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

Theorem trgtmd 21778
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 21777 . 2 (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd))
32simp3bi 1071 1 (𝑅 ∈ TopRing → 𝑀 ∈ TopMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  cfv 5804  mulGrpcmgp 18312  Ringcrg 18370  TopMndctmd 21684  TopGrpctgp 21685  TopRingctrg 21769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-trg 21773
This theorem is referenced by:  mulrcn  21792  cnmpt1mulr  21795  cnmpt2mulr  21796  nrgtdrg  22307  iistmd  29276
  Copyright terms: Public domain W3C validator