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Theorem drngnidl 17688
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 17215 . . . . . . . . . . 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 17687 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  a  e.  U  /\  a  =/=  {  .0.  } )  ->  E. b  e.  a  b  =/=  .0.  )
104, 5, 6, 9syl3anc 1228 . . . . . . . . 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 17669 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  DivRing  /\  a  e.  U )  ->  a  C_  B )
1413sselda 3504 . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . 14  |-  ( .r
`  R )  =  ( .r `  R
)
18 eqid 2467 . . . . . . . . . . . . . 14  |-  ( 1r
`  R )  =  ( 1r `  R
)
19 eqid 2467 . . . . . . . . . . . . . 14  |-  ( invr `  R )  =  (
invr `  R )
2012, 8, 17, 18, 19drnginvrl 17227 . . . . . . . . . . . . 13  |-  ( ( R  e.  DivRing  /\  b  e.  B  /\  b  =/=  .0.  )  ->  (
( ( invr `  R
) `  b )
( .r `  R
) b )  =  ( 1r `  R
) )
2111, 15, 16, 20syl3anc 1228 . . . . . . . . . . . 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 17225 . . . . . . . . . . . . . 14  |-  ( ( R  e.  DivRing  /\  b  e.  B  /\  b  =/=  .0.  )  ->  (
( invr `  R ) `  b )  e.  B
)
2511, 15, 16, 24syl3anc 1228 . . . . . . . . . . . . 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 17676 . . . . . . . . . . . . 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 1229 . . . . . . . . . . . 12  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  (
( ( invr `  R
) `  b )
( .r `  R
) b )  e.  a )
2921, 28eqeltrrd 2556 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  ( 1r `  R )  e.  a )
3029rexlimdvaa 2956 . . . . . . . . . 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 17677 . . . . . . . . . 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 2785 . . . . 5  |-  ( ( R  e.  DivRing  /\  a  e.  U )  ->  (
a  =  {  .0.  }  \/  a  =  B ) )
39 vex 3116 . . . . . 6  |-  a  e. 
_V
4039elpr 4045 . . . . 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 3510 . 2  |-  ( R  e.  DivRing  ->  U  C_  { {  .0.  } ,  B }
)
447, 8lidl0 17678 . . . 4  |-  ( R  e.  Ring  ->  {  .0.  }  e.  U )
457, 12lidl1 17679 . . . 4  |-  ( R  e.  Ring  ->  B  e.  U )
46 snex 4688 . . . . . 6  |-  {  .0.  }  e.  _V
47 fvex 5876 . . . . . . 7  |-  ( Base `  R )  e.  _V
4812, 47eqeltri 2551 . . . . . 6  |-  B  e. 
_V
4946, 48prss 4181 . . . . 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 3521 1  |-  ( R  e.  DivRing  ->  U  =  { {  .0.  } ,  B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   _Vcvv 3113    C_ wss 3476   {csn 4027   {cpr 4029   ` cfv 5588  (class class class)co 6285   Basecbs 14493   .rcmulr 14559   0gc0g 14698   1rcur 16967   Ringcrg 17012   invrcinvr 17133   DivRingcdr 17208  LIdealclidl 17628
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6956  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-sca 14574  df-vsca 14575  df-ip 14576  df-0g 14700  df-mnd 15735  df-grp 15871  df-minusg 15872  df-sbg 15873  df-subg 16012  df-mgp 16956  df-ur 16968  df-rng 17014  df-oppr 17085  df-dvdsr 17103  df-unit 17104  df-invr 17134  df-drng 17210  df-subrg 17239  df-lmod 17326  df-lss 17391  df-sra 17630  df-rgmod 17631  df-lidl 17632
This theorem is referenced by:  drnglpir  17712
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