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Theorem trgtmd 17863
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 17862 . 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 1632    e. wcel 1696   ` cfv 5271  mulGrpcmgp 15341   Ringcrg 15353  TopMndctmd 17769   TopGrpctgp 17770   TopRingctrg 17854
This theorem is referenced by:  mulrcn  17877  cnmpt1mulr  17880  cnmpt2mulr  17881  nrgtdrg  18219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-trg 17858
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