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Theorem isdrng2 17985
Description: A division ring can equivalently be defined as a ring such that the nonzero elements form a group under multiplication (from which it follows that this is the same group as the group of units). (Contributed by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
isdrng2.b  |-  B  =  ( Base `  R
)
isdrng2.z  |-  .0.  =  ( 0g `  R )
isdrng2.g  |-  G  =  ( (mulGrp `  R
)s  ( B  \  {  .0.  } ) )
Assertion
Ref Expression
isdrng2  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  G  e.  Grp )
)

Proof of Theorem isdrng2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 isdrng2.b . . 3  |-  B  =  ( Base `  R
)
2 eqid 2451 . . 3  |-  (Unit `  R )  =  (Unit `  R )
3 isdrng2.z . . 3  |-  .0.  =  ( 0g `  R )
41, 2, 3isdrng 17979 . 2  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) ) )
5 oveq2 6298 . . . . . . 7  |-  ( (Unit `  R )  =  ( B  \  {  .0.  } )  ->  ( (mulGrp `  R )s  (Unit `  R )
)  =  ( (mulGrp `  R )s  ( B  \  {  .0.  } ) ) )
65adantl 468 . . . . . 6  |-  ( ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )  ->  (
(mulGrp `  R )s  (Unit `  R ) )  =  ( (mulGrp `  R
)s  ( B  \  {  .0.  } ) ) )
7 isdrng2.g . . . . . 6  |-  G  =  ( (mulGrp `  R
)s  ( B  \  {  .0.  } ) )
86, 7syl6eqr 2503 . . . . 5  |-  ( ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )  ->  (
(mulGrp `  R )s  (Unit `  R ) )  =  G )
9 eqid 2451 . . . . . . 7  |-  ( (mulGrp `  R )s  (Unit `  R )
)  =  ( (mulGrp `  R )s  (Unit `  R )
)
102, 9unitgrp 17895 . . . . . 6  |-  ( R  e.  Ring  ->  ( (mulGrp `  R )s  (Unit `  R )
)  e.  Grp )
1110adantr 467 . . . . 5  |-  ( ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )  ->  (
(mulGrp `  R )s  (Unit `  R ) )  e. 
Grp )
128, 11eqeltrrd 2530 . . . 4  |-  ( ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )  ->  G  e.  Grp )
131, 2unitcl 17887 . . . . . . . . 9  |-  ( x  e.  (Unit `  R
)  ->  x  e.  B )
1413adantl 468 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  x  e.  B )
15 difss 3560 . . . . . . . . . . . . . . 15  |-  ( B 
\  {  .0.  }
)  C_  B
16 eqid 2451 . . . . . . . . . . . . . . . . 17  |-  (mulGrp `  R )  =  (mulGrp `  R )
1716, 1mgpbas 17729 . . . . . . . . . . . . . . . 16  |-  B  =  ( Base `  (mulGrp `  R ) )
187, 17ressbas2 15180 . . . . . . . . . . . . . . 15  |-  ( ( B  \  {  .0.  } )  C_  B  ->  ( B  \  {  .0.  } )  =  ( Base `  G ) )
1915, 18ax-mp 5 . . . . . . . . . . . . . 14  |-  ( B 
\  {  .0.  }
)  =  ( Base `  G )
20 eqid 2451 . . . . . . . . . . . . . 14  |-  ( 0g
`  G )  =  ( 0g `  G
)
2119, 20grpidcl 16694 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  ( B  \  {  .0.  } ) )
2221ad2antlr 733 . . . . . . . . . . . 12  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( 0g `  G
)  e.  ( B 
\  {  .0.  }
) )
23 eldifsn 4097 . . . . . . . . . . . 12  |-  ( ( 0g `  G )  e.  ( B  \  {  .0.  } )  <->  ( ( 0g `  G )  e.  B  /\  ( 0g
`  G )  =/= 
.0.  ) )
2422, 23sylib 200 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( ( 0g `  G )  e.  B  /\  ( 0g `  G
)  =/=  .0.  )
)
2524simprd 465 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( 0g `  G
)  =/=  .0.  )
26 simpll 760 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  R  e.  Ring )
2722eldifad 3416 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( 0g `  G
)  e.  B )
