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

Theorem trgtgp 19844
Description: A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
trgtgp  |-  ( R  e.  TopRing  ->  R  e.  TopGrp )

Proof of Theorem trgtgp
StepHypRef Expression
1 eqid 2450 . . 3  |-  (mulGrp `  R )  =  (mulGrp `  R )
21istrg 19840 . 2  |-  ( R  e.  TopRing 
<->  ( R  e.  TopGrp  /\  R  e.  Ring  /\  (mulGrp `  R )  e. TopMnd )
)
32simp1bi 1003 1  |-  ( R  e.  TopRing  ->  R  e.  TopGrp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1757   ` cfv 5502  mulGrpcmgp 16682   Ringcrg 16737  TopMndctmd 19743   TopGrpctgp 19744   TopRingctrg 19832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-rex 2798  df-rab 2801  df-v 3056  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-iota 5465  df-fv 5510  df-trg 19836
This theorem is referenced by:  trgtmd2  19845  trgtps  19846  pl1cn  26505
  Copyright terms: Public domain W3C validator