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Theorem isdrng2 17206
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 2467 . . 3  |-  (Unit `  R )  =  (Unit `  R )
3 isdrng2.z . . 3  |-  .0.  =  ( 0g `  R )
41, 2, 3isdrng 17200 . 2  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) ) )
5 oveq2 6292 . . . . . . 7  |-  ( (Unit `  R )  =  ( B  \  {  .0.  } )  ->  ( (mulGrp `  R )s  (Unit `  R )
)  =  ( (mulGrp `  R )s  ( B  \  {  .0.  } ) ) )
65adantl 466 . . . . . 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 2526 . . . . 5  |-  ( ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )  ->  (
(mulGrp `  R )s  (Unit `  R ) )  =  G )
9 eqid 2467 . . . . . . 7  |-  ( (mulGrp `  R )s  (Unit `  R )
)  =  ( (mulGrp `  R )s  (Unit `  R )
)
102, 9unitgrp 17117 . . . . . 6  |-  ( R  e.  Ring  ->  ( (mulGrp `  R )s  (Unit `  R )
)  e.  Grp )
1110adantr 465 . . . . 5  |-  ( ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )  ->  (
(mulGrp `  R )s  (Unit `  R ) )  e. 
Grp )
128, 11eqeltrrd 2556 . . . 4  |-  ( ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )  ->  G  e.  Grp )
131, 2unitcl 17109 . . . . . . . . 9  |-  ( x  e.  (Unit `  R
)  ->  x  e.  B )
1413adantl 466 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  x  e.  B )
15 difss 3631 . . . . . . . . . . . . . . 15  |-  ( B 
\  {  .0.  }
)  C_  B
16 eqid 2467 . . . . . . . . . . . . . . . . 17  |-  (mulGrp `  R )  =  (mulGrp `  R )
1716, 1mgpbas 16949 . . . . . . . . . . . . . . . 16  |-  B  =  ( Base `  (mulGrp `  R ) )
187, 17ressbas2 14546 . . . . . . . . . . . . . . 15  |-  ( ( B  \  {  .0.  } )  C_  B  ->  ( B  \  {  .0.  } )  =  ( Base `  G ) )
1915, 18ax-mp 5 . . . . . . . . . . . . . 14  |-  ( B 
\  {  .0.  }
)  =  ( Base `  G )
20 eqid 2467 . . . . . . . . . . . . . 14  |-  ( 0g
`  G )  =  ( 0g `  G
)
2119, 20grpidcl 15888 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  ( B  \  {  .0.  } ) )
2221ad2antlr 726 . . . . . . . . . . . 12  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( 0g `  G
)  e.  ( B 
\  {  .0.  }
) )
23 eldifsn 4152 . . . . . . . . . . . 12  |-  ( ( 0g `  G )  e.  ( B  \  {  .0.  } )  <->  ( ( 0g `  G )  e.  B  /\  ( 0g
`  G )  =/= 
.0.  ) )
2422, 23sylib 196 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( ( 0g `  G )  e.  B  /\  ( 0g `  G
)  =/=  .0.  )
)
2524simprd 463 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( 0g `  G
)  =/=  .0.  )
26 simpll 753 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  R  e.  Ring )
2722eldifad 3488 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( 0g `  G
)  e.  B )
28 simpr 461 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  x  e.  (Unit `  R
) )
29 eqid 2467 . . . . . . . . . . . 12  |-  (/r `  R
)  =  (/r `  R
)
30 eqid 2467 . . . . . . . . . . . 12  |-  ( .r
`  R )  =  ( .r `  R
)
311, 2, 29, 30dvrcan1 17141 . . . . . . . . . . 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 1228 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( ( ( 0g
`  G ) (/r `  R ) x ) ( .r `  R
) x )  =  ( 0g `  G
) )
331, 2, 29dvrcl 17136 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  ( 0g `  G )  e.  B  /\  x  e.  (Unit `  R )
)  ->  ( ( 0g `  G ) (/r `  R ) x )  e.  B )
3426, 27, 28, 33syl3anc 1228 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( ( 0g `  G ) (/r `  R
) x )  e.  B )
351, 30, 3rngrz 17037 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  (
( 0g `  G
) (/r `  R ) x )  e.  B )  ->  ( ( ( 0g `  G ) (/r `  R ) x ) ( .r `  R )  .0.  )  =  .0.  )
3626, 34, 35syl2anc 661 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( ( ( 0g
`  G ) (/r `  R ) x ) ( .r `  R
)  .0.  )  =  .0.  )
3725, 32, 363netr4d 2772 . . . . . . . . 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 6292 . . . . . . . . . 10  |-  ( x  =  .0.  ->  (
( ( 0g `  G ) (/r `  R
