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Theorem drngnidl 15997
Description: A division ring has only the two trivial ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
Hypotheses
Ref Expression
drngnidl.b  |-  B  =  ( Base `  R
)
drngnidl.z  |-  .0.  =  ( 0g `  R )
drngnidl.u  |-  U  =  (LIdeal `  R )
Assertion
Ref Expression
drngnidl  |-  ( R  e.  DivRing  ->  U  =  { {  .0.  } ,  B } )

Proof of Theorem drngnidl
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 447 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  a  =  {  .0.  } )  ->  a  =  {  .0.  } )
21orcd 381 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  a  =  {  .0.  } )  ->  (
a  =  {  .0.  }  \/  a  =  B ) )
3 drngrng 15535 . . . . . . . . . . 11  |-  ( R  e.  DivRing  ->  R  e.  Ring )
43ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  a  =/=  {  .0.  } )  ->  R  e.  Ring )
5 simplr 731 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  a  =/=  {  .0.  } )  ->  a  e.  U )
6 simpr 447 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  a  =/=  {  .0.  } )  ->  a  =/=  {  .0.  } )
7 drngnidl.u . . . . . . . . . . 11  |-  U  =  (LIdeal `  R )
8 drngnidl.z . . . . . . . . . . 11  |-  .0.  =  ( 0g `  R )
97, 8lidlnz 15996 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  a  e.  U  /\  a  =/=  {  .0.  } )  ->  E. b  e.  a  b  =/=  .0.  )
104, 5, 6, 9syl3anc 1182 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  a  =/=  {  .0.  } )  ->  E. b  e.  a  b  =/=  .0.  )
11 simpll 730 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  R  e.  DivRing )
12 drngnidl.b . . . . . . . . . . . . . . . . 17  |-  B  =  ( Base `  R
)
1312, 7lidlssOLD 15978 . . . . . . . . . . . . . . . 16  |-  ( ( R  e.  DivRing  /\  a  e.  U )  ->  a  C_  B )
1413sselda 3193 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  b  e.  a
)  ->  b  e.  B )
1514adantrr 697 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  b  e.  B )
16 simprr 733 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  b  =/=  .0.  )
17 eqid 2296 . . . . . . . . . . . . . . 15  |-  ( .r
`  R )  =  ( .r `  R
)
18 eqid 2296 . . . . . . . . . . . . . . 15  |-  ( 1r
`  R )  =  ( 1r `  R
)
19 eqid 2296 . . . . . . . . . . . . . . 15  |-  ( invr `  R )  =  (
invr `  R )
2012, 8, 17, 18, 19drnginvrl 15547 . . . . . . . . . . . . . 14  |-  ( ( R  e.  DivRing  /\  b  e.  B  /\  b  =/=  .0.  )  ->  (
( ( invr `  R
) `  b )
( .r `  R
) b )  =  ( 1r `  R
) )
2111, 15, 16, 20syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  (
( ( invr `  R
) `  b )
( .r `  R
) b )  =  ( 1r `  R
) )
223ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  R  e.  Ring )
23 simplr 731 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  a  e.  U )
2412, 8, 19drnginvrcl 15545 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  DivRing  /\  b  e.  B  /\  b  =/=  .0.  )  ->  (
( invr `  R ) `  b )  e.  B
)
2511, 15, 16, 24syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  (
( invr `  R ) `  b )  e.  B
)
26 simprl 732 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  b  e.  a )
277, 12, 17lidlmcl 15985 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  Ring  /\  a  e.  U )  /\  ( ( (
invr `  R ) `  b )  e.  B  /\  b  e.  a
