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Theorem distgp 21808
Description: Any group equipped with the discrete 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
distgp ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp)

Proof of Theorem distgp
StepHypRef Expression
1 simpl 473 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ Grp)
2 simpr 477 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐽 = 𝒫 𝐵)
3 distgp.1 . . . . . 6 𝐵 = (Base‘𝐺)
4 fvex 6160 . . . . . 6 (Base‘𝐺) ∈ V
53, 4eqeltri 2700 . . . . 5 𝐵 ∈ V
6 distopon 20706 . . . . 5 (𝐵 ∈ V → 𝒫 𝐵 ∈ (TopOn‘𝐵))
75, 6ax-mp 5 . . . 4 𝒫 𝐵 ∈ (TopOn‘𝐵)
82, 7syl6eqel 2712 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐽 ∈ (TopOn‘𝐵))
9 distgp.2 . . . 4 𝐽 = (TopOpen‘𝐺)
103, 9istps 20646 . . 3 (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵))
118, 10sylibr 224 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopSp)
12 eqid 2626 . . . . . 6 (-g𝐺) = (-g𝐺)
133, 12grpsubf 17410 . . . . 5 (𝐺 ∈ Grp → (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
1413adantr 481 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
155, 5xpex 6916 . . . . 5 (𝐵 × 𝐵) ∈ V
165, 15elmap 7831 . . . 4 ((-g𝐺) ∈ (𝐵𝑚 (𝐵 × 𝐵)) ↔ (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
1714, 16sylibr 224 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g𝐺) ∈ (𝐵𝑚 (𝐵 × 𝐵)))
182, 2oveq12d 6623 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝐽 ×t 𝐽) = (𝒫 𝐵 ×t 𝒫 𝐵))
19 txdis 21340 . . . . . . 7 ((𝐵 ∈ V ∧ 𝐵 ∈ V) → (𝒫 𝐵 ×t 𝒫 𝐵) = 𝒫 (𝐵 × 𝐵))
205, 5, 19mp2an 707 . . . . . 6 (𝒫 𝐵 ×t 𝒫 𝐵) = 𝒫 (𝐵 × 𝐵)
2118, 20syl6eq 2676 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝐽 ×t 𝐽) = 𝒫 (𝐵 × 𝐵))
2221oveq1d 6620 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝒫 (𝐵 × 𝐵) Cn 𝐽))
23 cndis 21000 . . . . 5 (((𝐵 × 𝐵) ∈ V ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝒫 (𝐵 × 𝐵) Cn 𝐽) = (𝐵𝑚 (𝐵 × 𝐵)))
2415, 8, 23sylancr 694 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝒫 (𝐵 × 𝐵) Cn 𝐽) = (𝐵𝑚 (𝐵 × 𝐵)))
2522, 24eqtrd 2660 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝐵𝑚 (𝐵 × 𝐵)))
2617, 25eleqtrrd 2707 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
279, 12istgp2 21800 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
281, 11, 26, 27syl3anbrc 1244 1 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  Vcvv 3191  𝒫 cpw 4135   × cxp 5077  wf 5846  cfv 5850  (class class class)co 6605  𝑚 cmap 7803  Basecbs 15776  TopOpenctopn 15998  Grpcgrp 17338  -gcsg 17340  TopOnctopon 20613  TopSpctps 20614   Cn ccn 20933   ×t ctx 21268  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
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  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-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  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-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-1st 7116  df-2nd 7117  df-map 7805  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-top 20616  df-bases 20617  df-topon 20618  df-topsp 20619  df-cn 20936  df-cnp 20937  df-tx 21270  df-tmd 21781  df-tgp 21782
This theorem is referenced by: (None)
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