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Theorem trgtmd 22773
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 22772 . 2 (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd))
32simp3bi 1144 1 (𝑅 ∈ TopRing → 𝑀 ∈ TopMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  cfv 6328  mulGrpcmgp 19235  Ringcrg 19293  TopMndctmd 22678  TopGrpctgp 22679  TopRingctrg 22764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rab 3118  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-trg 22768
This theorem is referenced by:  mulrcn  22787  cnmpt1mulr  22790  cnmpt2mulr  22791  nrgtdrg  23302  iistmd  31253
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