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Theorem issubdrg 15893
Description: Characterize the subfields of a division ring. (Contributed by Mario Carneiro, 3-Dec-2014.)
Hypotheses
Ref Expression
issubdrg.s  |-  S  =  ( Rs  A )
issubdrg.z  |-  .0.  =  ( 0g `  R )
issubdrg.i  |-  I  =  ( invr `  R
)
Assertion
Ref Expression
issubdrg  |-  ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R )
)  ->  ( S  e.  DivRing 
<-> 
A. x  e.  ( A  \  {  .0.  } ) ( I `  x )  e.  A
) )
Distinct variable groups:    x, A    x, R    x, S    x,  .0.
Allowed substitution hint:    I( x)

Proof of Theorem issubdrg
StepHypRef Expression
1 simpllr 736 . . . . . 6  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  A  e.  (SubRing `  R ) )
2 issubdrg.s . . . . . . 7  |-  S  =  ( Rs  A )
32subrgrng 15871 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
41, 3syl 16 . . . . 5  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  S  e.  Ring )
5 simpr 448 . . . . . . . . 9  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  x  e.  ( A  \  {  .0.  } ) )
6 eldifsn 3927 . . . . . . . . 9  |-  ( x  e.  ( A  \  {  .0.  } )  <->  ( x  e.  A  /\  x  =/=  .0.  ) )
75, 6sylib 189 . . . . . . . 8  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  ( x  e.  A  /\  x  =/=  .0.  ) )
87simpld 446 . . . . . . 7  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  x  e.  A )
92subrgbas 15877 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
101, 9syl 16 . . . . . . 7  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  A  =  ( Base `  S )
)
118, 10eleqtrd 2512 . . . . . 6  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  x  e.  ( Base `  S )
)
127simprd 450 . . . . . . 7  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  x  =/=  .0.  )
13 issubdrg.z . . . . . . . . 9  |-  .0.  =  ( 0g `  R )
142, 13subrg0 15875 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  .0.  =  ( 0g `  S ) )
151, 14syl 16 . . . . . . 7  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  .0.  =  ( 0g `  S ) )
1612, 15neeqtrd 2623 . . . . . 6  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  x  =/=  ( 0g `  S ) )
17 eqid 2436 . . . . . . . 8  |-  ( Base `  S )  =  (
Base `  S )
18 eqid 2436 . . . . . . . 8  |-  (Unit `  S )  =  (Unit `  S )
19 eqid 2436 . . . . . . . 8  |-  ( 0g
`  S )  =  ( 0g `  S
)
2017, 18, 19drngunit 15840 . . . . . . 7  |-  ( S  e.  DivRing  ->  ( x  e.  (Unit `  S )  <->  ( x  e.  ( Base `  S )  /\  x  =/=  ( 0g `  S
) ) ) )
2120ad2antlr 708 . . . . . 6  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  ( x  e.  (Unit `  S )  <->  ( x  e.  ( Base `  S )  /\  x  =/=  ( 0g `  S
) ) ) )
2211, 16, 21mpbir2and 889 . . . . 5  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  x  e.  (Unit `  S ) )
23 eqid 2436 . . . . . 6  |-  ( invr `  S )  =  (
invr `  S )
2418, 23, 17rnginvcl 15781 . . . . 5  |-  ( ( S  e.  Ring  /\  x  e.  (Unit `  S )
)  ->  ( ( invr `  S ) `  x )  e.  (
Base `  S )
)
254, 22, 24syl2anc 643 . . . 4  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  ( ( invr `  S ) `  x )  e.  (
Base `  S )
)
26 issubdrg.i . . . . . 6  |-  I  =  ( invr `  R
)
272, 26, 18, 23subrginv 15884 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  (Unit `  S ) )  ->  ( I `  x )  =  ( ( invr `  S
) `  x )
)
281, 22, 27syl2anc 643 . . . 4  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  ( I `  x )  =  ( ( invr `  S
) `  x )
)
2925, 28, 103eltr4d 2517 . . 3  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  ( I `  x )  e.  A
)
3029ralrimiva 2789 . 2  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  S  e.  DivRing )  ->  A. x  e.  ( A  \  {  .0.  } ) ( I `
 x )  e.  A )
313ad2antlr 708 . . 3  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  S  e.  Ring )
32 eqid 2436 . . . . . . . . . 10  |-  (Unit `  R )  =  (Unit `  R )
332, 32, 18subrguss 15883 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  (Unit `  S
)  C_  (Unit `  R
) )
3433ad2antlr 708 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  (Unit `  S
)  C_  (Unit `  R
) )
35 eqid 2436 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
3635, 32, 13isdrng 15839 . . . . . . . . . 10  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  (Unit `  R )  =  ( ( Base `  R )  \  {  .0.  } ) ) )
3736simprbi 451 . . . . . . . . 9  |-  ( R  e.  DivRing  ->  (Unit `  R
)  =  ( (
Base `  R )  \  {  .0.  } ) )
3837ad2antrr 707 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  (Unit `  R
)  =  ( (
Base `  R )  \  {  .0.  } ) )
3934, 38sseqtrd 3384 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  (Unit `  S
)  C_  ( ( Base `  R )  \  {  .0.  } ) )
4017, 18unitss 15765 . . . . . . . 8  |-  (Unit `  S )  C_  ( Base `  S )
