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Theorem subgntr 21815
Description: A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 21817, 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 5092 . . . . . 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 2626 . . . . . . . . . . . 12 (Base‘𝐺) = (Base‘𝐺)
42, 3tgptopon 21791 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
543ad2ant1 1080 . . . . . . . . . 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 20636 . . . . . . . . . . . 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 1063 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
113subgss 17511 . . . . . . . . . . . 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 20637 . . . . . . . . . . . 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 3625 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝑆 𝐽)
16 eqid 2626 . . . . . . . . . . 11 𝐽 = 𝐽
1716ntropn 20758 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
189, 15, 17syl2anc 692 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
19 toponss 20639 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ ((int‘𝐽)‘𝑆) ∈ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
206, 18, 19syl2anc 692 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ((int‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
2120resmptd 5415 . . . . . . 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 5317 . . . . . 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 2672 . . . . 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 1062 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐺 ∈ TopGrp)
25 simpr 477 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝑥𝑆)
2616ntrss2 20766 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
279, 15, 26syl2anc 692 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
28 simpl3 1064 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐴 ∈ ((int‘𝐽)‘𝑆))
2927, 28sseldd 3589 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐴𝑆)
30 eqid 2626 . . . . . . . . . 10 (-g𝐺) = (-g𝐺)
3130subgsubcl 17521 . . . . . . . . 9 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆𝐴𝑆) → (𝑥(-g𝐺)𝐴) ∈ 𝑆)
3210, 25, 29, 31syl3anc 1323 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → (𝑥(-g𝐺)𝐴) ∈ 𝑆)
3312, 32sseldd 3589 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → (𝑥(-g𝐺)𝐴) ∈ (Base‘𝐺))
34 eqid 2626 . . . . . . . 8 (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) = (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦))
35 eqid 2626 . . . . . . . 8 (+g𝐺) = (+g𝐺)
3634, 3, 35, 2tgplacthmeo 21812 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ (𝑥(-g𝐺)𝐴) ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
3724, 33, 36syl2anc 692 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
38 hmeoima 21473 . . . . . 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 692 . . . . 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 2705 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ∈ 𝐽)
41 tgpgrp 21787 . . . . . . 7 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
4224, 41syl 17 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐺 ∈ Grp)
43113ad2ant2 1081 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ (Base‘𝐺))
4443sselda 3588 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝑥 ∈ (Base‘𝐺))
4520, 28sseldd 3589 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐴 ∈ (Base‘𝐺))
463, 35, 30grpnpcan 17423 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝐴 ∈ (Base‘𝐺)) → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝐴) = 𝑥)
4742, 44, 45, 46syl3anc 1323 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝐴) = 𝑥)
48 ovex 6633 . . . . . 6 ((𝑥(-g𝐺)𝐴)(+g𝐺)𝐴) ∈ V
49 eqid 2626 . . . . . . 7 (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) = (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦))
50 oveq2 6613 . . . . . . 7 (𝑦 = 𝐴 → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦) = ((𝑥(-g𝐺)𝐴)(+g𝐺)𝐴))
5149, 50elrnmpt1s 5337 . . . . . 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 2705 . . . 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 3588 . . . . . . 7 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → 𝑦𝑆)
5735subgcl 17520 . . . . . . 7 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑥(-g𝐺)𝐴) ∈ 𝑆𝑦𝑆) → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦) ∈ 𝑆)
5854, 55, 56, 57syl3anc 1323 . . . . . 6 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦) ∈ 𝑆)
5958, 49fmptd 6341 . . . . 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 6012 . . . . 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 2693 . . . . . 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 3610 . . . . . 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 746 . . . . 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 3300 . . . 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 1321 . . 3 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ∃𝑢𝐽 (𝑥𝑢𝑢𝑆))
6766ralrimiva 2965 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → ∀𝑥𝑆𝑢𝐽 (𝑥𝑢𝑢𝑆))
68 eltop2 20685 . . 3 (𝐽 ∈ Top → (𝑆𝐽 ↔ ∀𝑥𝑆𝑢𝐽 (𝑥𝑢𝑢𝑆)))
698, 68syl 17 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → (𝑆𝐽 ↔ ∀𝑥𝑆𝑢𝐽 (𝑥𝑢𝑢𝑆)))
7067, 69mpbird 247 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝑆𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912  wrex 2913  Vcvv 3191  wss 3560   cuni 4407  cmpt 4678  ran crn 5080  cres 5081  cima 5082  wf 5846  cfv 5850  (class class class)co 6605  Basecbs 15776  +gcplusg 15857  TopOpenctopn 15998  Grpcgrp 17338  -gcsg 17340  SubGrpcsubg 17504  Topctop 20612  TopOnctopon 20613  intcnt 20726  Homeochmeo 21461  TopGrpctgp 21780
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  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 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-map 7805  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-0g 16018  df-topgen 16020  df-plusf 17157  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-grp 17341  df-minusg 17342  df-sbg 17343  df-subg 17507  df-top 20616  df-bases 20617  df-topon 20618  df-topsp 20619  df-ntr 20729  df-cn 20936  df-cnp 20937  df-tx 21270  df-hmeo 21463  df-tmd 21781  df-tgp 21782
This theorem is referenced by: (None)
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