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Theorem drngnidl 17745
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 drngring 17271 . . . . . . . . . . 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 17744 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  a  e.  U  /\  a  =/=  {  .0.  } )  ->  E. b  e.  a  b  =/=  .0.  )
104, 5, 6, 9syl3anc 1227 . . . . . . . . 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 17725 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  DivRing  /\  a  e.  U )  ->  a  C_  B )
1413sselda 3486 . . . . . . . . . . . . . 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 2441 . . . . . . . . . . . . . 14  |-  ( .r
`  R )  =  ( .r `  R
)
18 eqid 2441 . . . . . . . . . . . . . 14  |-  ( 1r
`  R )  =  ( 1r `  R
)
19 eqid 2441 . . . . . . . . . . . . . 14  |-  ( invr `  R )  =  (
invr `  R )
2012, 8, 17, 18, 19drnginvrl 17283 . . . . . . . . . . . . 13  |-  ( ( R  e.  DivRing  /\  b  e.  B  /\  b  =/=  .0.  )  ->  (
( ( invr `  R
) `  b )
( .r `  R
) b )  =  ( 1r `  R
) )
2111, 15, 16, 20syl3anc 1227 . . . . . . . . . . . 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 17281 . . . . . . . . . . . . . 14  |-  ( ( R  e.  DivRing  /\  b  e.  B  /\  b  =/=  .0.  )  ->  (
( invr `  R ) `  b )  e.  B
)
2511, 15, 16, 24syl3anc 1227 . . . . . . . . . . . . 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 17733 . . . . . . . . . . . . 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 1228 . . . . . . . . . . . 12  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  (
( ( invr `  R
) `  b )
( .r `  R
) b )  e.  a )
2921, 28eqeltrrd 2530 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  a  e.  U )  /\  ( b  e.  a  /\  b  =/=  .0.  ) )  ->  ( 1r `  R )  e.  a )
3029rexlimdvaa 2934 . . . . . . . . . 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 17734 . . . . . . . . . 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 2759 . . . . 5  |-  ( ( R  e.  DivRing  /\  a  e.  U )  ->  (
a  =  {  .0.  }  \/  a  =  B ) )
39 vex 3096 . . . . . 6  |-  a  e. 
_V
4039elpr 4028 . . . . 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 3492 . 2  |-  ( R  e.  DivRing  ->  U  C_  { {  .0.  } ,  B }
)
447, 8lidl0 17735 . . . 4  |-  ( R  e.  Ring  ->  {  .0.  }  e.  U )
457, 12lidl1 17736 . . . 4  |-  ( R  e.  Ring  ->  B  e.  U )
46 snex 4674 . . . . . 6  |-  {  .0.  }  e.  _V
47 fvex 5862 . . . . . . 7  |-  ( Base `  R )  e.  _V
4812, 47eqeltri 2525 . . . . . 6  |-  B  e. 
_V
4946, 48prss 4165 . . . . 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 3503 1  |-  ( R  e.  DivRing  ->  U  =  { {  .0.  } ,  B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1381    e. wcel 1802    =/= wne 2636   E.wrex 2792   _Vcvv 3093    C_ wss 3458   {csn 4010   {cpr 4012   ` cfv 5574  (class class class)co 6277   Basecbs 14504   .rcmulr 14570   0gc0g 14709   1rcur 17021   Ringcrg 17066   invrcinvr 17188   DivRingcdr 17264  LIdealclidl 17684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-tpos 6953  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  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 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-sca 14585  df-vsca 14586  df-ip 14587  df-0g 14711  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-grp 15926  df-minusg 15927  df-sbg 15928  df-subg 16067  df-mgp 17010  df-ur 17022  df-ring 17068  df-oppr 17140  df-dvdsr 17158  df-unit 17159  df-invr 17189  df-drng 17266  df-subrg 17295  df-lmod 17382  df-lss 17447  df-sra 17686  df-rgmod 17687  df-lidl 17688
This theorem is referenced by:  drnglpir  17769
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