419ad2antlr 708 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  A  =  (
Base `  S )
)
4240, 41syl5sseqr 3397 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  (Unit `  S
)  C_  A )
4339, 42ssind 3565 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  (Unit `  S
)  C_  ( (
( Base `  R )  \  {  .0.  } )  i^i  A ) )
4435subrgss 15869 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
4544ad2antlr 708 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  A  C_  ( Base `  R ) )
46 difin2 3603 . . . . . . 7  |-  ( A 
C_  ( Base `  R
)  ->  ( A  \  {  .0.  } )  =  ( ( (
Base `  R )  \  {  .0.  } )  i^i  A ) )
4745, 46syl 16 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  ( A  \  {  .0.  } )  =  ( ( ( Base `  R )  \  {  .0.  } )  i^i  A
) )
4843, 47sseqtr4d 3385 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  (Unit `  S
)  C_  ( A  \  {  .0.  } ) )
4944ad2antlr 708 . . . . . . . . . . . 12  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  A  C_  ( Base `  R
) )
50 simprl 733 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  x  e.  ( A  \  {  .0.  } ) )
5150, 6sylib 189 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  (
x  e.  A  /\  x  =/=  .0.  ) )
5251simpld 446 . . . . . . . . . . . 12  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  x  e.  A )
5349, 52sseldd 3349 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  x  e.  ( Base `  R
) )
5451simprd 450 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  x  =/=  .0.  )
5535, 32, 13drngunit 15840 . . . . . . . . . . . 12  |-  ( R  e.  DivRing  ->  ( x  e.  (Unit `  R )  <->  ( x  e.  ( Base `  R )  /\  x  =/=  .0.  ) ) )
5655ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  (
x  e.  (Unit `  R )  <->  ( x  e.  ( Base `  R
)  /\  x  =/=  .0.  ) ) )
5753, 54, 56mpbir2and 889 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  x  e.  (Unit `  R )
)
58 simprr 734 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  (
I `  x )  e.  A )
592, 32, 18, 26subrgunit 15886 . . . . . . . . . . 11  |-  ( A  e.  (SubRing `  R
)  ->  ( x  e.  (Unit `  S )  <->  ( x  e.  (Unit `  R )  /\  x  e.  A  /\  (
I `  x )  e.  A ) ) )
6059ad2antlr 708 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  (
x  e.  (Unit `  S )  <->  ( x  e.  (Unit `  R )  /\  x  e.  A  /\  ( I `  x
)  e.  A ) ) )
6157, 52, 58, 60mpbir3and 1137 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  x  e.  (Unit `  S )
)
6261expr 599 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  x  e.  ( A  \  {  .0.  } ) )  -> 
( ( I `  x )  e.  A  ->  x  e.  (Unit `  S ) ) )
6362ralimdva 2784 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R )
)  ->  ( A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A  ->  A. x  e.  ( A  \  {  .0.  }
) x  e.  (Unit `  S ) ) )
6463imp 419 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  A. x  e.  ( A  \  {  .0.  } ) x  e.  (Unit `  S ) )
65 dfss3 3338 . . . . . 6  |-  ( ( A  \  {  .0.  } )  C_  (Unit `  S
)  <->  A. x  e.  ( A  \  {  .0.  } ) x  e.  (Unit `  S ) )
6664, 65sylibr 204 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  ( A  \  {  .0.  } )  C_  (Unit `  S ) )
6748, 66eqssd 3365 . . . 4  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  (Unit `  S
)  =  ( A 
\  {  .0.  }
) )
6814ad2antlr 708 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  .0.  =  ( 0g `  S ) )
6968sneqd 3827 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  {  .0.  }  =  { ( 0g `  S ) } )
7041, 69difeq12d 3466 . . . 4  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  ( A  \  {  .0.  } )  =  ( ( Base `  S
)  \  { ( 0g `  S ) } ) )
7167, 70eqtrd 2468 . . 3  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  (Unit `  S
)  =  ( (
Base `  S )  \  { ( 0g `  S ) } ) )
7217, 18, 19isdrng 15839 . . 3  |-  ( S  e.  DivRing 
<->  ( S  e.  Ring  /\  (Unit `  S )  =  ( ( Base `  S )  \  {
( 0g `  S
) } ) ) )
7331, 71, 72sylanbrc 646 . 2  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  S  e.  DivRing )
7430, 73impbida 806 1  |-  ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R )
)  ->  ( S  e.  DivRing 
<-> 
A. x  e.  ( A  \  {  .0.  } ) ( I `  x )  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705    \ cdif 3317    i^i cin 3319    C_ wss 3320   {csn 3814   ` cfv 5454  (class class class)co 6081   Basecbs 13469   ↾s cress 13470   0gc0g 13723   Ringcrg 15660  Unitcui 15744   invrcinvr 15776   DivRingcdr 15835  SubRingcsubrg 15864
This theorem is referenced by:  cnsubdrglem  16749  issdrg2  27483
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-subg 14941  df-mgp 15649  df-rng 15663  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-drng 15837  df-subrg 15866
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