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

Theorem trgtmd 20775
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 20774 . 2  |-  ( R  e.  TopRing 
<->  ( R  e.  TopGrp  /\  R  e.  Ring  /\  M  e. TopMnd ) )
32simp3bi 1011 1  |-  ( R  e.  TopRing  ->  M  e. TopMnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1836   ` cfv 5513  mulGrpcmgp 17277   Ringcrg 17334  TopMndctmd 20677   TopGrpctgp 20678   TopRingctrg 20766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-rex 2752  df-rab 2755  df-v 3053  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-br 4385  df-iota 5477  df-fv 5521  df-trg 20770
This theorem is referenced by:  mulrcn  20789  cnmpt1mulr  20792  cnmpt2mulr  20793  nrgtdrg  21309  iistmd  28073
  Copyright terms: Public domain W3C validator