MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  trgtgp Structured version   Visualization version   GIF version

Theorem trgtgp 23672
Description: A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
trgtgp (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)

Proof of Theorem trgtgp
StepHypRef Expression
1 eqid 2733 . . 3 (mulGrp‘𝑅) = (mulGrp‘𝑅)
21istrg 23668 . 2 (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ TopMnd))
32simp1bi 1146 1 (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  cfv 6544  mulGrpcmgp 19987  Ringcrg 20056  TopMndctmd 23574  TopGrpctgp 23575  TopRingctrg 23660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-trg 23664
This theorem is referenced by:  trgtmd2  23673  trgtps  23674  pl1cn  32935
  Copyright terms: Public domain W3C validator