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Theorem divrngidl 28825
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 2441 . . 3  |-  (GId `  H )  =  (GId
`  H )
61, 2, 3, 4, 5isdrngo2 28761 . 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 28815 . . . . . . . . . . 11  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  Z  e.  i )
87adantr 465 . . . . . . . . . 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 5699 . . . . . . . . . . . . . 14  |-  (GId `  G )  e.  _V
103, 9eqeltri 2511 . . . . . . . . . . . . 13  |-  Z  e. 
_V
1110snss 3997 . . . . . . . . . . . 12  |-  ( Z  e.  i  <->  { Z }  C_  i )
12 necom 2691 . . . . . . . . . . . 12  |-  ( i  =/=  { Z }  <->  { Z }  =/=  i
)
13 pssdifn0 3738 . . . . . . . . . . . . 13  |-  ( ( { Z }  C_  i  /\  { Z }  =/=  i )  ->  (
i  \  { Z } )  =/=  (/) )
14 n0 3644 . . . . . . . . . . . . 13  |-  ( ( i  \  { Z } )  =/=  (/)  <->  E. z 
z  e.  ( i 
\  { Z }
) )
1513, 14sylib 196 . . . . . . . . . . . 12  |-  ( ( { Z }  C_  i  /\  { Z }  =/=  i )  ->  E. z 
z  e.  ( i 
\  { Z }
) )
1611, 12, 15syl2anb 479 . . . . . . . . . . 11  |-  ( ( Z  e.  i  /\  i  =/=  { Z }
)  ->  E. z 
z  e.  ( i 
\  { Z }
) )
171, 4idlss 28813 . . . . . . . . . . . . . . . . 17  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  i  C_  X )
18 ssdif 3489 . . . . . . . . . . . . . . . . . 18  |-  ( i 
C_  X  ->  (
i  \  { Z } )  C_  ( X  \  { Z }
) )
1918sselda 3354 . . . . . . . . . . . . . . . . 17  |-  ( ( i  C_  X  /\  z  e.  ( i  \  { Z } ) )  ->  z  e.  ( X  \  { Z } ) )
2017, 19sylan 471 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  z  e.  ( i  \  { Z } ) )  ->  z  e.  ( X  \  { Z } ) )
21 oveq2 6097 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  z  ->  (
y H x )  =  ( y H z ) )
2221eqeq1d 2449 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  z  ->  (
( y H x )  =  (GId `  H )  <->  ( y H z )  =  (GId `  H )
) )
2322rexbidv 2734 . . . . . . . . . . . . . . . . 17  |-  ( x  =  z  ->  ( E. y  e.  ( X  \  { Z }
) ( y H x )  =  (GId
`  H )  <->  E. y  e.  ( X  \  { Z } ) ( y H z )  =  (GId `  H )
) )
2423rspcva 3069 . . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . . . 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 3476 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  e.  ( i  \  { Z } )  -> 
z  e.  i )
27 eldifi 3476 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  ( X  \  { Z } )  -> 
y  e.  X )
2826, 27anim12i 566 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  e.  ( i 
\  { Z }
)  /\  y  e.  ( X  \  { Z } ) )  -> 
( z  e.  i  /\  y  e.  X
) )
291, 2, 4idllmulcl 28817 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( z  e.  i  /\  y  e.  X
) )  ->  (
y H z )  e.  i )
301, 2, 4, 51idl 28823 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  (
(GId `  H )  e.  i  <->  i  =  X ) )
3130biimpd 207 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  (
(GId `  H )  e.  i  ->  i  =  X ) )
3231adantr 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( z  e.  i  /\  y  e.  X
) )  ->  (
(GId `  H )  e.  i  ->  i  =  X ) )
33 eleq1 2501 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( y H z )  =  (GId `  H
)  ->  ( (
y H z )  e.  i  <->  (GId `  H
)  e.  i ) )
3433imbi1d 317 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( y H z )  =  (GId `  H
)  ->  ( (
( y H z )  e.  i  -> 
i  =  X )  <-> 
( (GId `  H
)  e.  i  -> 
i  =  X ) ) )
3532, 34syl5ibrcom 222 . . . . . . . . . . . . . . . . . . . 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 41 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( z  e.  i  /\  y  e.  X
