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Theorem drngnidl 17333
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 461 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  a  =  {  .0.  } )  ->  a  =  {  .0.  } )
21orcd 392 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  a  =  {  .0.  } )  ->  (
a  =  {  .0.  }  \/  a  =  B ) )
3 drngrng 16861 . . . . . . . . . . 11  |-  ( R  e.  DivRing  ->  R  e.  Ring )
43ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  a  =/=  {  .0.  } )  ->  R  e.  Ring )
5 simplr 754 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  a  =/=  {  .0.  } )  ->  a  e.  U )
6 simpr 461 . . . . . . . . . 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 17332 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  a  e.  U  /\  a  =/=  {  .0.  } )  ->  E. b  e.  a  b  =/=  .0.  )
104, 5, 6, 9syl3anc 1218 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  a  =/=  {  .0.  } )  ->  E. b  e.  a  b  =/=  .0.  )
11 simpll 753 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  R  e.  DivRing )
12 drngnidl.b . . . . . . . . . . . . . . . 16  |-  B  =  ( Base `  R
)
1312, 7lidlssOLD 17314 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  DivRing  /\  a  e.  U )  ->  a  C_  B )
1413sselda 3377 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  b  e.  a
)  ->  b  e.  B )
1514adantrr 716 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  b  e.  B )
16 simprr 756 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  b  =/=  .0.  )
17 eqid 2443 . . . . . . . . . . . . . 14  |-  ( .r
`  R )  =  ( .r `  R
)
18 eqid 2443 . . . . . . . . . . . . . 14  |-  ( 1r
`  R )  =  ( 1r `  R
)
19 eqid 2443 . . . . . . . . . . . . . 14  |-  ( invr `  R )  =  (
invr `  R )
2012, 8, 17, 18, 19drnginvrl 16873 . . . . . . . . . . . . 13  |-  ( ( R  e.  DivRing  /\  b  e.  B  /\  b  =/=  .0.  )  ->  (
( ( invr `  R
) `  b )
( .r `  R
) b )  =  ( 1r `  R
) )
2111, 15, 16, 20syl3anc 1218 . . . . . . . . . . . 12  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  (
( ( invr `  R
) `  b )
( .r `  R
) b )  =  ( 1r `  R
) )
223ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  R  e.  Ring )
23 simplr 754 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  a  e.  U )
2412, 8, 19drnginvrcl 16871 . . . . . . . . . . . . . 14  |-  ( ( R  e.  DivRing  /\  b  e.  B  /\  b  =/=  .0.  )  ->  (
( invr `  R ) `  b )  e.  B
)
2511, 15, 16, 24syl3anc 1218 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  (
( invr `  R ) `  b )  e.  B
)
26 simprl 755 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  b  e.  a )
277, 12, 17lidlmcl 17321 . . . . . . . . . . . . 13  |-  ( ( ( 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 1219 . . . . . . . . . . . 12  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  (
( ( invr `  R
) `  b )
( .r `  R
) b )  e.  a )
2921, 28eqeltrrd 2518 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  ( 1r `  R )  e.  a )
3029rexlimdvaa 2863 . . . . . . . . . 10  |-  ( ( R  e.  DivRing  /\  a  e.  U )  ->  ( E. b  e.  a 
b  =/=  .0.  ->  ( 1r `  R )  e.  a ) )
3130imp 429 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  E. b  e.  a  b  =/=  .0.  )  ->  ( 1r `  R
)  e.  a )
3210, 31syldan 470 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  a  =/=  {  .0.  } )  ->  ( 1r `  R )  e.  a )
337, 12, 18lidl1el 17322 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  a  e.  U )  ->  (
( 1r `  R
)  e.  a  <->  a  =  B ) )
343, 33sylan 471 . . . . . . . . 9  |-  ( ( R  e.  DivRing  /\  a  e.  U )  ->  (
( 1r `  R
)  e.  a  <->  a  =  B ) )
3534adantr 465 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  a  =/=  {  .0.  } )  ->  ( ( 1r `  R )  e.  a  <->  a  =  B ) )
3632, 35mpbid 210 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  a  =/=  {  .0.  } )  ->  a  =  B )
3736olcd 393 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  a  =/=  {  .0.  } )  ->  ( a  =  {  .0.  }  \/  a  =  B )
)
382, 37pm2.61dane 2713 . . . . 5  |-  ( ( R  e.  DivRing  /\  a  e.  U )  ->  (
a  =  {  .0.  }  \/  a  =  B ) )
39 vex 2996 . . . . . 6  |-  a  e. 
_V
4039elpr 3916 . . . . 5  |-  ( a  e.  { {  .0.  } ,  B }  <->  ( a  =  {  .0.  }  \/  a  =  B )
)
4138, 40sylibr 212 . . . 4  |-  ( ( R  e.  DivRing  /\  a  e.  U )  ->  a  e.  { {  .0.  } ,  B } )
4241ex 434 . . 3  |-  ( R  e.  DivRing  ->  ( a  e.  U  ->  a  e.  { {  .0.  } ,  B } ) )
4342ssrdv 3383 . 2  |-  ( R  e.  DivRing  ->  U  C_  { {  .0.  } ,  B }
)
447, 8lidl0 17323 . . . 4  |-  ( R  e.  Ring  ->  {  .0.  }  e.  U )
457, 12lidl1 17324 . . . 4  |-  ( R  e.  Ring  ->  B  e.  U )
46 snex 4554 . . . . . 6  |-  {  .0.  }  e.  _V
47 fvex 5722 . . . . . . 7  |-  ( Base `  R )  e.  _V
4812, 47eqeltri 2513 . . . . . 6  |-  B  e. 
_V
4946, 48prss 4048 . . . . 5  |-  ( ( {  .0.  }  e.  U  /\  B  e.  U
)  <->  { {  .0.  } ,  B }  C_  U
)
5049bicomi 202 . . . 4  |-  ( { {  .0.  } ,  B }  C_  U  <->  ( {  .0.  }  e.  U  /\  B  e.  U )
)
5144, 45, 50sylanbrc 664 . . 3  |-  ( R  e.  Ring  ->  { {  .0.  } ,  B }  C_  U )
523, 51syl 16 . 2  |-  ( R  e.  DivRing  ->  { {  .0.  } ,  B }  C_  U )
5343, 52eqssd 3394 1  |-  ( R  e.  DivRing  ->  U  =  { {  .0.  } ,  B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2620   E.wrex 2737   _Vcvv 2993    C_ wss 3349   {csn 3898   {cpr 3900   ` cfv 5439  (class class class)co 6112   Basecbs 14195   .rcmulr 14260   0gc0g 14399   1rcur 16625   Ringcrg 16667   invrcinvr 16785   DivRingcdr 16854  LIdealclidl 17273
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 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
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 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-tpos 6766  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-mulr 14273  df-sca 14275  df-vsca 14276  df-ip 14277  df-0g 14401  df-mnd 15436  df-grp 15566  df-minusg 15567  df-sbg 15568  df-subg 15699  df-mgp 16614  df-ur 16626  df-rng 16669  df-oppr 16737  df-dvdsr 16755  df-unit 16756  df-invr 16786  df-drng 16856  df-subrg 16885  df-lmod 16972  df-lss 17036  df-sra 17275  df-rgmod 17276  df-lidl 17277
This theorem is referenced by:  drnglpir  17357
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