28 simpr 463 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  x  e.  (Unit `  R
) )
29 eqid 2451 . . . . . . . . . . . 12  |-  (/r `  R
)  =  (/r `  R
)
30 eqid 2451 . . . . . . . . . . . 12  |-  ( .r
`  R )  =  ( .r `  R
)
311, 2, 29, 30dvrcan1 17919 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  ( 0g `  G )  e.  B  /\  x  e.  (Unit `  R )
)  ->  ( (
( 0g `  G
) (/r `  R ) x ) ( .r `  R ) x )  =  ( 0g `  G ) )
3226, 27, 28, 31syl3anc 1268 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( ( ( 0g
`  G ) (/r `  R ) x ) ( .r `  R
) x )  =  ( 0g `  G
) )
331, 2, 29dvrcl 17914 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  ( 0g `  G )  e.  B  /\  x  e.  (Unit `  R )
)  ->  ( ( 0g `  G ) (/r `  R ) x )  e.  B )
3426, 27, 28, 33syl3anc 1268 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( ( 0g `  G ) (/r `  R
) x )  e.  B )
351, 30, 3ringrz 17818 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  (
( 0g `  G
) (/r `  R ) x )  e.  B )  ->  ( ( ( 0g `  G ) (/r `  R ) x ) ( .r `  R )  .0.  )  =  .0.  )
3626, 34, 35syl2anc 667 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( ( ( 0g
`  G ) (/r `  R ) x ) ( .r `  R
)  .0.  )  =  .0.  )
3725, 32, 363netr4d 2701 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( ( ( 0g
`  G ) (/r `  R ) x ) ( .r `  R
) x )  =/=  ( ( ( 0g
`  G ) (/r `  R ) x ) ( .r `  R
)  .0.  ) )
38 oveq2 6298 . . . . . . . . . 10  |-  ( x  =  .0.  ->  (
( ( 0g `  G ) (/r `  R
) x ) ( .r `  R ) x )  =  ( ( ( 0g `  G ) (/r `  R
) x ) ( .r `  R )  .0.  ) )
3938necon3i 2656 . . . . . . . . 9  |-  ( ( ( ( 0g `  G ) (/r `  R
) x ) ( .r `  R ) x )  =/=  (
( ( 0g `  G ) (/r `  R
) x ) ( .r `  R )  .0.  )  ->  x  =/=  .0.  )
4037, 39syl 17 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  x  =/=  .0.  )
41 eldifsn 4097 . . . . . . . 8  |-  ( x  e.  ( B  \  {  .0.  } )  <->  ( x  e.  B  /\  x  =/=  .0.  ) )
4214, 40, 41sylanbrc 670 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  x  e.  ( B  \  {  .0.  } ) )
4342ex 436 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (
x  e.  (Unit `  R )  ->  x  e.  ( B  \  {  .0.  } ) ) )
4443ssrdv 3438 . . . . 5  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (Unit `  R )  C_  ( B  \  {  .0.  }
) )
45 eldifi 3555 . . . . . . . . . . 11  |-  ( x  e.  ( B  \  {  .0.  } )  ->  x  e.  B )
4645adantl 468 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x  e.  B )
47 eqid 2451 . . . . . . . . . . . . 13  |-  ( invg `  G )  =  ( invg `  G )
4819, 47grpinvcl 16711 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  x  e.  ( B  \  {  .0.  } ) )  ->  ( ( invg `  G ) `
 x )  e.  ( B  \  {  .0.  } ) )
4948adantll 720 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( invg `  G ) `  x
)  e.  ( B 
\  {  .0.  }
) )
5049eldifad 3416 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( invg `  G ) `  x
)  e.  B )
51 eqid 2451 . . . . . . . . . . 11  |-  ( ||r `  R
)  =  ( ||r `  R
)
521, 51, 30dvdsrmul 17876 . . . . . . . . . 10  |-  ( ( x  e.  B  /\  ( ( invg `  G ) `  x
)  e.  B )  ->  x ( ||r `  R
) ( ( ( invg `  G
) `  x )
( .r `  R
) x ) )
5346, 50, 52syl2anc 667 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x
( ||r `
 R ) ( ( ( invg `  G ) `  x
) ( .r `  R ) x ) )
54 fvex 5875 . . . . . . . . . . . . . 14  |-  ( Base `  R )  e.  _V
551, 54eqeltri 2525 . . . . . . . . . . . . 13  |-  B  e. 