) x ) ( .r `  R ) x )  =  ( ( ( 0g `  G ) (/r `  R
) x ) ( .r `  R )  .0.  ) )
3938necon3i 2707 . . . . . . . . 9  |-  ( ( ( ( 0g `  G ) (/r `  R
) x ) ( .r `  R ) x )  =/=  (
( ( 0g `  G ) (/r `  R
) x ) ( .r `  R )  .0.  )  ->  x  =/=  .0.  )
4037, 39syl 16 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  x  =/=  .0.  )
41 eldifsn 4152 . . . . . . . 8  |-  ( x  e.  ( B  \  {  .0.  } )  <->  ( x  e.  B  /\  x  =/=  .0.  ) )
4214, 40, 41sylanbrc 664 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  x  e.  ( B  \  {  .0.  } ) )
4342ex 434 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (
x  e.  (Unit `  R )  ->  x  e.  ( B  \  {  .0.  } ) ) )
4443ssrdv 3510 . . . . 5  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (Unit `  R )  C_  ( B  \  {  .0.  }
) )
45 eldifi 3626 . . . . . . . . . . 11  |-  ( x  e.  ( B  \  {  .0.  } )  ->  x  e.  B )
4645adantl 466 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x  e.  B )
47 eqid 2467 . . . . . . . . . . . . 13  |-  ( invg `  G )  =  ( invg `  G )
4819, 47grpinvcl 15905 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  x  e.  ( B  \  {  .0.  } ) )  ->  ( ( invg `  G ) `
 x )  e.  ( B  \  {  .0.  } ) )
4948adantll 713 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( invg `  G ) `  x
)  e.  ( B 
\  {  .0.  }
) )
5049eldifad 3488 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( invg `  G ) `  x
)  e.  B )
51 eqid 2467 . . . . . . . . . . 11  |-  ( ||r `  R
)  =  ( ||r `  R
)
521, 51, 30dvdsrmul 17098 . . . . . . . . . 10  |-  ( ( x  e.  B  /\  ( ( invg `  G ) `  x
)  e.  B )  ->  x ( ||r `  R
) ( ( ( invg `  G
) `  x )
( .r `  R
) x ) )
5346, 50, 52syl2anc 661 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x
( ||r `
 R ) ( ( ( invg `  G ) `  x
) ( .r `  R ) x ) )
54 fvex 5876 . . . . . . . . . . . . . 14  |-  ( Base `  R )  e.  _V
551, 54eqeltri 2551 . . . . . . . . . . . . 13  |-  B  e. 
_V
56 difexg 4595 . . . . . . . . . . . . 13  |-  ( B  e.  _V  ->  ( B  \  {  .0.  }
)  e.  _V )
5716, 30mgpplusg 16947 . . . . . . . . . . . . . 14  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
587, 57ressplusg 14597 . . . . . . . . . . . . 13  |-  ( ( B  \  {  .0.  } )  e.  _V  ->  ( .r `  R )  =  ( +g  `  G
) )
5955, 56, 58mp2b 10 . . . . . . . . . . . 12  |-  ( .r
`  R )  =  ( +g  `  G
)
6019, 59, 20, 47grplinv 15906 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  x  e.  ( B  \  {  .0.  } ) )  ->  ( (
( invg `  G ) `  x
) ( .r `  R ) x )  =  ( 0g `  G ) )
6160adantll 713 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( ( invg `  G ) `  x
) ( .r `  R ) x )  =  ( 0g `  G ) )
62 eqid 2467 . . . . . . . . . . . . . . 15  |-  ( 1r
`  R )  =  ( 1r `  R
)
631, 62rngidcl 17020 . . . . . . . . . . . . . 14  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  B )
641, 30, 62rnglidm 17023 . . . . . . . . . . . . . 14  |-  ( ( R  e.  Ring  /\  ( 1r `  R )  e.  B )  ->  (
( 1r `  R
) ( .r `  R ) ( 1r
`  R ) )  =  ( 1r `  R ) )
6563, 64mpdan 668 . . . . . . . . . . . . 13  |-  ( R  e.  Ring  ->  ( ( 1r `  R ) ( .r `  R
) ( 1r `  R ) )  =  ( 1r `  R
) )
6665adantr 465 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (
( 1r `  R
) ( .r `  R ) ( 1r
`  R ) )  =  ( 1r `  R ) )
67 simpr 461 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  G  e.  Grp )
682, 621unit 17108 . . . . . . . . . . . . . . 15  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  (Unit `  R )
)
6968adantr 465 . . . . . . . . . . . . . 14  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  ( 1r `  R )  e.  (Unit `  R )
)
7044, 69sseldd 3505 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  ( 1r `  R )  e.  ( B  \  {  .0.  } ) )
7119, 59, 20grpid 15895 . . . . . . . . . . . . 13  |-  ( ( G  e.  Grp  /\  ( 1r `  R )  e.  ( B  \  {  .0.  } ) )  ->  ( ( ( 1r `  R ) ( .r `  R
) ( 1r `  R ) )  =  ( 1r `  R
