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Theorem isdrng2 17920
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 2429 . . 3  |-  (Unit `  R )  =  (Unit `  R )
3 isdrng2.z . . 3  |-  .0.  =  ( 0g `  R )
41, 2, 3isdrng 17914 . 2  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) ) )
5 oveq2 6313 . . . . . . 7  |-  ( (Unit `  R )  =  ( B  \  {  .0.  } )  ->  ( (mulGrp `  R )s  (Unit `  R )
)  =  ( (mulGrp `  R )s  ( B  \  {  .0.  } ) ) )
65adantl 467 . . . . . 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 2488 . . . . 5  |-  ( ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )  ->  (
(mulGrp `  R )s  (Unit `  R ) )  =  G )
9 eqid 2429 . . . . . . 7  |-  ( (mulGrp `  R )s  (Unit `  R )
)  =  ( (mulGrp `  R )s  (Unit `  R )
)
102, 9unitgrp 17830 . . . . . 6  |-  ( R  e.  Ring  ->  ( (mulGrp `  R )s  (Unit `  R )
)  e.  Grp )
1110adantr 466 . . . . 5  |-  ( ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )  ->  (
(mulGrp `  R )s  (Unit `  R ) )  e. 
Grp )
128, 11eqeltrrd 2518 . . . 4  |-  ( ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )  ->  G  e.  Grp )
131, 2unitcl 17822 . . . . . . . . 9  |-  ( x  e.  (Unit `  R
)  ->  x  e.  B )
1413adantl 467 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  x  e.  B )
15 difss 3598 . . . . . . . . . . . . . . 15  |-  ( B 
\  {  .0.  }
)  C_  B
16 eqid 2429 . . . . . . . . . . . . . . . . 17  |-  (mulGrp `  R )  =  (mulGrp `  R )
1716, 1mgpbas 17664 . . . . . . . . . . . . . . . 16  |-  B  =  ( Base `  (mulGrp `  R ) )
187, 17ressbas2 15142 . . . . . . . . . . . . . . 15  |-  ( ( B  \  {  .0.  } )  C_  B  ->  ( B  \  {  .0.  } )  =  ( Base `  G ) )
1915, 18ax-mp 5 . . . . . . . . . . . . . 14  |-  ( B 
\  {  .0.  }
)  =  ( Base `  G )
20 eqid 2429 . . . . . . . . . . . . . 14  |-  ( 0g
`  G )  =  ( 0g `  G
)
2119, 20grpidcl 16645 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  ( B  \  {  .0.  } ) )
2221ad2antlr 731 . . . . . . . . . . . 12  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( 0g `  G
)  e.  ( B 
\  {  .0.  }
) )
23 eldifsn 4128 . . . . . . . . . . . 12  |-  ( ( 0g `  G )  e.  ( B  \  {  .0.  } )  <->  ( ( 0g `  G )  e.  B  /\  ( 0g
`  G )  =/= 
.0.  ) )
2422, 23sylib 199 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( ( 0g `  G )  e.  B  /\  ( 0g `  G
)  =/=  .0.  )
)
2524simprd 464 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( 0g `  G
)  =/=  .0.  )
26 simpll 758 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  R  e.  Ring )
2722eldifad 3454 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( 0g `  G
)  e.  B )
28 simpr 462 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  x  e.  (Unit `  R
) )
29 eqid 2429 . . . . . . . . . . . 12  |-  (/r `  R
)  =  (/r `  R
)
30 eqid 2429 . . . . . . . . . . . 12  |-  ( .r
`  R )  =  ( .r `  R
)
311, 2, 29, 30dvrcan1 17854 . . . . . . . . . . 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 1264 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( ( ( 0g
`  G ) (/r `  R ) x ) ( .r `  R
) x )  =  ( 0g `  G
) )
331, 2, 29dvrcl 17849 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  ( 0g `  G )  e.  B  /\  x  e.  (Unit `  R )
)  ->  ( ( 0g `  G ) (/r `  R ) x )  e.  B )
3426, 27, 28, 33syl3anc 1264 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( ( 0g `  G ) (/r `  R
) x )  e.  B )
351, 30, 3ringrz 17753 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  (
( 0g `  G
) (/r `  R ) x )  e.  B )  ->  ( ( ( 0g `  G ) (/r `  R ) x ) ( .r `  R )  .0.  )  =  .0.  )
3626, 34, 35syl2anc 665 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( ( ( 0g
`  G ) (/r `  R ) x ) ( .r `  R
)  .0.  )  =  .0.  )
3725, 32, 363netr4d 2736 . . . . . . . . 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 6313 . . . . . . . . . 10  |-  ( x  =  .0.  ->  (
