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Theorem issubdrg 16890
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 758 . . . . . 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 16868 . . . . . 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 4000 . . . . . . . . 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 16874 . . . . . . . 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 2519 . . . . . 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 16872 . . . . . . . 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 2630 . . . . . 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 16837 . . . . . . 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 913 . . . . 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, 17rnginvcl 16768 . . . . 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 16881 . . . . 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 2524 . . 3  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  ( I `  x )  e.  A
)
3029ralrimiva 2799 . 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 16880 . . . . . . . . 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 16836 . . . . . . . . . 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 3392 . . . . . . 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 16752 . . . . . . . 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 3405 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  (Unit `  S
)  C_  A )
4339, 42ssind 3574 . . . . . 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 16866 . . . . . . . 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 3612 . . . . . . 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 3393 . . . . 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 755 . . . . . . . . . . . . . 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 3357 . . . . . . . . . . 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 16837 . . . . . . . . . . . 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 913 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  x  e.  (Unit `  R )
)
58 simprr 756 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  (
I `  x )  e.  A )
592, 32, 18, 26subrgunit 16883 . . . . . . . . . . 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 1171 . . . . . . . . 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 2794 . . . . . . 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 3346 . . . . . 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 3373 . . . 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 3889 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  {  .0.  }  =  { ( 0g `  S ) } )
7041, 69difeq12d 3475 . . . 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 2475 . . 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 16836 . . 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 828 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 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715    \ cdif 3325    i^i cin 3327    C_ wss 3328   {csn 3877   ` cfv 5418  (class class class)co 6091   Basecbs 14174   ↾s cress 14175   0gc0g 14378   Ringcrg 16645  Unitcui 16731   invrcinvr 16763   DivRingcdr 16832  SubRingcsubrg 16861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-tpos 6745  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-0g 14380  df-mnd 15415  df-grp 15545  df-minusg 15546  df-subg 15678  df-mgp 16592  df-ur 16604  df-rng 16647  df-oppr 16715  df-dvdsr 16733  df-unit 16734  df-invr 16764  df-drng 16834  df-subrg 16863
This theorem is referenced by:  cnsubdrglem  17864  issdrg2  29555
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