ran crn 5634  cres 5635  cima 5636  cfv 6496  (class class class)co 7357  Basecbs 17083  +gcplusg 17133  TopOpenctopn 17303  Grpcgrp 18748  -gcsg 18750  SubGrpcsubg 18922  Topctop 22242  TopOnctopon 22259  intcnt 22368  Homeochmeo 23104  TopGrpctgp 23422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-topgen 17325  df-plusf 18496  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-sbg 18753  df-subg 18925  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-ntr 22371  df-cn 22578  df-cnp 22579  df-tx 22913  df-hmeo 23106  df-tmd 23423  df-tgp 23424
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator