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Theorem indistgp 22105
Description: Any group equipped with the indiscrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
distgp.1 𝐵 = (Base‘𝐺)
distgp.2 𝐽 = (TopOpen‘𝐺)
Assertion
Ref Expression
indistgp ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ TopGrp)

Proof of Theorem indistgp
StepHypRef Expression
1 simpl 474 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ Grp)
2 simpr 479 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐽 = {∅, 𝐵})
3 distgp.1 . . . . . 6 𝐵 = (Base‘𝐺)
4 fvex 6362 . . . . . 6 (Base‘𝐺) ∈ V
53, 4eqeltri 2835 . . . . 5 𝐵 ∈ V
6 indistopon 21007 . . . . 5 (𝐵 ∈ V → {∅, 𝐵} ∈ (TopOn‘𝐵))
75, 6ax-mp 5 . . . 4 {∅, 𝐵} ∈ (TopOn‘𝐵)
82, 7syl6eqel 2847 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐽 ∈ (TopOn‘𝐵))
9 distgp.2 . . . 4 𝐽 = (TopOpen‘𝐺)
103, 9istps 20940 . . 3 (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵))
118, 10sylibr 224 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ TopSp)
12 eqid 2760 . . . . . 6 (-g𝐺) = (-g𝐺)
133, 12grpsubf 17695 . . . . 5 (𝐺 ∈ Grp → (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
1413adantr 472 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
155, 5xpex 7127 . . . . 5 (𝐵 × 𝐵) ∈ V
165, 15elmap 8052 . . . 4 ((-g𝐺) ∈ (𝐵𝑚 (𝐵 × 𝐵)) ↔ (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
1714, 16sylibr 224 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (-g𝐺) ∈ (𝐵𝑚 (𝐵 × 𝐵)))
182oveq2d 6829 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → ((𝐽 ×t 𝐽) Cn 𝐽) = ((𝐽 ×t 𝐽) Cn {∅, 𝐵}))
19 txtopon 21596 . . . . . 6 ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)))
208, 8, 19syl2anc 696 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)))
21 cnindis 21298 . . . . 5 (((𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)) ∧ 𝐵 ∈ V) → ((𝐽 ×t 𝐽) Cn {∅, 𝐵}) = (𝐵𝑚 (𝐵 × 𝐵)))
2220, 5, 21sylancl 697 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → ((𝐽 ×t 𝐽) Cn {∅, 𝐵}) = (𝐵𝑚 (𝐵 × 𝐵)))
2318, 22eqtrd 2794 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝐵𝑚 (𝐵 × 𝐵)))
2417, 23eleqtrrd 2842 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
259, 12istgp2 22096 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
261, 11, 24, 25syl3anbrc 1429 1 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  c0 4058  {cpr 4323   × cxp 5264  wf 6045  cfv 6049  (class class class)co 6813  𝑚 cmap 8023  Basecbs 16059  TopOpenctopn 16284  Grpcgrp 17623  -gcsg 17625  TopOnctopon 20917  TopSpctps 20938   Cn ccn 21230   ×t ctx 21565  TopGrpctgp 22076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025  df-0g 16304  df-topgen 16306  df-plusf 17442  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-grp 17626  df-minusg 17627  df-sbg 17628  df-top 20901  df-topon 20918  df-topsp 20939  df-bases 20952  df-cn 21233  df-cnp 21234  df-tx 21567  df-tmd 22077  df-tgp 22078
This theorem is referenced by: (None)
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