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Theorem isdrng2 18063
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 2471 . . 3  |-  (Unit `  R )  =  (Unit `  R )
3 isdrng2.z . . 3  |-  .0.  =  ( 0g `  R )
41, 2, 3isdrng 18057 . 2  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) ) )
5 oveq2 6316 . . . . . . 7  |-  ( (Unit `  R )  =  ( B  \  {  .0.  } )  ->  ( (mulGrp `  R )s  (Unit `  R )
)  =  ( (mulGrp `  R )s  ( B  \  {  .0.  } ) ) )
65adantl 473 . . . . . 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 2523 . . . . 5  |-  ( ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )  ->  (
(mulGrp `  R )s  (Unit `  R ) )  =  G )
9 eqid 2471 . . . . . . 7  |-  ( (mulGrp `  R )s  (Unit `  R )
)  =  ( (mulGrp `  R )s  (Unit `  R )
)
102, 9unitgrp 17973 . . . . . 6  |-  ( R  e.  Ring  ->  ( (mulGrp `  R )s  (Unit `  R )
)  e.  Grp )
1110adantr 472 . . . . 5  |-  ( ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )  ->  (
(mulGrp `  R )s  (Unit `  R ) )  e. 
Grp )
128, 11eqeltrrd 2550 . . . 4  |-  ( ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )  ->  G  e.  Grp )
131, 2unitcl 17965 . . . . . . . . 9  |-  ( x  e.  (Unit `  R
)  ->  x  e.  B )
1413adantl 473 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  x  e.  B )
15 difss 3549 . . . . . . . . . . . . . . 15  |-  ( B 
\  {  .0.  }
)  C_  B
16 eqid 2471 . . . . . . . . . . . . . . . . 17  |-  (mulGrp `  R )  =  (mulGrp `  R )
1716, 1mgpbas 17807 . . . . . . . . . . . . . . . 16  |-  B  =  ( Base `  (mulGrp `  R ) )
187, 17ressbas2 15258 . . . . . . . . . . . . . . 15  |-  ( ( B  \  {  .0.  } )  C_  B  ->  ( B  \  {  .0.  } )  =  ( Base `  G ) )
1915, 18ax-mp 5 . . . . . . . . . . . . . 14  |-  ( B 
\  {  .0.  }
)  =  ( Base `  G )
20 eqid 2471 . . . . . . . . . . . . . 14  |-  ( 0g
`  G )  =  ( 0g `  G
)
2119, 20grpidcl 16772 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  ( B  \  {  .0.  } ) )
2221ad2antlr 741 . . . . . . . . . . . 12  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( 0g `  G
)  e.  ( B 
\  {  .0.  }
) )
23 eldifsn 4088 . . . . . . . . . . . 12  |-  ( ( 0g `  G )  e.  ( B  \  {  .0.  } )  <->  ( ( 0g `  G )  e.  B  /\  ( 0g
`  G )  =/= 
.0.  ) )
2422, 23sylib 201 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( ( 0g `  G )  e.  B  /\  ( 0g `  G
)  =/=  .0.  )
)
2524simprd 470 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( 0g `  G
)  =/=  .0.  )
26 simpll 768 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  R  e.  Ring )
2722eldifad 3402 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( 0g `  G
)  e.  B )
28 simpr 468 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  x  e.  (Unit `  R
) )
29 eqid 2471 . . . . . . . . . . . 12  |-  (/r `  R
)  =  (/r `  R
)
30 eqid 2471 . . . . . . . . . . . 12  |-  ( .r
`  R )  =  ( .r `  R
)
311, 2, 29, 30dvrcan1 17997 . . . . . . . . . . 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 1292 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( ( ( 0g
`  G ) (/r `  R ) x ) ( .r `  R
) x )  =  ( 0g `  G
) )
331, 2, 29dvrcl 17992 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  ( 0g `  G )  e.  B  /\  x  e.  (Unit `  R )
)  ->  ( ( 0g `  G ) (/r `  R ) x )  e.  B )
3426, 27, 28, 33syl3anc 1292 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( ( 0g `  G ) (/r `  R
) x )  e.  B )
351, 30, 3ringrz 17896 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  (
( 0g `  G
) (/r `  R ) x )  e.  B )  ->  ( ( ( 0g `  G ) (/r `  R ) x ) ( .r `  R )  .0.  )  =  .0.  )
3626, 34, 35syl2anc 673 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( ( ( 0g
`  G ) (/r `  R ) x ) ( .r `  R
)  .0.  )  =  .0.  )
3725, 32, 363netr4d 2720 . . . . . . . . 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 6316 . . . . . . . . . 10  |-  ( x  =  .0.  ->  (