) )  ->  (
( y H z )  =  (GId `  H )  ->  i  =  X ) )
3728, 36sylan2 474 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( z  e.  ( i  \  { Z } )  /\  y  e.  ( X  \  { Z } ) ) )  ->  ( ( y H z )  =  (GId `  H )  ->  i  =  X ) )
3837anassrs 648 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( R  e.  RingOps 
/\  i  e.  ( Idl `  R ) )  /\  z  e.  ( i  \  { Z } ) )  /\  y  e.  ( X  \  { Z } ) )  ->  ( (
y H z )  =  (GId `  H
)  ->  i  =  X ) )
3938rexlimdva 2839 . . . . . . . . . . . . . . . 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 429 . . . . . . . . . . . . . . 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 470 . . . . . . . . . . . . . 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 802 . . . . . . . . . . . . 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 434 . . . . . . . . . . . 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 1690 . . . . . . . . . . 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 32 . . . . . . . . . 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 675 . . . . . . . . 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 802 . . . . . . . 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 2694 . . . . . . . 8  |-  ( ( i  =  { Z }  \/  i  =  X )  <->  ( i  =/=  { Z }  ->  i  =  X ) )
4947, 48sylibr 212 . . . . . . 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 434 . . . . . 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 28822 . . . . . . . . 9  |-  ( R  e.  RingOps  ->  { Z }  e.  ( Idl `  R
) )
52 eleq1 2501 . . . . . . . . 9  |-  ( i  =  { Z }  ->  ( i  e.  ( Idl `  R )  <->  { Z }  e.  ( Idl `  R ) ) )
5351, 52syl5ibrcom 222 . . . . . . . 8  |-  ( R  e.  RingOps  ->  ( i  =  { Z }  ->  i  e.  ( Idl `  R
) ) )
541, 4rngoidl 28821 . . . . . . . . 9  |-  ( R  e.  RingOps  ->  X  e.  ( Idl `  R ) )
55 eleq1 2501 . . . . . . . . 9  |-  ( i  =  X  ->  (
i  e.  ( Idl `  R )  <->  X  e.  ( Idl `  R ) ) )
5654, 55syl5ibrcom 222 . . . . . . . 8  |-  ( R  e.  RingOps  ->  ( i  =  X  ->  i  e.  ( Idl `  R ) ) )
5753, 56jaod 380 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( ( i  =  { Z }  \/  i  =  X
)  ->  i  e.  ( Idl `  R ) ) )
5857adantr 465 . . . . . 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 191 . . . . 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 2973 . . . . . 6  |-  i  e. 
_V
6160elpr 3893 . . . . 5  |-  ( i  e.  { { Z } ,  X }  <->  ( i  =  { Z }  \/  i  =  X ) )
6259, 61syl6bbr 263 . . . 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 2439 . . 3  |-  ( ( R  e.  RingOps  /\  A. x  e.  ( X  \  { Z } ) E. y  e.  ( X  \  { Z } ) ( y H x )  =  (GId `  H )
)  ->  ( Idl `  R )  =  { { Z } ,  X } )
6463adantrl 715 . 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 195 1  |-  ( R  e.  DivRingOps  ->  ( Idl `  R
)  =  { { Z } ,  X }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2604   A.wral 2713   E.wrex 2714   _Vcvv 2970    \ cdif 3323    C_ wss 3326   (/)c0 3635   {csn 3875   {cpr 3877   ran crn 4839   ` cfv 5416  (class class class)co 6089   1stc1st 6573   2ndc2nd 6574  GIdcgi 23672   RingOpscrngo 23860   DivRingOpscdrng 23890   Idlcidl 28804
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-om 6475  df-1st 6575  df-2nd 6576  df-1o 6918  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-grpo 23676  df-gid 23677  df-ginv 23678  df-ablo 23767  df-ass 23798  df-exid 23800  df-mgm 23804  df-sgr 23816  df-mndo 23823  df-rngo 23861  df-drngo 23891  df-idl 28807
This theorem is referenced by:  divrngpr  28850  isfldidl  28865
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