) )  ->  (
( ( invr `  R
) `  b )
( .r `  R
) b )  e.  a )
2822, 23, 25, 26, 27syl22anc 1183 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  (
( ( invr `  R
) `  b )
( .r `  R
) b )  e.  a )
2921, 28eqeltrrd 2371 . . . . . . . . . . . 12  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  ( 1r `  R )  e.  a )
3029exp32 588 . . . . . . . . . . 11  |-  ( ( R  e.  DivRing  /\  a  e.  U )  ->  (
b  e.  a  -> 
( b  =/=  .0.  ->  ( 1r `  R
)  e.  a ) ) )
3130rexlimdv 2679 . . . . . . . . . 10  |-  ( ( R  e.  DivRing  /\  a  e.  U )  ->  ( E. b  e.  a 
b  =/=  .0.  ->  ( 1r `  R )  e.  a ) )
3231imp 418 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  E. b  e.  a  b  =/=  .0.  )  ->  ( 1r `  R
)  e.  a )
3310, 32syldan 456 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  a  =/=  {  .0.  } )  ->  ( 1r `  R )  e.  a )
347, 12, 18lidl1el 15986 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  a  e.  U )  ->  (
( 1r `  R
)  e.  a  <->  a  =  B ) )
353, 34sylan 457 . . . . . . . . 9  |-  ( ( R  e.  DivRing  /\  a  e.  U )  ->  (
( 1r `  R
)  e.  a  <->  a  =  B ) )
3635adantr 451 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  a  =/=  {  .0.  } )  ->  ( ( 1r `  R )  e.  a  <->  a  =  B ) )
3733, 36mpbid 201 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  a  =/=  {  .0.  } )  ->  a  =  B )
3837olcd 382 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  a  =/=  {  .0.  } )  ->  ( a  =  {  .0.  }  \/  a  =  B )
)
392, 38pm2.61dane 2537 . . . . 5  |-  ( ( R  e.  DivRing  /\  a  e.  U )  ->  (
a  =  {  .0.  }  \/  a  =  B ) )
40 vex 2804 . . . . . 6  |-  a  e. 
_V
4140elpr 3671 . . . . 5  |-  ( a  e.  { {  .0.  } ,  B }  <->  ( a  =  {  .0.  }  \/  a  =  B )
)
4239, 41sylibr 203 . . . 4  |-  ( ( R  e.  DivRing  /\  a  e.  U )  ->  a  e.  { {  .0.  } ,  B } )
4342ex 423 . . 3  |-  ( R  e.  DivRing  ->  ( a  e.  U  ->  a  e.  { {  .0.  } ,  B } ) )
4443ssrdv 3198 . 2  |-  ( R  e.  DivRing  ->  U  C_  { {  .0.  } ,  B }
)
457, 8lidl0 15987 . . . 4  |-  ( R  e.  Ring  ->  {  .0.  }  e.  U )
467, 12lidl1 15988 . . . 4  |-  ( R  e.  Ring  ->  B  e.  U )
47 snex 4232 . . . . . 6  |-  {  .0.  }  e.  _V
48 fvex 5555 . . . . . . 7  |-  ( Base `  R )  e.  _V
4912, 48eqeltri 2366 . . . . . 6  |-  B  e. 
_V
5047, 49prss 3785 . . . . 5  |-  ( ( {  .0.  }  e.  U  /\  B  e.  U
)  <->  { {  .0.  } ,  B }  C_  U
)
5150bicomi 193 . . . 4  |-  ( { {  .0.  } ,  B }  C_  U  <->  ( {  .0.  }  e.  U  /\  B  e.  U )
)
5245, 46, 51sylanbrc 645 . . 3  |-  ( R  e.  Ring  ->  { {  .0.  } ,  B }  C_  U )
533, 52syl 15 . 2  |-  ( R  e.  DivRing  ->  { {  .0.  } ,  B }  C_  U )
5444, 53eqssd 3209 1  |-  ( R  e.  DivRing  ->  U  =  { {  .0.  } ,  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   _Vcvv 2801    C_ wss 3165   {csn 3653   {cpr 3654   ` cfv 5271  (class class class)co 5874   Basecbs 13164   .rcmulr 13225   0gc0g 13416   Ringcrg 15353   1rcur 15355   invrcinvr 15469   DivRingcdr 15528  LIdealclidl 15939
This theorem is referenced by:  drnglpir  16021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-drng 15530  df-subrg 15559  df-lmod 15645  df-lss 15706  df-sra 15941  df-rgmod 15942  df-lidl 15943
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