( ( 0g `  G ) (/r `  R
) x ) ( .r `  R ) x )  =  ( ( ( 0g `  G ) (/r `  R
) x ) ( .r `  R )  .0.  ) )
3938necon3i 2675 . . . . . . . . 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 4088 . . . . . . . 8  |-  ( x  e.  ( B  \  {  .0.  } )  <->  ( x  e.  B  /\  x  =/=  .0.  ) )
4214, 40, 41sylanbrc 677 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  x  e.  ( B  \  {  .0.  } ) )
4342ex 441 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (
x  e.  (Unit `  R )  ->  x  e.  ( B  \  {  .0.  } ) ) )
4443ssrdv 3424 . . . . 5  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (Unit `  R )  C_  ( B  \  {  .0.  }
) )
45 eldifi 3544 . . . . . . . . . . 11  |-  ( x  e.  ( B  \  {  .0.  } )  ->  x  e.  B )
4645adantl 473 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x  e.  B )
47 eqid 2471 . . . . . . . . . . . . 13  |-  ( invg `  G )  =  ( invg `  G )
4819, 47grpinvcl 16789 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  x  e.  ( B  \  {  .0.  } ) )  ->  ( ( invg `  G ) `
 x )  e.  ( B  \  {  .0.  } ) )
4948adantll 728 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( invg `  G ) `  x
)  e.  ( B 
\  {  .0.  }
) )
5049eldifad 3402 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( invg `  G ) `  x
)  e.  B )
51 eqid 2471 . . . . . . . . . . 11  |-  ( ||r `  R
)  =  ( ||r `  R
)
521, 51, 30dvdsrmul 17954 . . . . . . . . . 10  |-  ( ( x  e.  B  /\  ( ( invg `  G ) `  x
)  e.  B )  ->  x ( ||r `  R
) ( ( ( invg `  G
) `  x )
( .r `  R
) x ) )
5346, 50, 52syl2anc 673 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x
( ||r `
 R ) ( ( ( invg `  G ) `  x
) ( .r `  R ) x ) )
54 fvex 5889 . . . . . . . . . . . . . 14  |-  ( Base `  R )  e.  _V
551, 54eqeltri 2545 . . . . . . . . . . . . 13  |-  B  e. 
_V
56 difexg 4545 . . . . . . . . . . . . 13  |-  ( B  e.  _V  ->  ( B  \  {  .0.  }
)  e.  _V )
5716, 30mgpplusg 17805 . . . . . . . . . . . . . 14  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
587, 57ressplusg 15317 . . . . . . . . . . . . 13  |-  ( ( B  \  {  .0.  } )  e.  _V  ->  ( .r `  R )  =  ( +g  `  G
) )
5955, 56, 58mp2b 10 . . . . . . . . . . . 12  |-  ( .r
`  R )  =  ( +g  `  G
)
6019, 59, 20, 47grplinv 16790 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  x  e.  ( B  \  {  .0.  } ) )  ->  ( (
( invg `  G ) `  x
) ( .r `  R ) x )  =  ( 0g `  G ) )
6160adantll 728 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( ( invg `  G ) `  x
) ( .r `  R ) x )  =  ( 0g `  G ) )
62 eqid 2471 . . . . . . . . . . . . . . 15  |-  ( 1r
`  R )  =  ( 1r `  R
)
631, 62ringidcl 17879 . . . . . . . . . . . . . 14  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  B )
641, 30, 62ringlidm 17882 . . . . . . . . . . . . . 14  |-  ( ( R  e.  Ring  /\  ( 1r `  R )  e.  B )  ->  (
( 1r `  R
) ( .r `  R ) ( 1r
`  R ) )  =  ( 1r `  R ) )
6563, 64mpdan 681 . . . . . . . . . . . . 13  |-  ( R  e.  Ring  ->  ( ( 1r `  R ) ( .r `  R
) ( 1r `  R ) )  =  ( 1r `  R
) )
6665adantr 472 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (
( 1r `  R
) ( .r `  R ) ( 1r
`  R ) )  =  ( 1r `  R ) )
67 simpr 468 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  G  e.  Grp )
682, 621unit 17964 . . . . . . . . . . . . . . 15  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  (Unit `  R )
)
6968adantr 472 . . . . . . . . . . . . . 14  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  ( 1r `  R )  e.  (Unit `  R )
)
7044, 69sseldd 3419 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  ( 1r `  R )  e.  ( B  \  {  .0.  } ) )
7119, 59, 20grpid 16779 . . . . . . . . . . . . 13  |-  ( ( G  e.  Grp  /\  ( 1r `  R )  e.  ( B  \  {  .0.  } ) )  ->  ( ( ( 1r `  R ) ( .r `  R
) ( 1r `  R ) )  =  ( 1r `  R
