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Theorem issubdrg 17302
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 )
32subrgring 17280 . . . . . 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 4157 . . . . . . . . 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 17286 . . . . . . . 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 2557 . . . . . 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 17284 . . . . . . . 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 2762 . . . . . 6  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  x  =/=  ( 0g `  S ) )
17 eqid 2467 . . . . . . . 8  |-  ( Base `  S )  =  (
Base `  S )
18 eqid 2467 . . . . . . . 8  |-  (Unit `  S )  =  (Unit `  S )
19 eqid 2467 . . . . . . . 8  |-  ( 0g
`  S )  =  ( 0g `  S
)
2017, 18, 19drngunit 17249 . . . . . . 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 920 . . . . 5  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  x  e.  (Unit `  S ) )
23 eqid 2467 . . . . . 6  |-  ( invr `  S )  =  (
invr `  S )
2418, 23, 17ringinvcl 17174 . . . . 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 17293 . . . . 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 2570 . . 3  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  ( I `  x )  e.  A
)
3029ralrimiva 2881 . 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 2467 . . . . . . . . . 10  |-  (Unit `  R )  =  (Unit `  R )
332, 32, 18subrguss 17292 . . . . . . . . 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 2467 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
3635, 32, 13isdrng 17248 . . . . . . . . . 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 3545 . . . . . . 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 17158 . . . . . . . 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 3558 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  (Unit `  S
)  C_  A )
4339, 42ssind 3727 . . . . . 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 17278 . . . . . . . 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 3765 . . . . . . 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 3546 . . . . 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 3510 . . . . . . . . . . 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 17249 . . . . . . . . . . . 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 920 . . . . . . . . . 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 17295 . . . . . . . . . . 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 1179 . . . . . . . . 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 2875 . . . . . . 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 3499 . . . . . 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 3526 . . . 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 4044 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  {  .0.  }  =  { ( 0g `  S ) } )
7041, 69difeq12d 3628 . . . 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 2508 . . 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 17248 . . 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 830 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817    \ cdif 3478    i^i cin 3480    C_ wss 3481   {csn 4032   ` cfv 5593  (class class class)co 6294   Basecbs 14502   ↾s cress 14503   0gc0g 14707   Ringcrg 17047  Unitcui 17137   invrcinvr 17169   DivRingcdr 17244  SubRingcsubrg 17273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-tpos 6965  df-recs 7052  df-rdg 7086  df-er 7321  df-en 7527  df-dom 7528  df-sdom 7529  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-2 10604  df-3 10605  df-ndx 14505  df-slot 14506  df-base 14507  df-sets 14508  df-ress 14509  df-plusg 14580  df-mulr 14581  df-0g 14709  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-grp 15906  df-minusg 15907  df-subg 16047  df-mgp 16991  df-ur 17003  df-ring 17049  df-oppr 17121  df-dvdsr 17139  df-unit 17140  df-invr 17170  df-drng 17246  df-subrg 17275
This theorem is referenced by:  cnsubdrglem  18316  issdrg2  31044
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