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Theorem divrngidl 26756
Description: The only ideals in a division ring are the zero ideal and the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
divrngidl.1  |-  G  =  ( 1st `  R
)
divrngidl.2  |-  H  =  ( 2nd `  R
)
divrngidl.3  |-  X  =  ran  G
divrngidl.4  |-  Z  =  (GId `  G )
Assertion
Ref Expression
divrngidl  |-  ( R  e.  DivRingOps  ->  ( Idl `  R
)  =  { { Z } ,  X }
)

Proof of Theorem divrngidl
Dummy variables  i  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 divrngidl.1 . . 3  |-  G  =  ( 1st `  R
)
2 divrngidl.2 . . 3  |-  H  =  ( 2nd `  R
)
3 divrngidl.4 . . 3  |-  Z  =  (GId `  G )
4 divrngidl.3 . . 3  |-  X  =  ran  G
5 eqid 2296 . . 3  |-  (GId `  H )  =  (GId
`  H )
61, 2, 3, 4, 5isdrngo2 26692 . 2  |-  ( R  e.  DivRingOps 
<->  ( R  e.  RingOps  /\  ( (GId `  H )  =/=  Z  /\  A. x  e.  ( X  \  { Z } ) E. y  e.  ( X  \  { Z } ) ( y H x )  =  (GId `  H )
) ) )
71, 3idl0cl 26746 . . . . . . . . . . 11  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  Z  e.  i )
87adantr 451 . . . . . . . . . 10  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  A. x  e.  ( X 
\  { Z }
) E. y  e.  ( X  \  { Z } ) ( y H x )  =  (GId `  H )
)  ->  Z  e.  i )
9 fvex 5555 . . . . . . . . . . . . . 14  |-  (GId `  G )  e.  _V
103, 9eqeltri 2366 . . . . . . . . . . . . 13  |-  Z  e. 
_V
1110snss 3761 . . . . . . . . . . . 12  |-  ( Z  e.  i  <->  { Z }  C_  i )
12 necom 2540 . . . . . . . . . . . 12  |-  ( i  =/=  { Z }  <->  { Z }  =/=  i
)
13 pssdifn0 3528 . . . . . . . . . . . . 13  |-  ( ( { Z }  C_  i  /\  { Z }  =/=  i )  ->  (
i  \  { Z } )  =/=  (/) )
14 n0 3477 . . . . . . . . . . . . 13  |-  ( ( i  \  { Z } )  =/=  (/)  <->  E. z 
z  e.  ( i 
\  { Z }
) )
1513, 14sylib 188 . . . . . . . . . . . 12  |-  ( ( { Z }  C_  i  /\  { Z }  =/=  i )  ->  E. z 
z  e.  ( i 
\  { Z }
) )
1611, 12, 15syl2anb 465 . . . . . . . . . . 11  |-  ( ( Z  e.  i  /\  i  =/=  { Z }
)  ->  E. z 
z  e.  ( i 
\  { Z }
) )
171, 4idlss 26744 . . . . . . . . . . . . . . . . 17  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  i  C_  X )
18 ssdif 3324 . . . . . . . . . . . . . . . . . 18  |-  ( i 
C_  X  ->  (
i  \  { Z } )  C_  ( X  \  { Z }
) )
1918sselda 3193 . . . . . . . . . . . . . . . . 17  |-  ( ( i  C_  X  /\  z  e.  ( i  \  { Z } ) )  ->  z  e.  ( X  \  { Z } ) )
2017, 19sylan 457 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  z  e.  ( i  \  { Z } ) )  ->  z  e.  ( X  \  { Z } ) )
21 oveq2 5882 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  z  ->  (
y H x )  =  ( y H z ) )
2221eqeq1d 2304 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  z  ->  (
( y H x )  =  (GId `  H )  <->  ( y H z )  =  (GId `  H )
) )
2322rexbidv 2577 . . . . . . . . . . . . . . . . 17  |-  ( x  =  z  ->  ( E. y  e.  ( X  \  { Z }
) ( y H x )  =  (GId
`  H )  <->  E. y  e.  ( X  \  { Z } ) ( y H z )  =  (GId `  H )
) )
2423rspcva 2895 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  ( X 
\  { Z }
)  /\  A. x  e.  ( X  \  { Z } ) E. y  e.  ( X  \  { Z } ) ( y H x )  =  (GId `  H )
)  ->  E. y  e.  ( X  \  { Z } ) ( y H z )  =  (GId `  H )
)
2520, 24sylan 457 . . . . . . . . . . . . . . 15  |-  ( ( ( ( R  e.  RingOps 
/\  i  e.  ( Idl `  R ) )  /\  z  e.  ( i  \  { Z } ) )  /\  A. x  e.  ( X 
\  { Z }
) E. y  e.  ( X  \  { Z } ) ( y H x )  =  (GId `  H )
)  ->  E. y  e.  ( X  \  { Z } ) ( y H z )  =  (GId `  H )
)
26 eldifi 3311 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  e.  ( i  \  { Z } )  -> 
