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Theorem drngrng 15519
Description: A division ring is a ring. (Contributed by NM, 8-Sep-2011.)
Assertion
Ref Expression
drngrng  |-  ( R  e.  DivRing  ->  R  e.  Ring )

Proof of Theorem drngrng
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2283 . . 3  |-  (Unit `  R )  =  (Unit `  R )
3 eqid 2283 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
41, 2, 3isdrng 15516 . 2  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  (Unit `  R )  =  ( ( Base `  R )  \  {
( 0g `  R
) } ) ) )
54simplbi 446 1  |-  ( R  e.  DivRing  ->  R  e.  Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    \ cdif 3149   {csn 3640   ` cfv 5255   Basecbs 13148   0gc0g 13400   Ringcrg 15337  Unitcui 15421   DivRingcdr 15512
This theorem is referenced by:  drnggrp  15520  drngid  15526  drngunz  15527  drnginvrcl  15529  drnginvrn0  15530  drnginvrl  15531  drnginvrr  15532  drngmul0or  15533  abvtriv  15606  rlmlvec  15958  drngnidl  15981  drnglpir  16005  drngnzr  16014  drngdomn  16044  qsssubdrg  16431  cphsubrglem  18613  drnguc1p  19556  ig1peu  19557  ig1pcl  19561  ig1pdvds  19562  ig1prsp  19563  ply1lpir  19564  padicabv  20779  sdrgacs  27509  cntzsdrg  27510  dvalveclem  31215  dvhlveclem  31298  hlhilsrnglem  32146
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-drng 15514
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