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Theorem issubdrg 17328
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 760 . . . . . 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 )
32subrgring 17306 . . . . . 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 461 . . . . . . . . 9  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  x  e.  ( A  \  {  .0.  } ) )
6 eldifsn 4140 . . . . . . . . 9  |-  ( x  e.  ( A  \  {  .0.  } )  <->  ( x  e.  A  /\  x  =/=  .0.  ) )
75, 6sylib 196 . . . . . . . 8  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  ( x  e.  A  /\  x  =/=  .0.  ) )
87simpld 459 . . . . . . 7  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  x  e.  A )
92subrgbas 17312 . . . . . . . 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 2533 . . . . . 6  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  x  e.  ( Base `  S )
)
127simprd 463 . . . . . . 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 17310 . . . . . . . 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 2738 . . . . . 6  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  x  =/=  ( 0g `  S ) )
17 eqid 2443 . . . . . . . 8  |-  ( Base `  S )  =  (
Base `  S )
18 eqid 2443 . . . . . . . 8  |-  (Unit `  S )  =  (Unit `  S )
19 eqid 2443 . . . . . . . 8  |-  ( 0g
`  S )  =  ( 0g `  S
)
2017, 18, 19drngunit 17275 . . . . . . 7  |-  ( S  e.  DivRing  ->  ( x  e.  (Unit `  S )  <->  ( x  e.  ( Base `  S )  /\  x  =/=  ( 0g `  S
) ) ) )
2120ad2antlr 726 . . . . . 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 922 . . . . 5  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  x  e.  (Unit `  S ) )
23 eqid 2443 . . . . . 6  |-  ( invr `  S )  =  (
invr `  S )
2418, 23, 17ringinvcl 17199 . . . . 5  |-  ( ( S  e.  Ring  /\  x  e.  (Unit `  S )
)  ->  ( ( invr `  S ) `  x )  e.  (
Base `  S )
)
254, 22, 24syl2anc 661 . . . 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 17319 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  (Unit `  S ) )  ->  ( I `  x )  =  ( ( invr `  S
) `  x )
)
281, 22, 27syl2anc 661 . . . 4  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  ( I `  x )  =  ( ( invr `  S
) `  x )
)
2925, 28, 103eltr4d 2546 . . 3  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  ( I `  x )  e.  A
)
3029ralrimiva 2857 . 2  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  S  e.  DivRing )  ->  A. x  e.  ( A  \  {  .0.  } ) ( I `
 x )  e.  A )
313ad2antlr 726 . . 3  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  S  e.  Ring )
32 eqid 2443 . . . . . . . . . 10  |-  (Unit `  R )  =  (Unit `  R )
332, 32, 18subrguss 17318 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  (Unit `  S
)  C_  (Unit `  R
) )
3433ad2antlr 726 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  (Unit `  S
)  C_  (Unit `  R
) )
35 eqid 2443 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
3635, 32, 13isdrng 17274 . . . . . . . . . 10  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  (Unit `  R )  =  ( ( Base `  R )  \  {  .0.  } ) ) )
3736simprbi 464 . . . . . . . . 9  |-  ( R  e.  DivRing  ->  (Unit `  R
)  =  ( (
Base `  R )  \  {  .0.  } ) )
3837ad2antrr 725 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  (Unit `  R
)  =  ( (
Base `  R )  \  {  .0.  } ) )
3934, 38sseqtrd 3525 . . . . . . 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 17183 . . . . . . . 8  |-  (Unit `  S )  C_  ( Base `  S )
419ad2antlr 726 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  A  =  (
Base `  S )
)
4240, 41syl5sseqr 3538 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  (Unit `  S
)  C_  A )
4339, 42ssind 3707 . . . . . 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 17304 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
4544ad2antlr 726 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  A  C_  ( Base `  R ) )
46 difin2 3745 . . . . . . 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 3526 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  (Unit `  S
)  C_  ( A  \  {  .0.  } ) )
4944ad2antlr 726 . . . . . . . . . . . 12  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  A  C_  ( Base `  R
) )
50 simprl 756 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  x  e.  ( A  \  {  .0.  } ) )
5150, 6sylib 196 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  (
x  e.  A  /\  x  =/=  .0.  ) )
5251simpld 459 . . . . . . . . . . . 12  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  x  e.  A )
5349, 52sseldd 3490 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  x  e.  ( Base `  R
) )
5451simprd 463 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  x  =/=  .0.  )
5535, 32, 13drngunit 17275 . . . . . . . . . . . 12  |-  ( R  e.  DivRing  ->  ( x  e.  (Unit `  R )  <->  ( x  e.  ( Base `  R )  /\  x  =/=  .0.  ) ) )
5655ad2antrr 725 . . . . . . . . . . 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 922 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  x  e.  (Unit `  R )
)
58 simprr 757 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  (
I `  x )  e.  A )
592, 32, 18, 26subrgunit 17321 . . . . . . . . . . 11  |-  ( A  e.  (SubRing `  R
)  ->  ( x  e.  (Unit `  S )  <->  ( x  e.  (Unit `  R )  /\  x  e.  A  /\  (
I `  x )  e.  A ) ) )
6059ad2antlr 726 . . . . . . . . . 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 1180 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  x  e.  (Unit `  S )
)
6261expr 615 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  x  e.  ( A  \  {  .0.  } ) )  -> 
( ( I `  x )  e.  A  ->  x  e.  (Unit `  S ) ) )
6362ralimdva 2851 . . . . . . 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 429 . . . . . 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 3479 . . . . . 6  |-  ( ( A  \  {  .0.  } )  C_  (Unit `  S
)  <->  A. x  e.  ( A  \  {  .0.  } ) x  e.  (Unit `  S ) )
6664, 65sylibr 212 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  ( A  \  {  .0.  } )  C_  (Unit `  S ) )
6748, 66eqssd 3506 . . . 4  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  (Unit `  S
)  =  ( A 
\  {  .0.  }
) )
6814ad2antlr 726 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  .0.  =  ( 0g `  S ) )
6968sneqd 4026 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  {  .0.  }  =  { ( 0g `  S ) } )
7041, 69difeq12d 3608 . . . 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 2484 . . 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 17274 . . 3  |-  ( S  e.  DivRing 
<->  ( S  e.  Ring  /\  (Unit `  S )  =  ( ( Base `  S )  \  {
( 0g `  S
) } ) ) )
7331, 71, 72sylanbrc 664 . 2  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  S  e.  DivRing )
7430, 73impbida 832 1  |-  ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R )
)  ->  ( S  e.  DivRing 
<-> 
A. x  e.  ( A  \  {  .0.  } ) ( I `  x )  e.  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793    \ cdif 3458    i^i cin 3460    C_ wss 3461   {csn 4014   ` cfv 5578  (class class class)co 6281   Basecbs 14509   ↾s cress 14510   0gc0g 14714   Ringcrg 17072  Unitcui 17162   invrcinvr 17194   DivRingcdr 17270  SubRingcsubrg 17299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-tpos 6957  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-minusg 15932  df-subg 16072  df-mgp 17016  df-ur 17028  df-ring 17074  df-oppr 17146  df-dvdsr 17164  df-unit 17165  df-invr 17195  df-drng 17272  df-subrg 17301
This theorem is referenced by:  cnsubdrglem  18343  issdrg2  31123
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