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Theorem indistgp 21814
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 473 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ Grp)
2 simpr 477 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐽 = {∅, 𝐵})
3 distgp.1 . . . . . 6 𝐵 = (Base‘𝐺)
4 fvex 6158 . . . . . 6 (Base‘𝐺) ∈ V
53, 4eqeltri 2694 . . . . 5 𝐵 ∈ V
6 indistopon 20715 . . . . 5 (𝐵 ∈ V → {∅, 𝐵} ∈ (TopOn‘𝐵))
75, 6ax-mp 5 . . . 4 {∅, 𝐵} ∈ (TopOn‘𝐵)
82, 7syl6eqel 2706 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐽 ∈ (TopOn‘𝐵))
9 distgp.2 . . . 4 𝐽 = (TopOpen‘𝐺)
103, 9istps 20651 . . 3 (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵))
118, 10sylibr 224 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ TopSp)
12 eqid 2621 . . . . . 6 (-g𝐺) = (-g𝐺)
133, 12grpsubf 17415 . . . . 5 (𝐺 ∈ Grp → (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
1413adantr 481 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
155, 5xpex 6915 . . . . 5 (𝐵 × 𝐵) ∈ V
165, 15elmap 7830 . . . 4 ((-g𝐺) ∈ (𝐵𝑚 (𝐵 × 𝐵)) ↔ (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
1714, 16sylibr 224 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (-g𝐺) ∈ (𝐵𝑚 (𝐵 × 𝐵)))
182oveq2d 6620 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → ((𝐽 ×t 𝐽) Cn 𝐽) = ((𝐽 ×t 𝐽) Cn {∅, 𝐵}))
19 txtopon 21304 . . . . . 6 ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)))
208, 8, 19syl2anc 692 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)))
21 cnindis 21006 . . . . 5 (((𝐽 ×t 𝐽) ∈ (TopOn‘(𝐵 × 𝐵)) ∧ 𝐵 ∈ V) → ((𝐽 ×t 𝐽) Cn {∅, 𝐵}) = (𝐵𝑚 (𝐵 × 𝐵)))
2220, 5, 21sylancl 693 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → ((𝐽 ×t 𝐽) Cn {∅, 𝐵}) = (𝐵𝑚 (𝐵 × 𝐵)))
2318, 22eqtrd 2655 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝐵𝑚 (𝐵 × 𝐵)))
2417, 23eleqtrrd 2701 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
259, 12istgp2 21805 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
261, 11, 24, 25syl3anbrc 1244 1 ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  c0 3891  {cpr 4150   × cxp 5072  wf 5843  cfv 5847  (class class class)co 6604  𝑚 cmap 7802  Basecbs 15781  TopOpenctopn 16003  Grpcgrp 17343  -gcsg 17345  TopOnctopon 20618  TopSpctps 20619   Cn ccn 20938   ×t ctx 21273  TopGrpctgp 21785
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804  df-0g 16023  df-topgen 16025  df-plusf 17162  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347  df-sbg 17348  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cn 20941  df-cnp 20942  df-tx 21275  df-tmd 21786  df-tgp 21787
This theorem is referenced by: (None)
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