MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isdrng2 Unicode version

Theorem isdrng2 15522
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 2283 . . 3  |-  (Unit `  R )  =  (Unit `  R )
3 isdrng2.z . . 3  |-  .0.  =  ( 0g `  R )
41, 2, 3isdrng 15516 . 2  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) ) )
5 oveq2 5866 . . . . . . 7  |-  ( (Unit `  R )  =  ( B  \  {  .0.  } )  ->  ( (mulGrp `  R )s  (Unit `  R )
)  =  ( (mulGrp `  R )s  ( B  \  {  .0.  } ) ) )
65adantl 452 . . . . . 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 2333 . . . . 5  |-  ( ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )  ->  (
(mulGrp `  R )s  (Unit `  R ) )  =  G )
9 eqid 2283 . . . . . . 7  |-  ( (mulGrp `  R )s  (Unit `  R )
)  =  ( (mulGrp `  R )s  (Unit `  R )
)
102, 9unitgrp 15449 . . . . . 6  |-  ( R  e.  Ring  ->  ( (mulGrp `  R )s  (Unit `  R )
)  e.  Grp )
1110adantr 451 . . . . 5  |-  ( ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )  ->  (
(mulGrp `  R )s  (Unit `  R ) )  e. 
Grp )
128, 11eqeltrrd 2358 . . . 4  |-  ( ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )  ->  G  e.  Grp )
131, 2unitcl 15441 . . . . . . . . 9  |-  ( x  e.  (Unit `  R
)  ->  x  e.  B )
1413adantl 452 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  x  e.  B )
15 difss 3303 . . . . . . . . . . . . . . 15  |-  ( B 
\  {  .0.  }
)  C_  B
16 eqid 2283 . . . . . . . . . . . . . . . . 17  |-  (mulGrp `  R )  =  (mulGrp `  R )
1716, 1mgpbas 15331 . . . . . . . . . . . . . . . 16  |-  B  =  ( Base `  (mulGrp `  R ) )
187, 17ressbas2 13199 . . . . . . . . . . . . . . 15  |-  ( ( B  \  {  .0.  } )  C_  B  ->  ( B  \  {  .0.  } )  =  ( Base `  G ) )
1915, 18ax-mp 8 . . . . . . . . . . . . . 14  |-  ( B 
\  {  .0.  }
)  =  ( Base `  G )
20 eqid 2283 . . . . . . . . . . . . . 14  |-  ( 0g
`  G )  =  ( 0g `  G
)
2119, 20grpidcl 14510 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  ( B  \  {  .0.  } ) )
2221ad2antlr 707 . . . . . . . . . . . 12  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( 0g `  G
)  e.  ( B 
\  {  .0.  }
) )
23 eldifsn 3749 . . . . . . . . . . . 12  |-  ( ( 0g `  G )  e.  ( B  \  {  .0.  } )  <->  ( ( 0g `  G )  e.  B  /\  ( 0g
`  G )  =/= 
.0.  ) )
2422, 23sylib 188 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( ( 0g `  G )  e.  B  /\  ( 0g `  G
)  =/=  .0.  )
)
2524simprd 449 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( 0g `  G
)  =/=  .0.  )
26 simpll 730 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  R  e.  Ring )
2715, 22sseldi 3178 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( 0g `  G
)  e.  B )
28 simpr 447 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  x  e.  (Unit `  R
) )
29 eqid 2283 . . . . . . . . . . . 12  |-  (/r `  R
)  =  (/r `  R
)
30 eqid 2283 . . . . . . . . . . . 12  |-  ( .r
`  R )  =  ( .r `  R
)
311, 2, 29, 30dvrcan1 15473 . . . . . . . . . . 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 1182 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( ( ( 0g
`  G ) (/r `  R ) x ) ( .r `  R
) x )  =  ( 0g `  G
) )
331, 2, 29dvrcl 15468 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  ( 0g `  G )  e.  B  /\  x  e.  (Unit `  R )
)  ->  ( ( 0g `  G ) (/r `  R ) x )  e.  B )
3426, 27, 28, 33syl3anc 1182 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( ( 0g `  G ) (/r `  R
) x )  e.  B )
351, 30, 3rngrz 15378 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  (
( 0g `  G
) (/r `  R ) x )  e.  B )  ->  ( ( ( 0g `  G ) (/r `  R ) x ) ( .r `  R )  .0.  )  =  .0.  )
3626, 34, 35syl2anc 642 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  -> 
( ( ( 0g
`  G ) (/r `  R ) x ) ( .r `  R
)  .0.  )  =  .0.  )
3725, 32, 363netr4d 2473 . . . . . . . . 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 5866 . . . . . . . . . 10  |-  ( x  =  .0.  ->  (
