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Theorem drngrng 15805
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 2412 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2412 . . 3  |-  (Unit `  R )  =  (Unit `  R )
3 eqid 2412 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
41, 2, 3isdrng 15802 . 2  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  (Unit `  R )  =  ( ( Base `  R )  \  {
( 0g `  R
) } ) ) )
54simplbi 447 1  |-  ( R  e.  DivRing  ->  R  e.  Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    \ cdif 3285   {csn 3782   ` cfv 5421   Basecbs 13432   0gc0g 13686   Ringcrg 15623  Unitcui 15707   DivRingcdr 15798
This theorem is referenced by:  drnggrp  15806  drngid  15812  drngunz  15813  drnginvrcl  15815  drnginvrn0  15816  drnginvrl  15817  drnginvrr  15818  drngmul0or  15819  abvtriv  15892  rlmlvec  16240  drngnidl  16263  drnglpir  16287  drngnzr  16296  drngdomn  16326  qsssubdrg  16721  cphsubrglem  19101  drnguc1p  20054  ig1peu  20055  ig1pcl  20059  ig1pdvds  20060  ig1prsp  20061  ply1lpir  20062  padicabv  21285  ofldsqr  24201  ofldchr  24205  zrhunitpreima  24323  elzrhunit  24324  qqhval2lem  24326  qqh0  24329  qqh1  24330  qqhf  24331  qqhghm  24333  qqhrhm  24334  qqhnm  24335  qqhucn  24337  zrhre  24346  qqhre  24347  sdrgacs  27385  cntzsdrg  27386  dvalveclem  31520  dvhlveclem  31603  hlhilsrnglem  32451
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
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-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  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-drng 15800
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