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Theorem trgtmd 22188
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 22187 . 2 (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd))
32simp3bi 1141 1 (𝑅 ∈ TopRing → 𝑀 ∈ TopMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  cfv 6030  mulGrpcmgp 18697  Ringcrg 18755  TopMndctmd 22094  TopGrpctgp 22095  TopRingctrg 22179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-iota 5993  df-fv 6038  df-trg 22183
This theorem is referenced by:  mulrcn  22202  cnmpt1mulr  22205  cnmpt2mulr  22206  nrgtdrg  22717  iistmd  30288
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