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Theorem trgtmd 17847
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 17846 . 2  |-  ( R  e.  TopRing 
<->  ( R  e.  TopGrp  /\  R  e.  Ring  /\  M  e. TopMnd ) )
32simp3bi 972 1  |-  ( R  e.  TopRing  ->  M  e. TopMnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5255  mulGrpcmgp 15325   Ringcrg 15337  TopMndctmd 17753   TopGrpctgp 17754   TopRingctrg 17838
This theorem is referenced by:  mulrcn  17861  cnmpt1mulr  17864  cnmpt2mulr  17865  nrgtdrg  18203
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-trg 17842
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