)  <->  ( 0g `  G )  =  ( 1r `  R ) ) )
7267, 70, 71syl2anc 661 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (
( ( 1r `  R ) ( .r
`  R ) ( 1r `  R ) )  =  ( 1r
`  R )  <->  ( 0g `  G )  =  ( 1r `  R ) ) )
7366, 72mpbid 210 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  ( 0g `  G )  =  ( 1r `  R
) )
7473adantr 465 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  ( 0g `  G )  =  ( 1r `  R
) )
7561, 74eqtrd 2508 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( ( invg `  G ) `  x
) ( .r `  R ) x )  =  ( 1r `  R ) )
7653, 75breqtrd 4471 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x
( ||r `
 R ) ( 1r `  R ) )
77 eqid 2467 . . . . . . . . . . . 12  |-  (oppr `  R
)  =  (oppr `  R
)
7877, 1opprbas 17079 . . . . . . . . . . 11  |-  B  =  ( Base `  (oppr `  R
) )
79 eqid 2467 . . . . . . . . . . 11  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
80 eqid 2467 . . . . . . . . . . 11  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
8178, 79, 80dvdsrmul 17098 . . . . . . . . . 10  |-  ( ( x  e.  B  /\  ( ( invg `  G ) `  x
)  e.  B )  ->  x ( ||r `  (oppr `  R
) ) ( ( ( invg `  G ) `  x
) ( .r `  (oppr `  R ) ) x ) )
8246, 50, 81syl2anc 661 . . . . . . . . 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 17076 . . . . . . . . . 10  |-  ( ( ( invg `  G ) `  x
) ( .r `  (oppr `  R ) ) x )  =  ( x ( .r `  R
) ( ( invg `  G ) `
 x ) )
8419, 59, 20, 47grprinv 15907 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  x  e.  ( B  \  {  .0.  } ) )  ->  ( x
( .r `  R
) ( ( invg `  G ) `
 x ) )  =  ( 0g `  G ) )
8584adantll 713 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
x ( .r `  R ) ( ( invg `  G
) `  x )
)  =  ( 0g
`  G ) )
8685, 74eqtrd 2508 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
x ( .r `  R ) ( ( invg `  G
) `  x )
)  =  ( 1r
`  R ) )
8783, 86syl5eq 2520 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( ( invg `  G ) `  x
) ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) )
8882, 87breqtrd 4471 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
892, 62, 51, 77, 79isunit 17107 . . . . . . . 8  |-  ( x  e.  (Unit `  R
)  <->  ( x (
||r `  R ) ( 1r
`  R )  /\  x ( ||r `
 (oppr
`  R ) ) ( 1r `  R
) ) )
9076, 88, 89sylanbrc 664 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x  e.  (Unit `  R )
)
9190ex 434 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (
x  e.  ( B 
\  {  .0.  }
)  ->  x  e.  (Unit `  R ) ) )
9291ssrdv 3510 . . . . 5  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  ( B  \  {  .0.  }
)  C_  (Unit `  R
) )
9344, 92eqssd 3521 . . . 4  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (Unit `  R )  =  ( B  \  {  .0.  } ) )
9412, 93impbida 830 . . 3  |-  ( R  e.  Ring  ->  ( (Unit `  R )  =  ( B  \  {  .0.  } )  <->  G  e.  Grp ) )
9594pm5.32i 637 . 2  |-  ( ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )  <->  ( R  e.  Ring  /\  G  e.  Grp ) )
964, 95bitri 249 1  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  G  e.  Grp )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    \ cdif 3473    C_ wss 3476   {csn 4027   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   Basecbs 14490   ↾s cress 14491   +g cplusg 14555   .rcmulr 14556   0gc0g 14695   Grpcgrp 15727   invgcminusg 15728  mulGrpcmgp 16943   1rcur 16955   Ringcrg 17000  opprcoppr 17072   ||rcdsr 17088  Unitcui 17089  /rcdvr 17132   DivRingcdr 17196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-tpos 6955  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-0g 14697  df-mnd 15732  df-grp 15867  df-minusg 15868  df-mgp 16944  df-ur 16956  df-rng 17002  df-oppr 17073  df-dvdsr 17091  df-unit 17092  df-invr 17122  df-dvr 17133  df-drng 17198
This theorem is referenced by:  drngmgp  17208  isdrngd  17221
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