( ( 0g `  G ) (/r `  R
) x ) ( .r `  R ) x )  =  ( ( ( 0g `  G ) (/r `  R
) x ) ( .r `  R )  .0.  ) )
3938necon3i 2485 . . . . . . . . 9  |-  ( ( ( ( 0g `  G ) (/r `  R
) x ) ( .r `  R ) x )  =/=  (
( ( 0g `  G ) (/r `  R
) x ) ( .r `  R )  .0.  )  ->  x  =/=  .0.  )
4037, 39syl 15 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  x  =/=  .0.  )
41 eldifsn 3749 . . . . . . . 8  |-  ( x  e.  ( B  \  {  .0.  } )  <->  ( x  e.  B  /\  x  =/=  .0.  ) )
4214, 40, 41sylanbrc 645 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  (Unit `  R ) )  ->  x  e.  ( B  \  {  .0.  } ) )
4342ex 423 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (
x  e.  (Unit `  R )  ->  x  e.  ( B  \  {  .0.  } ) ) )
4443ssrdv 3185 . . . . 5  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (Unit `  R )  C_  ( B  \  {  .0.  }
) )
45 eldifi 3298 . . . . . . . . . . 11  |-  ( x  e.  ( B  \  {  .0.  } )  ->  x  e.  B )
4645adantl 452 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x  e.  B )
47 eqid 2283 . . . . . . . . . . . . 13  |-  ( inv g `  G )  =  ( inv g `  G )
4819, 47grpinvcl 14527 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  x  e.  ( B  \  {  .0.  } ) )  ->  ( ( inv g `  G ) `
 x )  e.  ( B  \  {  .0.  } ) )
4948adantll 694 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( inv g `  G ) `  x
)  e.  ( B 
\  {  .0.  }
) )
5015, 49sseldi 3178 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( inv g `  G ) `  x
)  e.  B )
51 eqid 2283 . . . . . . . . . . 11  |-  ( ||r `  R
)  =  ( ||r `  R
)
521, 51, 30dvdsrmul 15430 . . . . . . . . . 10  |-  ( ( x  e.  B  /\  ( ( inv g `  G ) `  x
)  e.  B )  ->  x ( ||r `  R
) ( ( ( inv g `  G
) `  x )
( .r `  R
) x ) )
5346, 50, 52syl2anc 642 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x
( ||r `
 R ) ( ( ( inv g `  G ) `  x
) ( .r `  R ) x ) )
54 fvex 5539 . . . . . . . . . . . . . 14  |-  ( Base `  R )  e.  _V
551, 54eqeltri 2353 . . . . . . . . . . . . 13  |-  B  e. 
_V
56 difexg 4162 . . . . . . . . . . . . 13  |-  ( B  e.  _V  ->  ( B  \  {  .0.  }
)  e.  _V )
5716, 30mgpplusg 15329 . . . . . . . . . . . . . 14  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
587, 57ressplusg 13250 . . . . . . . . . . . . 13  |-  ( ( B  \  {  .0.  } )  e.  _V  ->  ( .r `  R )  =  ( +g  `  G
) )
5955, 56, 58mp2b 9 . . . . . . . . . . . 12  |-  ( .r
`  R )  =  ( +g  `  G
)
6019, 59, 20, 47grplinv 14528 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  x  e.  ( B  \  {  .0.  } ) )  ->  ( (
( inv g `  G ) `  x
) ( .r `  R ) x )  =  ( 0g `  G ) )
6160adantll 694 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( ( inv g `  G ) `  x
) ( .r `  R ) x )  =  ( 0g `  G ) )
62 eqid 2283 . . . . . . . . . . . . . . 15  |-  ( 1r
`  R )  =  ( 1r `  R
)
631, 62rngidcl 15361 . . . . . . . . . . . . . 14  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  B )
641, 30, 62rnglidm 15364 . . . . . . . . . . . . . 14  |-  ( ( R  e.  Ring  /\  ( 1r `  R )  e.  B )  ->  (
( 1r `  R
) ( .r `  R ) ( 1r
`  R ) )  =  ( 1r `  R ) )
6563, 64mpdan 649 . . . . . . . . . . . . 13  |-  ( R  e.  Ring  ->  ( ( 1r `  R ) ( .r `  R
) ( 1r `  R ) )  =  ( 1r `  R
) )
6665adantr 451 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (
( 1r `  R
) ( .r `  R ) ( 1r
`  R ) )  =  ( 1r `  R ) )
67 simpr 447 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  G  e.  Grp )
682, 621unit 15440 . . . . . . . . . . . . . . 15  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  (Unit `  R )
)
6968adantr 451 . . . . . . . . . . . . . 14  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  ( 1r `  R )  e.  (Unit `  R )
)
7044, 69sseldd 3181 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  ( 1r `  R )  e.  ( B  \  {  .0.  } ) )