z  e.  i )
27 eldifi 3311 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  ( X  \  { Z } )  -> 
y  e.  X )
2826, 27anim12i 549 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  e.  ( i 
\  { Z }
)  /\  y  e.  ( X  \  { Z } ) )  -> 
( z  e.  i  /\  y  e.  X
) )
291, 2, 4idllmulcl 26748 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( z  e.  i  /\  y  e.  X
) )  ->  (
y H z )  e.  i )
301, 2, 4, 51idl 26754 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  (
(GId `  H )  e.  i  <->  i  =  X ) )
3130biimpd 198 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  (
(GId `  H )  e.  i  ->  i  =  X ) )
3231adantr 451 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( z  e.  i  /\  y  e.  X
) )  ->  (
(GId `  H )  e.  i  ->  i  =  X ) )
33 eleq1 2356 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( y H z )  =  (GId `  H
)  ->  ( (
y H z )  e.  i  <->  (GId `  H
)  e.  i ) )
3433imbi1d 308 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( y H z )  =  (GId `  H
)  ->  ( (
( y H z )  e.  i  -> 
i  =  X )  <-> 
( (GId `  H
)  e.  i  -> 
i  =  X ) ) )
3532, 34syl5ibrcom 213 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( z  e.  i  /\  y  e.  X
) )  ->  (
( y H z )  =  (GId `  H )  ->  (
( y H z )  e.  i  -> 
i  =  X ) ) )
3629, 35mpid 37 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( z  e.  i  /\  y  e.  X
) )  ->  (
( y H z )  =  (GId `  H )  ->  i  =  X ) )
3728, 36sylan2 460 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( z  e.  ( i  \  { Z } )  /\  y  e.  ( X  \  { Z } ) ) )  ->  ( ( y H z )  =  (GId `  H )  ->  i  =  X ) )
3837anassrs 629 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( R  e.  RingOps 
/\  i  e.  ( Idl `  R ) )  /\  z  e.  ( i  \  { Z } ) )  /\  y  e.  ( X  \  { Z } ) )  ->  ( (
y H z )  =  (GId `  H
)  ->  i  =  X ) )
3938rexlimdva 2680 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  z  e.  ( i  \  { Z } ) )  ->  ( E. y  e.  ( X  \  { Z } ) ( y H z )  =  (GId `  H )  ->  i  =  X ) )
4039imp 418 . . . . . . . . . . . . . . 15  |-  ( ( ( ( R  e.  RingOps 
/\  i  e.  ( Idl `  R ) )  /\  z  e.  ( i  \  { Z } ) )  /\  E. y  e.  ( X 
\  { Z }
) ( y H z )  =  (GId
`  H ) )  ->  i  =  X )
4125, 40syldan 456 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e.  RingOps 
/\  i  e.  ( Idl `  R ) )  /\  z  e.  ( i  \  { Z } ) )  /\  A. x  e.  ( X 
\  { Z }
) E. y  e.  ( X  \  { Z } ) ( y H x )  =  (GId `  H )
)  ->  i  =  X )
4241an32s 779 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  RingOps 
/\  i  e.  ( Idl `  R ) )  /\  A. x  e.  ( X  \  { Z } ) E. y  e.  ( X  \  { Z } ) ( y H x )  =  (GId `  H )
)  /\  z  e.  ( i  \  { Z } ) )  -> 
i  =  X )
4342ex 423 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  A. x  e.  ( X 
\  { Z }
) E. y  e.  ( X  \  { Z } ) ( y H x )  =  (GId `  H )
)  ->  ( z  e.  ( i  \  { Z } )  ->  i  =  X ) )
4443exlimdv 1626 . . . . . . . . . . 11  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  A. x  e.  ( X 
\  { Z }
) E. y  e.  ( X  \  { Z } ) ( y H x )  =  (GId `  H )
)  ->  ( E. z  z  e.  (
i  \  { Z } )  ->  i  =  X ) )
4516, 44syl5 28 . . . . . . . . . 10  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  A. x  e.  ( X 
\  { Z }
) E. y  e.  ( X  \  { Z } ) ( y H x )  =  (GId `  H )
)  ->  ( ( Z  e.  i  /\  i  =/=  { Z }
)  ->  i  =  X ) )
468, 45mpand 656 . . . . . . . . 9  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  A. x  e.  ( X 
\  { Z }
) E. y  e.  ( X  \  { Z } ) ( y H x )  =  (GId `  H )
