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Theorem indistgp 22714
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 486 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ Grp)
2 simpr 488 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐽 = {∅, 𝐵})
3 distgp.1 . . . . . 6 𝐵 = (Base‘𝐺)
43fvexi 6677 . . . . 5 𝐵 ∈ V
5 indistopon 21615 . . . . 5 (𝐵 ∈ V → {∅, 𝐵} ∈ (TopOn‘𝐵))
64, 5ax-mp 5 . . . 4 {∅, 𝐵} ∈ (TopOn‘𝐵)
72, 6eqeltrdi 2924 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐽 ∈ (TopOn‘𝐵))
8 distgp.2 . . . 4 𝐽 = (TopOpen‘𝐺)
93, 8istps 21548 . . 3 (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵))
107, 9sylibr 237 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ TopSp)
11 eqid 2824 . . . . . 6 (-g𝐺) = (-g𝐺)
123, 11grpsubf 18180 . . . . 5 (𝐺 ∈ Grp → (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
1312adantr 484 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
144, 4xpex 7472 . . . . 5 (𝐵 × 𝐵) ∈ V
154, 14elmap 8433 . . . 4 ((-g𝐺) ∈ (𝐵m (𝐵 × 𝐵)) ↔ (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
1613, 15sylibr 237 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (-g𝐺) ∈ (𝐵m (𝐵 × 𝐵)))
172oveq2d 7167 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → ((𝐽 ×t 𝐽) Cn 𝐽) = ((𝐽 ×t 𝐽) Cn {∅, 𝐵}))
18 txtopon 22205 . . . . . 6 ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)))
197, 7, 18syl2anc 587 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)))
20 cnindis 21906 . . . . 5 (((𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)) ∧ 𝐵 ∈ V) → ((𝐽 ×t 𝐽) Cn {∅, 𝐵}) = (𝐵m (𝐵 × 𝐵)))
2119, 4, 20sylancl 589 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → ((𝐽 ×t 𝐽) Cn {∅, 𝐵}) = (𝐵m (𝐵 × 𝐵)))
2217, 21eqtrd 2859 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝐵m (𝐵 × 𝐵)))
2316, 22eleqtrrd 2919 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
248, 11istgp2 22705 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
251, 10, 23, 24syl3anbrc 1340 1 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  c0 4276  {cpr 4552   × cxp 5541  wf 6341  cfv 6345  (class class class)co 7151  m cmap 8404  Basecbs 16485  TopOpenctopn 16697  Grpcgrp 18105  -gcsg 18107  TopOnctopon 21524  TopSpctps 21546   Cn ccn 21838   ×t ctx 22174  TopGrpctgp 22685
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7686  df-2nd 7687  df-map 8406  df-0g 16717  df-topgen 16719  df-plusf 17853  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-grp 18108  df-minusg 18109  df-sbg 18110  df-top 21508  df-topon 21525  df-topsp 21547  df-bases 21560  df-cn 21841  df-cnp 21842  df-tx 22176  df-tmd 22686  df-tgp 22687
This theorem is referenced by: (None)
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