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Theorem divrngidl 26653
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 2283 . . 3  |-  (GId `  H )  =  (GId
`  H )
61, 2, 3, 4, 5isdrngo2 26589 . 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 26643 . . . . . . . . . . 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 5539 . . . . . . . . . . . . . 14  |-  (GId `  G )  e.  _V
103, 9eqeltri 2353 . . . . . . . . . . . . 13  |-  Z  e. 
_V
1110snss 3748 . . . . . . . . . . . 12  |-  ( Z  e.  i  <->  { Z }  C_  i )
12 necom 2527 . . . . . . . . . . . 12  |-  ( i  =/=  { Z }  <->  { Z }  =/=  i
)
13 pssdifn0 3515 . . . . . . . . . . . . 13  |-  ( ( { Z }  C_  i  /\  { Z }  =/=  i )  ->  (
i  \  { Z } )  =/=  (/) )
14 n0 3464 . . . . . . . . . . . . 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 26641 . . . . . . . . . . . . . . . . 17  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  i  C_  X )
18 ssdif 3311 . . . . . . . . . . . . . . . . . 18  |-  ( i 
C_  X  ->  (
i  \  { Z } )  C_  ( X  \  { Z }
) )
1918sselda 3180 . . . . . . . . . . . . . . . . 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 5866 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  z  ->  (
y H x )  =  ( y H z ) )
2221eqeq1d 2291 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  z  ->  (
( y H x )  =  (GId `  H )  <->  ( y H z )  =  (GId `  H )
) )
2322rexbidv 2564 . . . . . . . . . . . . . . . . 17  |-  ( x  =  z  ->  ( E. y  e.  ( X  \  { Z }
) ( y H x )  =  (GId
`  H )  <->  E. y  e.  ( X  \  { Z } ) ( y H z )  =  (GId `  H )
) )
2423rspcva 2882 . . . . . . . . . . . . . . . 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 3298 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  e.  ( i  \  { Z } )  -> 
z  e.  i )
27 eldifi 3298 . . . . . . . . . . . . . . . . . . . 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 26645 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( z  e.  i  /\  y  e.  X
) )  ->  (
y H z )  e.  i )
301, 2, 4, 51idl 26651 . . . . . . . . . . . . . . . . . . . . . . 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 2343 . . . . . . . . . . . . . . . . . . . . . 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 2667 . . . . . . . . . . . . . . . 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 1664 . . . . . . . . . . 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 2530 . . . . . . . 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 26650 . . . . . . . . 9  |-  ( R  e.  RingOps  ->  { Z }  e.  ( Idl `  R
) )
52 eleq1 2343 . . . . . . . . 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 26649 . . . . . . . . 9  |-  ( R  e.  RingOps  ->  X  e.  ( Idl `  R ) )
55 eleq1 2343 . . . . . . . . 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 2791 . . . . . 6  |-  i  e. 
_V
6160elpr 3658 . . . . 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 2281 . . 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 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455   {csn 3640   {cpr 3641   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121  GIdcgi 20854   RingOpscrngo 21042   DivRingOpscdrng 21072   Idlcidl 26632
This theorem is referenced by:  divrngpr  26678  isfldidl  26693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-riota 6304  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-grpo 20858  df-gid 20859  df-ginv 20860  df-ablo 20949  df-ass 20980  df-exid 20982  df-mgm 20986  df-sgr 20998  df-mndo 21005  df-rngo 21043  df-drngo 21073  df-idl 26635
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