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Theorem trgtmd 18194
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 18193 . 2  |-  ( R  e.  TopRing 
<->  ( R  e.  TopGrp  /\  R  e.  Ring  /\  M  e. TopMnd ) )
32simp3bi 974 1  |-  ( R  e.  TopRing  ->  M  e. TopMnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5454  mulGrpcmgp 15648   Ringcrg 15660  TopMndctmd 18100   TopGrpctgp 18101   TopRingctrg 18185
This theorem is referenced by:  mulrcn  18208  cnmpt1mulr  18211  cnmpt2mulr  18212  nrgtdrg  18728  iistmd  24300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-trg 18189
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