( ( 0g `  G ) (/r `  R
) x ) ( .r `  R ) x )  =  ( ( ( 0g `  G ) (/r `  R
) x ) ( .r `  R )  .0.  ) )
3938necon3i 2671 . . . . . . . . 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 4128 . . . . . . . 8  |-  ( x  e.  ( B  \  {  .0.  } )  <->  ( x  e.  B  /\  x  =/=  .0.  ) )
4214, 40, 41sylanbrc 668 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  x  e.  ( B  \  {  .0.  } ) )
4342ex 435 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (
x  e.  (Unit `  R )  ->  x  e.  ( B  \  {  .0.  } ) ) )
4443ssrdv 3476 . . . . 5  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (Unit `  R )  C_  ( B  \  {  .0.  }
) )
45 eldifi 3593 . . . . . . . . . . 11  |-  ( x  e.  ( B  \  {  .0.  } )  ->  x  e.  B )
4645adantl 467 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x  e.  B )
47 eqid 2429 . . . . . . . . . . . . 13  |-  ( invg `  G )  =  ( invg `  G )
4819, 47grpinvcl 16662 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  x  e.  ( B  \  {  .0.  } ) )  ->  ( ( invg `  G ) `
 x )  e.  ( B  \  {  .0.  } ) )
4948adantll 718 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( invg `  G ) `  x
)  e.  ( B 
\  {  .0.  }
) )
5049eldifad 3454 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( invg `  G ) `  x
)  e.  B )
51 eqid 2429 . . . . . . . . . . 11  |-  ( ||r `  R
)  =  ( ||r `  R
)
521, 51, 30dvdsrmul 17811 . . . . . . . . . 10  |-  ( ( x  e.  B  /\  ( ( invg `  G ) `  x
)  e.  B )  ->  x ( ||r `  R
) ( ( ( invg `  G
) `  x )
( .r `  R
) x ) )
5346, 50, 52syl2anc 665 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x
( ||r `
 R ) ( ( ( invg `  G ) `  x
) ( .r `  R ) x ) )
54 fvex 5891 . . . . . . . . . . . . . 14  |-  ( Base `  R )  e.  _V
551, 54eqeltri 2513 . . . . . . . . . . . . 13  |-  B  e. 
_V
56 difexg 4573 . . . . . . . . . . . . 13  |-  ( B  e.  _V  ->  ( B  \  {  .0.  }
)  e.  _V )
5716, 30mgpplusg 17662 . . . . . . . . . . . . . 14  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
587, 57ressplusg 15198 . . . . . . . . . . . . 13  |-  ( ( B  \  {  .0.  } )  e.  _V  ->  ( .r `  R )  =  ( +g  `  G
) )
5955, 56, 58mp2b 10 . . . . . . . . . . . 12  |-  ( .r
`  R )  =  ( +g  `  G
)
6019, 59, 20, 47grplinv 16663 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  x  e.  ( B  \  {  .0.  } ) )  ->  ( (
( invg `  G ) `  x
) ( .r `  R ) x )  =  ( 0g `  G ) )
6160adantll 718 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( ( invg `  G ) `  x
) ( .r `  R ) x )  =  ( 0g `  G ) )
62 eqid 2429 . . . . . . . . . . . . . . 15  |-  ( 1r
`  R )  =  ( 1r `  R
)
631, 62ringidcl 17736 . . . . . . . . . . . . . 14  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  B )
641, 30, 62ringlidm 17739 . . . . . . . . . . . . . 14  |-  ( ( R  e.  Ring  /\  ( 1r `  R )  e.  B )  ->  (
( 1r `  R
) ( .r `  R ) ( 1r
`  R ) )  =  ( 1r `  R ) )
6563, 64mpdan 672 . . . . . . . . . . . . 13  |-  ( R  e.  Ring  ->  ( ( 1r `  R ) ( .r `  R
) ( 1r `  R ) )  =  ( 1r `  R
) )
6665adantr 466 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (
( 1r `  R
) ( .r `  R ) ( 1r
`  R ) )  =  ( 1r `  R ) )
67 simpr 462 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  G  e.  Grp )
682, 621unit 17821 . . . . . . . . . . . . . . 15  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  (Unit `  R )
)
6968adantr 466 . . . . . . . . . . . . . 14  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  ( 1r `  R )  e.  (Unit `  R )
)
7044, 69sseldd 3471 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  ( 1r `  R )  e.  ( B  \  {  .0.  } ) )
7119, 59, 20grpid 16652 . . . . . . . . . . . . 13  |-  ( ( G  e.  Grp  /\  ( 1r `  R )  e.  ( B  \  {  .0.  } ) )  ->  ( ( ( 1r `  R ) ( .r `  R
) ( 1r `  R ) )  =  ( 1r `  R
