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

Theorem trgtmd 22345
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 ∈ TopRing → 𝑀 ∈ TopMnd)

Proof of Theorem trgtmd
StepHypRef Expression
1 istrg.1 . . 3 𝑀 = (mulGrp‘𝑅)
21istrg 22344 . 2 (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd))
32simp3bi 1181 1 (𝑅 ∈ TopRing → 𝑀 ∈ TopMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1656  wcel 2164  cfv 6127  mulGrpcmgp 18850  Ringcrg 18908  TopMndctmd 22251  TopGrpctgp 22252  TopRingctrg 22336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-iota 6090  df-fv 6135  df-trg 22340
This theorem is referenced by:  mulrcn  22359  cnmpt1mulr  22362  cnmpt2mulr  22363  nrgtdrg  22874  iistmd  30489
  Copyright terms: Public domain W3C validator