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Theorem trgtgp 20402
Description: A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
trgtgp  |-  ( R  e.  TopRing  ->  R  e.  TopGrp )

Proof of Theorem trgtgp
StepHypRef Expression
1 eqid 2467 . . 3  |-  (mulGrp `  R )  =  (mulGrp `  R )
21istrg 20398 . 2  |-  ( R  e.  TopRing 
<->  ( R  e.  TopGrp  /\  R  e.  Ring  /\  (mulGrp `  R )  e. TopMnd )
)
32simp1bi 1011 1  |-  ( R  e.  TopRing  ->  R  e.  TopGrp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   ` cfv 5586  mulGrpcmgp 16928   Ringcrg 16983  TopMndctmd 20301   TopGrpctgp 20302   TopRingctrg 20390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5549  df-fv 5594  df-trg 20394
This theorem is referenced by:  trgtmd2  20403  trgtps  20404  pl1cn  27570
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