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Theorem trggrp 17950
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 17949 . 2  |-  ( R  e.  TopRing  ->  R  e.  Ring )
2 rnggrp 15439 . 2  |-  ( R  e.  Ring  ->  R  e. 
Grp )
31, 2syl 15 1  |-  ( R  e.  TopRing  ->  R  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1710   Grpcgrp 14455   Ringcrg 15430   TopRingctrg 17934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-nul 4228
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-iota 5298  df-fv 5342  df-ov 5945  df-rng 15433  df-trg 17938
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