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

Theorem issubdrg 18111
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 777 . . . . . 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 18089 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
41, 3syl 17 . . . . 5  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  S  e.  Ring )
5 simpr 468 . . . . . . . . 9  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  x  e.  ( A  \  {  .0.  } ) )
6 eldifsn 4088 . . . . . . . . 9  |-  ( x  e.  ( A  \  {  .0.  } )  <->  ( x  e.  A  /\  x  =/=  .0.  ) )
75, 6sylib 201 . . . . . . . 8  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  ( x  e.  A  /\  x  =/=  .0.  ) )
87simpld 466 . . . . . . 7  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  x  e.  A )
92subrgbas 18095 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
101, 9syl 17 . . . . . . 7  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  A  =  ( Base `  S )
)
118, 10eleqtrd 2551 . . . . . 6  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  x  e.  ( Base `  S )
)
127simprd 470 . . . . . . 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 18093 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  .0.  =  ( 0g `  S ) )
151, 14syl 17 . . . . . . 7  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  .0.  =  ( 0g `  S ) )
1612, 15neeqtrd 2712 . . . . . 6  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  x  =/=  ( 0g `  S ) )
17 eqid 2471 . . . . . . . 8  |-  ( Base `  S )  =  (
Base `  S )
18 eqid 2471 . . . . . . . 8  |-  (Unit `  S )  =  (Unit `  S )
19 eqid 2471 . . . . . . . 8  |-  ( 0g
`  S )  =  ( 0g `  S
)
2017, 18, 19drngunit 18058 . . . . . . 7  |-  ( S  e.  DivRing  ->  ( x  e.  (Unit `  S )  <->  ( x  e.  ( Base `  S )  /\  x  =/=  ( 0g `  S
) ) ) )
2120ad2antlr 741 . . . . . 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 936 . . . . 5  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  x  e.  (Unit `  S ) )
23 eqid 2471 . . . . . 6  |-  ( invr `  S )  =  (
invr `  S )
2418, 23, 17ringinvcl 17982 . . . . 5  |-  ( ( S  e.  Ring  /\  x  e.  (Unit `  S )
)  ->  ( ( invr `  S ) `  x )  e.  (
Base `  S )
)
254, 22, 24syl2anc 673 . . . 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 18102 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  (Unit `  S ) )  ->  ( I `  x )  =  ( ( invr `  S
) `  x )
)
281, 22, 27syl2anc 673 . . . 4  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  ( I `  x )  =  ( ( invr `  S
) `  x )
)
2925, 28, 103eltr4d 2564 . . 3  |-  ( ( ( ( R  e.  DivRing 
/\  A  e.  (SubRing `  R ) )  /\  S  e.  DivRing )  /\  x  e.  ( A  \  {  .0.  } ) )  ->  ( I `  x )  e.  A
)
3029ralrimiva 2809 . 2  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  S  e.  DivRing )  ->  A. x  e.  ( A  \  {  .0.  } ) ( I `
 x )  e.  A )
313ad2antlr 741 . . 3  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  S  e.  Ring )
32 eqid 2471 . . . . . . . . . 10  |-  (Unit `  R )  =  (Unit `  R )
332, 32, 18subrguss 18101 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  (Unit `  S
)  C_  (Unit `  R
) )
3433ad2antlr 741 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  (Unit `  S
)  C_  (Unit `  R
) )
35 eqid 2471 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
3635, 32, 13isdrng 18057 . . . . . . . . . 10  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  (Unit `  R )  =  ( ( Base `  R )  \  {  .0.  } ) ) )
3736simprbi 471 . . . . . . . . 9  |-  ( R  e.  DivRing  ->  (Unit `  R
)  =  ( (
Base `  R )  \  {  .0.  } ) )
3837ad2antrr 740 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  (Unit `  R
)  =  ( (
Base `  R )  \  {  .0.  } ) )
3934, 38sseqtrd 3454 . . . . . . 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 17966 . . . . . . . 8  |-  (Unit `  S )  C_  ( Base `  S )
419ad2antlr 741 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  A  =  (
Base `  S )
)
4240, 41syl5sseqr 3467 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  (Unit `  S
)  C_  A )
