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Theorem drngnidl 15981
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 15519 . . . . . . . . . . 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 15980 . . . . . . . . . 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 15962 . . . . . . . . . . . . . . . 16  |-  ( ( R  e.  DivRing  /\  a  e.  U )  ->  a  C_  B )
1413sselda 3180 . . . . . . . . . . . . . . 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 2283 . . . . . . . . . . . . . . 15  |-  ( .r
`  R )  =  ( .r `  R
)
18 eqid 2283 . . . . . . . . . . . . . . 15  |-  ( 1r
`  R )  =  ( 1r `  R
)
19 eqid 2283 . . . . . . . . . . . . . . 15  |-  ( invr `  R )  =  (
invr `  R )
2012, 8, 17, 18, 19drnginvrl 15531 . . . . . . . . . . . . . 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 15529 . . . . . . . . . . . . . . 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 15969 . . . . . . . . . . . . . 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 2358 . . . . . . . . . . . 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 2666 . . . . . . . . . 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 15970 . . . . . . . . . 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 2524 . . . . 5  |-  ( ( R  e.  DivRing  /\  a  e.  U )  ->  (
a  =  {  .0.  }  \/  a  =  B ) )
40 vex 2791 . . . . . 6  |-  a  e. 
_V
4140elpr 3658 . . . . 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 3185 . 2  |-  ( R  e.  DivRing  ->  U  C_  { {  .0.  } ,  B }
)
457, 8lidl0 15971 . . . 4  |-  ( R  e.  Ring  ->  {  .0.  }  e.  U )
467, 12lidl1 15972 . . . 4  |-  ( R  e.  Ring  ->  B  e.  U )
47 snex 4216 . . . . . 6  |-  {  .0.  }  e.  _V
48 fvex 5539 . . . . . . 7  |-  ( Base `  R )  e.  _V
4912, 48eqeltri 2353 . . . . . 6  |-  B  e. 
_V
5047, 49prss 3769 . . . . 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 3196 1  |-  ( R  e.  DivRing  ->  U  =  { {  .0.  } ,  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   _Vcvv 2788    C_ wss 3152   {csn 3640   {cpr 3641   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209   0gc0g 13400   Ringcrg 15337   1rcur 15339   invrcinvr 15453   DivRingcdr 15512  LIdealclidl 15923
This theorem is referenced by:  drnglpir  16005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-subrg 15543  df-lmod 15629  df-lss 15690  df-sra 15925  df-rgmod 15926  df-lidl 15927
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