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Theorem drngrng 15535
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 2296 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2296 . . 3  |-  (Unit `  R )  =  (Unit `  R )
3 eqid 2296 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
41, 2, 3isdrng 15532 . 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 1632    e. wcel 1696    \ cdif 3162   {csn 3653   ` cfv 5271   Basecbs 13164   0gc0g 13416   Ringcrg 15353  Unitcui 15437   DivRingcdr 15528
This theorem is referenced by:  drnggrp  15536  drngid  15542  drngunz  15543  drnginvrcl  15545  drnginvrn0  15546  drnginvrl  15547  drnginvrr  15548  drngmul0or  15549  abvtriv  15622  rlmlvec  15974  drngnidl  15997  drnglpir  16021  drngnzr  16030  drngdomn  16060  qsssubdrg  16447  cphsubrglem  18629  drnguc1p  19572  ig1peu  19573  ig1pcl  19577  ig1pdvds  19578  ig1prsp  19579  ply1lpir  19580  padicabv  20795  sdrgacs  27612  cntzsdrg  27613  dvalveclem  31837  dvhlveclem  31920  hlhilsrnglem  32768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-drng 15530
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