)  ->  ( i  =/=  { Z }  ->  i  =  X ) )
4746an32s 779 . . . . . . . 8  |-  ( ( ( R  e.  RingOps  /\  A. x  e.  ( X 
\  { Z }
) E. y  e.  ( X  \  { Z } ) ( y H x )  =  (GId `  H )
)  /\  i  e.  ( Idl `  R ) )  ->  ( i  =/=  { Z }  ->  i  =  X ) )
48 neor 2543 . . . . . . . 8  |-  ( ( i  =  { Z }  \/  i  =  X )  <->  ( i  =/=  { Z }  ->  i  =  X ) )
4947, 48sylibr 203 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  A. x  e.  ( X 
\  { Z }
) E. y  e.  ( X  \  { Z } ) ( y H x )  =  (GId `  H )
)  /\  i  e.  ( Idl `  R ) )  ->  ( i  =  { Z }  \/  i  =  X )
)
5049ex 423 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A. x  e.  ( X  \  { Z } ) E. y  e.  ( X  \  { Z } ) ( y H x )  =  (GId `  H )
)  ->  ( i  e.  ( Idl `  R
)  ->  ( i  =  { Z }  \/  i  =  X )
) )
511, 30idl 26753 . . . . . . . . 9  |-  ( R  e.  RingOps  ->  { Z }  e.  ( Idl `  R
) )
52 eleq1 2356 . . . . . . . . 9  |-  ( i  =  { Z }  ->  ( i  e.  ( Idl `  R )  <->  { Z }  e.  ( Idl `  R ) ) )
5351, 52syl5ibrcom 213 . . . . . . . 8  |-  ( R  e.  RingOps  ->  ( i  =  { Z }  ->  i  e.  ( Idl `  R
) ) )
541, 4rngoidl 26752 . . . . . . . . 9  |-  ( R  e.  RingOps  ->  X  e.  ( Idl `  R ) )
55 eleq1 2356 . . . . . . . . 9  |-  ( i  =  X  ->  (
i  e.  ( Idl `  R )  <->  X  e.  ( Idl `  R ) ) )
5654, 55syl5ibrcom 213 . . . . . . . 8  |-  ( R  e.  RingOps  ->  ( i  =  X  ->  i  e.  ( Idl `  R ) ) )
5753, 56jaod 369 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( ( i  =  { Z }  \/  i  =  X
)  ->  i  e.  ( Idl `  R ) ) )
5857adantr 451 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A. x  e.  ( X  \  { Z } ) E. y  e.  ( X  \  { Z } ) ( y H x )  =  (GId `  H )
)  ->  ( (
i  =  { Z }  \/  i  =  X )  ->  i  e.  ( Idl `  R
) ) )
5950, 58impbid 183 . . . . 5  |-  ( ( R  e.  RingOps  /\  A. x  e.  ( X  \  { Z } ) E. y  e.  ( X  \  { Z } ) ( y H x )  =  (GId `  H )
)  ->  ( i  e.  ( Idl `  R
)  <->  ( i  =  { Z }  \/  i  =  X )
) )
60 vex 2804 . . . . . 6  |-  i  e. 
_V
6160elpr 3671 . . . . 5  |-  ( i  e.  { { Z } ,  X }  <->  ( i  =  { Z }  \/  i  =  X ) )
6259, 61syl6bbr 254 . . . 4  |-  ( ( R  e.  RingOps  /\  A. x  e.  ( X  \  { Z } ) E. y  e.  ( X  \  { Z } ) ( y H x )  =  (GId `  H )
)  ->  ( i  e.  ( Idl `  R
)  <->  i  e.  { { Z } ,  X } ) )
6362eqrdv 2294 . . 3  |-  ( ( R  e.  RingOps  /\  A. x  e.  ( X  \  { Z } ) E. y  e.  ( X  \  { Z } ) ( y H x )  =  (GId `  H )
)  ->  ( Idl `  R )  =  { { Z } ,  X } )
6463adantrl 696 . 2  |-  ( ( R  e.  RingOps  /\  (
(GId `  H )  =/=  Z  /\  A. x  e.  ( X  \  { Z } ) E. y  e.  ( X  \  { Z } ) ( y H x )  =  (GId `  H )
) )  ->  ( Idl `  R )  =  { { Z } ,  X } )
656, 64sylbi 187 1  |-  ( R  e.  DivRingOps  ->  ( Idl `  R
)  =  { { Z } ,  X }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801    \ cdif 3162    C_ wss 3165   (/)c0 3468   {csn 3653   {cpr 3654   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137  GIdcgi 20870   RingOpscrngo 21058   DivRingOpscdrng 21088   Idlcidl 26735
This theorem is referenced by:  divrngpr  26781  isfldidl  26796
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
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-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-1st 6138  df-2nd 6139  df-riota 6320  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-grpo 20874  df-gid 20875  df-ginv 20876  df-ablo 20965  df-ass 20996  df-exid 20998  df-mgm 21002  df-sgr 21014  df-mndo 21021  df-rngo 21059  df-drngo 21089  df-idl 26738
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