_V
56 difexg 4551 . . . . . . . . . . . . 13  |-  ( B  e.  _V  ->  ( B  \  {  .0.  }
)  e.  _V )
5716, 30mgpplusg 17727 . . . . . . . . . . . . . 14  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
587, 57ressplusg 15239 . . . . . . . . . . . . 13  |-  ( ( B  \  {  .0.  } )  e.  _V  ->  ( .r `  R )  =  ( +g  `  G
) )
5955, 56, 58mp2b 10 . . . . . . . . . . . 12  |-  ( .r
`  R )  =  ( +g  `  G
)
6019, 59, 20, 47grplinv 16712 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  x  e.  ( B  \  {  .0.  } ) )  ->  ( (
( invg `  G ) `  x
) ( .r `  R ) x )  =  ( 0g `  G ) )
6160adantll 720 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( ( invg `  G ) `  x
) ( .r `  R ) x )  =  ( 0g `  G ) )
62 eqid 2451 . . . . . . . . . . . . . . 15  |-  ( 1r
`  R )  =  ( 1r `  R
)
631, 62ringidcl 17801 . . . . . . . . . . . . . 14  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  B )
641, 30, 62ringlidm 17804 . . . . . . . . . . . . . 14  |-  ( ( R  e.  Ring  /\  ( 1r `  R )  e.  B )  ->  (
( 1r `  R
) ( .r `  R ) ( 1r
`  R ) )  =  ( 1r `  R ) )
6563, 64mpdan 674 . . . . . . . . . . . . 13  |-  ( R  e.  Ring  ->  ( ( 1r `  R ) ( .r `  R
) ( 1r `  R ) )  =  ( 1r `  R
) )
6665adantr 467 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (
( 1r `  R
) ( .r `  R ) ( 1r
`  R ) )  =  ( 1r `  R ) )
67 simpr 463 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  G  e.  Grp )
682, 621unit 17886 . . . . . . . . . . . . . . 15  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  (Unit `  R )
)
6968adantr 467 . . . . . . . . . . . . . 14  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  ( 1r `  R )  e.  (Unit `  R )
)
7044, 69sseldd 3433 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  ( 1r `  R )  e.  ( B  \  {  .0.  } ) )
7119, 59, 20grpid 16701 . . . . . . . . . . . . 13  |-  ( ( G  e.  Grp  /\  ( 1r `  R )  e.  ( B  \  {  .0.  } ) )  ->  ( ( ( 1r `  R ) ( .r `  R
) ( 1r `  R ) )  =  ( 1r `  R
)  <->  ( 0g `  G )  =  ( 1r `  R ) ) )
7267, 70, 71syl2anc 667 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (
( ( 1r `  R ) ( .r
`  R ) ( 1r `  R ) )  =  ( 1r
`  R )  <->  ( 0g `  G )  =  ( 1r `  R ) ) )
7366, 72mpbid 214 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  ( 0g `  G )  =  ( 1r `  R
) )
7473adantr 467 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  ( 0g `  G )  =  ( 1r `  R
) )
7561, 74eqtrd 2485 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( ( invg `  G ) `  x
) ( .r `  R ) x )  =  ( 1r `  R ) )
7653, 75breqtrd 4427 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x
( ||r `
 R ) ( 1r `  R ) )
77 eqid 2451 . . . . . . . . . . . 12  |-  (oppr `  R
)  =  (oppr `  R
)
7877, 1opprbas 17857 . . . . . . . . . . 11  |-  B  =  ( Base `  (oppr `  R
) )
79 eqid 2451 . . . . . . . . . . 11  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
80 eqid 2451 . . . . . . . . . . 11  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
