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

Theorem trgtmd 19870
Description: The multiplicative monoid of a topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypothesis
Ref Expression
istrg.1  |-  M  =  (mulGrp `  R )
Assertion
Ref Expression
trgtmd  |-  ( R  e.  TopRing  ->  M  e. TopMnd )

Proof of Theorem trgtmd
StepHypRef Expression
1 istrg.1 . . 3  |-  M  =  (mulGrp `  R )
21istrg 19869 . 2  |-  ( R  e.  TopRing 
<->  ( R  e.  TopGrp  /\  R  e.  Ring  /\  M  e. TopMnd ) )
32simp3bi 1005 1  |-  ( R  e.  TopRing  ->  M  e. TopMnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   ` cfv 5525  mulGrpcmgp 16712   Ringcrg 16767  TopMndctmd 19772   TopGrpctgp 19773   TopRingctrg 19861
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 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
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 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2804  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-iota 5488  df-fv 5533  df-trg 19865
This theorem is referenced by:  mulrcn  19884  cnmpt1mulr  19887  cnmpt2mulr  19888  nrgtdrg  20404  iistmd  26476
  Copyright terms: Public domain W3C validator