)  <->  ( 0g `  G )  =  ( 1r `  R ) ) )
7267, 70, 71syl2anc 665 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (
( ( 1r `  R ) ( .r
`  R ) ( 1r `  R ) )  =  ( 1r
`  R )  <->  ( 0g `  G )  =  ( 1r `  R ) ) )
7366, 72mpbid 213 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  ( 0g `  G )  =  ( 1r `  R
) )
7473adantr 466 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  ( 0g `  G )  =  ( 1r `  R
) )
7561, 74eqtrd 2470 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( ( invg `  G ) `  x
) ( .r `  R ) x )  =  ( 1r `  R ) )
7653, 75breqtrd 4450 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x
( ||r `
 R ) ( 1r `  R ) )
77 eqid 2429 . . . . . . . . . . . 12  |-  (oppr `  R
)  =  (oppr `  R
)
7877, 1opprbas 17792 . . . . . . . . . . 11  |-  B  =  ( Base `  (oppr `  R
) )
79 eqid 2429 . . . . . . . . . . 11  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
80 eqid 2429 . . . . . . . . . . 11  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
8178, 79, 80dvdsrmul 17811 . . . . . . . . . 10  |-  ( ( x  e.  B  /\  ( ( invg `  G ) `  x
)  e.  B )  ->  x ( ||r `  (oppr `  R
) ) ( ( ( invg `  G ) `  x
) ( .r `  (oppr `  R ) ) x ) )
8246, 50, 81syl2anc 665 . . . . . . . . 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 17789 . . . . . . . . . 10  |-  ( ( ( invg `  G ) `  x
) ( .r `  (oppr `  R ) ) x )  =  ( x ( .r `  R
) ( ( invg `  G ) `
 x ) )
8419, 59, 20, 47grprinv 16664 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  x  e.  ( B  \  {  .0.  } ) )  ->  ( x
( .r `  R
) ( ( invg `  G ) `
 x ) )  =  ( 0g `  G ) )
8584adantll 718 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
x ( .r `  R ) ( ( invg `  G
) `  x )
)  =  ( 0g
`  G ) )
8685, 74eqtrd 2470 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
x ( .r `  R ) ( ( invg `  G
) `  x )
)  =  ( 1r
`  R ) )
8783, 86syl5eq 2482 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( ( invg `  G ) `  x
) ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) )
8882, 87breqtrd 4450 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
892, 62, 51, 77, 79isunit 17820 . . . . . . . 8  |-  ( x  e.  (Unit `  R
)  <->  ( x (
||r `  R ) ( 1r
`  R )  /\  x ( ||r `
 (oppr
`  R ) ) ( 1r `  R
) ) )
9076, 88, 89sylanbrc 668 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x  e.  (Unit `  R )
)
9190ex 435 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (
x  e.  ( B 
\  {  .0.  }
)  ->  x  e.  (Unit `  R ) ) )
9291ssrdv 3476 . . . . 5  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  ( B  \  {  .0.  }
)  C_  (Unit `  R
) )
9344, 92eqssd 3487 . . . 4  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (Unit `  R )  =  ( B  \  {  .0.  } ) )
9412, 93impbida 840 . . 3  |-  ( R  e.  Ring  ->  ( (Unit `  R )  =  ( B  \  {  .0.  } )  <->  G  e.  Grp ) )
9594pm5.32i 641 . 2  |-  ( ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )  <->  ( R  e.  Ring  /\  G  e.  Grp ) )
964, 95bitri 252 1  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  G  e.  Grp )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   _Vcvv 3087    \ cdif 3439    C_ wss 3442   {csn 4002   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   ↾s cress 15085   +g cplusg 15152   .rcmulr 15153   0gc0g 15297   Grpcgrp 16620   invgcminusg 16621  mulGrpcmgp 17658   1rcur 17670   Ringcrg 17715  opprcoppr 17785   ||rcdsr 17801  Unitcui 17802  /rcdvr 17845   DivRingcdr 17910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-tpos 6981  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-minusg 16625  df-mgp 17659  df-ur 17671  df-ring 17717  df-oppr 17786  df-dvdsr 17804  df-unit 17805  df-invr 17835  df-dvr 17846  df-drng 17912
This theorem is referenced by:  drngmgp  17922  isdrngd  17935
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