7119, 59, 20grpid 14517 . . . . . . . . . . . . 13  |-  ( ( G  e.  Grp  /\  ( 1r `  R )  e.  ( B  \  {  .0.  } ) )  ->  ( ( ( 1r `  R ) ( .r `  R
) ( 1r `  R ) )  =  ( 1r `  R
)  <->  ( 0g `  G )  =  ( 1r `  R ) ) )
7267, 70, 71syl2anc 642 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (
( ( 1r `  R ) ( .r
`  R ) ( 1r `  R ) )  =  ( 1r
`  R )  <->  ( 0g `  G )  =  ( 1r `  R ) ) )
7366, 72mpbid 201 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  ( 0g `  G )  =  ( 1r `  R
) )
7473adantr 451 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  ( 0g `  G )  =  ( 1r `  R
) )
7561, 74eqtrd 2315 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( ( inv g `  G ) `  x
) ( .r `  R ) x )  =  ( 1r `  R ) )
7653, 75breqtrd 4047 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x
( ||r `
 R ) ( 1r `  R ) )
77 eqid 2283 . . . . . . . . . . . 12  |-  (oppr `  R
)  =  (oppr `  R
)
7877, 1opprbas 15411 . . . . . . . . . . 11  |-  B  =  ( Base `  (oppr `  R
) )
79 eqid 2283 . . . . . . . . . . 11  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
80 eqid 2283 . . . . . . . . . . 11  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
8178, 79, 80dvdsrmul 15430 . . . . . . . . . 10  |-  ( ( x  e.  B  /\  ( ( inv g `  G ) `  x
)  e.  B )  ->  x ( ||r `  (oppr `  R
) ) ( ( ( inv g `  G ) `  x
) ( .r `  (oppr `  R ) ) x ) )
8246, 50, 81syl2anc 642 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x
( ||r `
 (oppr
`  R ) ) ( ( ( inv g `  G ) `
 x ) ( .r `  (oppr `  R
) ) x ) )
831, 30, 77, 80opprmul 15408 . . . . . . . . . 10  |-  ( ( ( inv g `  G ) `  x
) ( .r `  (oppr `  R ) ) x )  =  ( x ( .r `  R
) ( ( inv g `  G ) `
 x ) )
8419, 59, 20, 47grprinv 14529 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  x  e.  ( B  \  {  .0.  } ) )  ->  ( x
( .r `  R
) ( ( inv g `  G ) `
 x ) )  =  ( 0g `  G ) )
8584adantll 694 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
x ( .r `  R ) ( ( inv g `  G
) `  x )
)  =  ( 0g
`  G ) )
8685, 74eqtrd 2315 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
x ( .r `  R ) ( ( inv g `  G
) `  x )
)  =  ( 1r
`  R ) )
8783, 86syl5eq 2327 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  (
( ( inv g `  G ) `  x
) ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) )
8882, 87breqtrd 4047 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
892, 62, 51, 77, 79isunit 15439 . . . . . . . 8  |-  ( x  e.  (Unit `  R
)  <->  ( x (
||r `  R ) ( 1r
`  R )  /\  x ( ||r `
 (oppr
`  R ) ) ( 1r `  R
) ) )
9076, 88, 89sylanbrc 645 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  G  e.  Grp )  /\  x  e.  ( B  \  {  .0.  }
) )  ->  x  e.  (Unit `  R )
)
9190ex 423 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (
x  e.  ( B 
\  {  .0.  }
)  ->  x  e.  (Unit `  R ) ) )
9291ssrdv 3185 . . . . 5  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  ( B  \  {  .0.  }
)  C_  (Unit `  R
) )
9344, 92eqssd 3196 . . . 4  |-  ( ( R  e.  Ring  /\  G  e.  Grp )  ->  (Unit `  R )  =  ( B  \  {  .0.  } ) )
9412, 93impbida 805 . . 3  |-  ( R  e.  Ring  ->  ( (Unit `  R )  =  ( B  \  {  .0.  } )  <->  G  e.  Grp ) )
9594pm5.32i 618 . 2  |-  ( ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )  <->  ( R  e.  Ring  /\  G  e.  Grp ) )
964, 95bitri 240 1  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  G  e.  Grp )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {csn 3640   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   +g cplusg 13208   .rcmulr 13209   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363  mulGrpcmgp 15325   Ringcrg 15337   1rcur 15339  opprcoppr 15404   ||rcdsr 15420  Unitcui 15421  /rcdvr 15464   DivRingcdr 15512
This theorem is referenced by:  drngmgp  15524  isdrngd  15537
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514
  Copyright terms: Public domain W3C validator