4339, 42ssind 3647 . . . . . 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 18087 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
4544ad2antlr 741 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  A  C_  ( Base `  R ) )
46 difin2 3696 . . . . . . 7  |-  ( A 
C_  ( Base `  R
)  ->  ( A  \  {  .0.  } )  =  ( ( (
Base `  R )  \  {  .0.  } )  i^i  A ) )
4745, 46syl 17 . . . . . 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 3455 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  (Unit `  S
)  C_  ( A  \  {  .0.  } ) )
4944ad2antlr 741 . . . . . . . . . . . 12  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  A  C_  ( Base `  R
) )
50 simprl 772 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  x  e.  ( A  \  {  .0.  } ) )
5150, 6sylib 201 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  (
x  e.  A  /\  x  =/=  .0.  ) )
5251simpld 466 . . . . . . . . . . . 12  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  x  e.  A )
5349, 52sseldd 3419 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  x  e.  ( Base `  R
) )
5451simprd 470 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  x  =/=  .0.  )
5535, 32, 13drngunit 18058 . . . . . . . . . . . 12  |-  ( R  e.  DivRing  ->  ( x  e.  (Unit `  R )  <->  ( x  e.  ( Base `  R )  /\  x  =/=  .0.  ) ) )
5655ad2antrr 740 . . . . . . . . . . 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 936 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  x  e.  (Unit `  R )
)
58 simprr 774 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  (
I `  x )  e.  A )
592, 32, 18, 26subrgunit 18104 . . . . . . . . . . 11  |-  ( A  e.  (SubRing `  R
)  ->  ( x  e.  (Unit `  S )  <->  ( x  e.  (Unit `  R )  /\  x  e.  A  /\  (
I `  x )  e.  A ) ) )
6059ad2antlr 741 . . . . . . . . . 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 1213 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  (
x  e.  ( A 
\  {  .0.  }
)  /\  ( I `  x )  e.  A
) )  ->  x  e.  (Unit `  S )
)
6261expr 626 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  x  e.  ( A  \  {  .0.  } ) )  -> 
( ( I `  x )  e.  A  ->  x  e.  (Unit `  S ) ) )
6362ralimdva 2805 . . . . . . 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 436 . . . . . 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 3408 . . . . . 6  |-  ( ( A  \  {  .0.  } )  C_  (Unit `  S
)  <->  A. x  e.  ( A  \  {  .0.  } ) x  e.  (Unit `  S ) )
6664, 65sylibr 217 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  ( A  \  {  .0.  } )  C_  (Unit `  S ) )
6748, 66eqssd 3435 . . . 4  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  (Unit `  S
)  =  ( A 
\  {  .0.  }
) )
6814ad2antlr 741 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  .0.  =  ( 0g `  S ) )
6968sneqd 3971 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  {  .0.  }  =  { ( 0g `  S ) } )
7041, 69difeq12d 3541 . . . 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 2505 . . 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 18057 . . 3  |-  ( S  e.  DivRing 
<->  ( S  e.  Ring  /\  (Unit `  S )  =  ( ( Base `  S )  \  {
( 0g `  S
) } ) ) )
7331, 71, 72sylanbrc 677 . 2  |-  ( ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R
) )  /\  A. x  e.  ( A  \  {  .0.  } ) ( I `  x
)  e.  A )  ->  S  e.  DivRing )
7430, 73impbida 850 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756    \ cdif 3387    i^i cin 3389    C_ wss 3390   {csn 3959   ` cfv 5589  (class class class)co 6308   Basecbs 15199   ↾s cress 15200   0gc0g 15416   Ringcrg 17858  Unitcui 17945   invrcinvr 17977   DivRingcdr 18053  SubRingcsubrg 18082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-0g 15418  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-grp 16751  df-minusg 16752  df-subg 16892  df-mgp 17802  df-ur 17814  df-ring 17860  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-drng 18055  df-subrg 18084
This theorem is referenced by:  cnsubdrglem  19096  issdrg2  36135
  Copyright terms: Public domain W3C validator