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Theorem trggrp 18206
Description: A topological ring is a group. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
trggrp  |-  ( R  e.  TopRing  ->  R  e.  Grp )

Proof of Theorem trggrp
StepHypRef Expression
1 trgrng 18205 . 2  |-  ( R  e.  TopRing  ->  R  e.  Ring )
2 rnggrp 15674 . 2  |-  ( R  e.  Ring  ->  R  e. 
Grp )
31, 2syl 16 1  |-  ( R  e.  TopRing  ->  R  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726   Grpcgrp 14690   Ringcrg 15665   TopRingctrg 18190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-nul 4341
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-ov 6087  df-rng 15668  df-trg 18194
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