8178, 79, 80dvdsrmul 17876 . . . . . . . . . 10  |-  ( ( x  e.  B  /\  ( ( invg `  G ) `  x
)  e.  B )  ->  x ( ||r `  (oppr `  R
) ) ( ( ( invg `  G ) `  x
) ( .r `  (oppr `  R ) ) x ) )
8246, 50, 81syl2anc 667 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x
( ||r `
 (oppr
`  R ) ) ( ( ( invg `  G ) `
 x ) ( .r `  (oppr `  R
) ) x ) )
831, 30, 77, 80opprmul 17854 . . . . . . . . . 10  |-  ( ( ( invg `  G ) `  x
) ( .r `  (oppr `  R ) ) x )  =  ( x ( .r `  R
) ( ( invg `  G ) `
 x ) )
8419, 59, 20, 47grprinv 16713 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  x  e.  ( B  \  {  .0.  } ) )  ->  ( x
( .r `  R
) ( ( invg `  G ) `
 x ) )  =  ( 0g `  G ) )
8584adantll 720 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
x ( .r `  R ) ( ( invg `  G
) `  x )
)  =  ( 0g
`  G ) )
8685, 74eqtrd 2485 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
x ( .r `  R ) ( ( invg `  G
) `  x )
)  =  ( 1r
`  R ) )
8783, 86syl5eq 2497 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( ( invg `  G ) `  x
) ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) )
8882, 87breqtrd 4427 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
892, 62, 51, 77, 79isunit 17885 . . . . . . . 8  |-  ( x  e.  (Unit `  R
)  <->  ( x (
||r `  R ) ( 1r
`  R )  /\  x ( ||r `
 (oppr
`  R ) ) ( 1r `  R
) ) )
9076, 88, 89sylanbrc 670 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x  e.  (Unit `  R )
)
9190ex 436 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (
x  e.  ( B 
\  {  .0.  }
)  ->  x  e.  (Unit `  R ) ) )
9291ssrdv 3438 . . . . 5  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  ( B  \  {  .0.  }
)  C_  (Unit `  R
) )
9344, 92eqssd 3449 . . . 4  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (Unit `  R )  =  ( B  \  {  .0.  } ) )
9412, 93impbida 843 . . 3  |-  ( R  e.  Ring  ->  ( (Unit `  R )  =  ( B  \  {  .0.  } )  <->  G  e.  Grp ) )
9594pm5.32i 643 . 2  |-  ( ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )  <->  ( R  e.  Ring  /\  G  e.  Grp ) )
964, 95bitri 253 1  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  G  e.  Grp )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   _Vcvv 3045    \ cdif 3401    C_ wss 3404   {csn 3968   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   Basecbs 15121   ↾s cress 15122   +g cplusg 15190   .rcmulr 15191   0gc0g 15338   Grpcgrp 16669   invgcminusg 16670  mulGrpcmgp 17723   1rcur 17735   Ringcrg 17780  opprcoppr 17850   ||rcdsr 17866  Unitcui 17867  /rcdvr 17910   DivRingcdr 17975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-tpos 6973  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-0g 15340  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-grp 16673  df-minusg 16674  df-mgp 17724  df-ur 17736  df-ring 17782  df-oppr 17851  df-dvdsr 17869  df-unit 17870  df-invr 17900  df-dvr 17911  df-drng 17977
This theorem is referenced by:  drngmgp  17987  isdrngd  18000
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