)  <->  ( 0g `  G )  =  ( 1r `  R ) ) )
7267, 70, 71syl2anc 673 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (
( ( 1r `  R ) ( .r
`  R ) ( 1r `  R ) )  =  ( 1r
`  R )  <->  ( 0g `  G )  =  ( 1r `  R ) ) )
7366, 72mpbid 215 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  ( 0g `  G )  =  ( 1r `  R
) )
7473adantr 472 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  ( 0g `  G )  =  ( 1r `  R
) )
7561, 74eqtrd 2505 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( ( invg `  G ) `  x
) ( .r `  R ) x )  =  ( 1r `  R ) )
7653, 75breqtrd 4420 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x
( ||r `
 R ) ( 1r `  R ) )
77 eqid 2471 . . . . . . . . . . . 12  |-  (oppr `  R
)  =  (oppr `  R
)
7877, 1opprbas 17935 . . . . . . . . . . 11  |-  B  =  ( Base `  (oppr `  R
) )
79 eqid 2471 . . . . . . . . . . 11  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
80 eqid 2471 . . . . . . . . . . 11  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
8178, 79, 80dvdsrmul 17954 . . . . . . . . . 10  |-  ( ( x  e.  B  /\  ( ( invg `  G ) `  x
)  e.  B )  ->  x ( ||r `  (oppr `  R
) ) ( ( ( invg `  G ) `  x
) ( .r `  (oppr `  R ) ) x ) )
8246, 50, 81syl2anc 673 . . . . . . . . 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 17932 . . . . . . . . . 10  |-  ( ( ( invg `  G ) `  x
) ( .r `  (oppr `  R ) ) x )  =  ( x ( .r `  R
) ( ( invg `  G ) `
 x ) )
8419, 59, 20, 47grprinv 16791 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  x  e.  ( B  \  {  .0.  } ) )  ->  ( x
( .r `  R
) ( ( invg `  G ) `
 x ) )  =  ( 0g `  G ) )
8584adantll 728 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
x ( .r `  R ) ( ( invg `  G
) `  x )
)  =  ( 0g
`  G ) )
8685, 74eqtrd 2505 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
x ( .r `  R ) ( ( invg `  G
) `  x )
)  =  ( 1r
`  R ) )
8783, 86syl5eq 2517 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( ( invg `  G ) `  x
) ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) )
8882, 87breqtrd 4420 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
892, 62, 51, 77, 79isunit 17963 . . . . . . . 8  |-  ( x  e.  (Unit `  R
)  <->  ( x (
||r `  R ) ( 1r
`  R )  /\  x ( ||r `
 (oppr
`  R ) ) ( 1r `  R
) ) )
9076, 88, 89sylanbrc 677 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x  e.  (Unit `  R )
)
9190ex 441 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (
x  e.  ( B 
\  {  .0.  }
)  ->  x  e.  (Unit `  R ) ) )
9291ssrdv 3424 . . . . 5  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  ( B  \  {  .0.  }
)  C_  (Unit `  R
) )
9344, 92eqssd 3435 . . . 4  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (Unit `  R )  =  ( B  \  {  .0.  } ) )
9412, 93impbida 850 . . 3  |-  ( R  e.  Ring  ->  ( (Unit `  R )  =  ( B  \  {  .0.  } )  <->  G  e.  Grp ) )
9594pm5.32i 649 . 2  |-  ( ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )  <->  ( R  e.  Ring  /\  G  e.  Grp ) )
964, 95bitri 257 1  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  G  e.  Grp )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031    \ cdif 3387    C_ wss 3390   {csn 3959   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Basecbs 15199   ↾s cress 15200   +g cplusg 15268   .rcmulr 15269   0gc0g 15416   Grpcgrp 16747   invgcminusg 16748  mulGrpcmgp 17801   1rcur 17813   Ringcrg 17858  opprcoppr 17928   ||rcdsr 17944  Unitcui 17945  /rcdvr 17988   DivRingcdr 18053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-0g 15418  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-grp 16751  df-minusg 16752  df-mgp 17802  df-ur 17814  df-ring 17860  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-dvr 17989  df-drng 18055
This theorem is referenced by:  drngmgp  18065  isdrngd  18078
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