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Theorem trggrp 18162
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 18161 . 2  |-  ( R  e.  TopRing  ->  R  e.  Ring )
2 rnggrp 15632 . 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 1721   Grpcgrp 14648   Ringcrg 15623   TopRingctrg 18146
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-nul 4306
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429  df-ov 6051  df-rng 15626  df-trg 18150
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