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Theorem subgntr 23931
Description: A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 23933, 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 5689 . . . . . 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 2731 . . . . . . . . . . . 12 (Base‘𝐺) = (Base‘𝐺)
42, 3tgptopon 23906 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
543ad2ant1 1132 . . . . . . . . . 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 22735 . . . . . . . . . . . 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 1191 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
113subgss 19050 . . . . . . . . . . . 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 22736 . . . . . . . . . . . 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 4022 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝑆 𝐽)
16 eqid 2731 . . . . . . . . . . 11 𝐽 = 𝐽
1716ntropn 22873 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
189, 15, 17syl2anc 583 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
19 toponss 22749 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ ((int‘𝐽)‘𝑆) ∈ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
206, 18, 19syl2anc 583 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ((int‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
2120resmptd 6040 . . . . . . 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 5937 . . . . . 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 2783 . . . . 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 1190 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐺 ∈ TopGrp)
25 simpr 484 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝑥𝑆)
2616ntrss2 22881 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
279, 15, 26syl2anc 583 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
28 simpl3 1192 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐴 ∈ ((int‘𝐽)‘𝑆))
2927, 28sseldd 3983 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐴𝑆)
30 eqid 2731 . . . . . . . . . 10 (-g𝐺) = (-g𝐺)
3130subgsubcl 19060 . . . . . . . . 9 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆𝐴𝑆) → (𝑥(-g𝐺)𝐴) ∈ 𝑆)
3210, 25, 29, 31syl3anc 1370 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → (𝑥(-g𝐺)𝐴) ∈ 𝑆)
3312, 32sseldd 3983 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → (𝑥(-g𝐺)𝐴) ∈ (Base‘𝐺))
34 eqid 2731 . . . . . . . 8 (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) = (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦))
35 eqid 2731 . . . . . . . 8 (+g𝐺) = (+g𝐺)
3634, 3, 35, 2tgplacthmeo 23927 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ (𝑥(-g𝐺)𝐴) ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
3724, 33, 36syl2anc 583 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
38 hmeoima 23589 . . . . . 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 583 . . . . 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 2833 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ∈ 𝐽)
41 tgpgrp 23902 . . . . . . 7 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
4224, 41syl 17 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐺 ∈ Grp)
43113ad2ant2 1133 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ (Base‘𝐺))
4443sselda 3982 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝑥 ∈ (Base‘𝐺))
4520, 28sseldd 3983 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → 𝐴 ∈ (Base‘𝐺))
463, 35, 30grpnpcan 18958 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝐴 ∈ (Base‘𝐺)) → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝐴) = 𝑥)
4742, 44, 45, 46syl3anc 1370 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝐴) = 𝑥)
48 ovex 7445 . . . . . 6 ((𝑥(-g𝐺)𝐴)(+g𝐺)𝐴) ∈ V
49 eqid 2731 . . . . . . 7 (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) = (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦))
50 oveq2 7420 . . . . . . 7 (𝑦 = 𝐴 → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦) = ((𝑥(-g𝐺)𝐴)(+g𝐺)𝐴))
5149, 50elrnmpt1s 5956 . . . . . 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 585 . . . . 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 2833 . . . 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 3982 . . . . . . 7 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → 𝑦𝑆)
5735subgcl 19059 . . . . . . 7 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑥(-g𝐺)𝐴) ∈ 𝑆𝑦𝑆) → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦) ∈ 𝑆)
5854, 55, 56, 57syl3anc 1370 . . . . . 6 ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦) ∈ 𝑆)
5958fmpttd 7116 . . . . 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 6725 . . . 4 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g𝐺)𝐴)(+g𝐺)𝑦)) ⊆ 𝑆)
61 eleq2 2821 . . . . . 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 4007 . . . . . 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 630 . . . . 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 3612 . . . 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 834 . . 3 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥𝑆) → ∃𝑢𝐽 (𝑥𝑢𝑢𝑆))
6665ralrimiva 3145 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → ∀𝑥𝑆𝑢𝐽 (𝑥𝑢𝑢𝑆))
67 eltop2 22798 . . 3 (𝐽 ∈ Top → (𝑆𝐽 ↔ ∀𝑥𝑆𝑢𝐽 (𝑥𝑢𝑢𝑆)))
688, 67syl 17 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → (𝑆𝐽 ↔ ∀𝑥𝑆𝑢𝐽 (𝑥𝑢𝑢𝑆)))
6966, 68mpbird 257 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝑆𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wral 3060  wrex 3069  Vcvv 3473  wss 3948   cuni 4908  cmpt 5231  ran crn 5677  cres 5678  cima 5679  cfv 6543  (class class class)co 7412  Basecbs 17151  +gcplusg 17204  TopOpenctopn 17374  Grpcgrp 18861  -gcsg 18863  SubGrpcsubg 19043  Topctop 22715  TopOnctopon 22732  intcnt 22841  Homeochmeo 23577  TopGrpctgp 23895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-er 8709  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-0g 17394  df-topgen 17396  df-plusf 18570  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-grp 18864  df-minusg 18865  df-sbg 18866  df-subg 19046  df-top 22716  df-topon 22733  df-topsp 22755  df-bases 22769  df-ntr 22844  df-cn 23051  df-cnp 23052  df-tx 23386  df-hmeo 23579  df-tmd 23896  df-tgp 23897
This theorem is referenced by: (None)
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