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Theorem trgtgp 20962
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 2402 . . 3  |-  (mulGrp `  R )  =  (mulGrp `  R )
21istrg 20958 . 2  |-  ( R  e.  TopRing 
<->  ( R  e.  TopGrp  /\  R  e.  Ring  /\  (mulGrp `  R )  e. TopMnd )
)
32simp1bi 1012 1  |-  ( R  e.  TopRing  ->  R  e.  TopGrp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1842   ` cfv 5569  mulGrpcmgp 17461   Ringcrg 17518  TopMndctmd 20861   TopGrpctgp 20862   TopRingctrg 20950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-iota 5533  df-fv 5577  df-trg 20954
This theorem is referenced by:  trgtmd2  20963  trgtps  20964  pl1cn  28390
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