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Theorem divrngidl 26680
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 2443 . . 3  |-  (GId `  H )  =  (GId
`  H )
61, 2, 3, 4, 5isdrngo2 26616 . 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 26670 . . . . . . . . . . 11  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  Z  e.  i )
87adantr 453 . . . . . . . . . 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 5773 . . . . . . . . . . . . . 14  |-  (GId `  G )  e.  _V
103, 9eqeltri 2513 . . . . . . . . . . . . 13  |-  Z  e. 
_V
1110snss 3955 . . . . . . . . . . . 12  |-  ( Z  e.  i  <->  { Z }  C_  i )
12 necom 2692 . . . . . . . . . . . 12  |-  ( i  =/=  { Z }  <->  { Z }  =/=  i
)
13 pssdifn0 3717 . . . . . . . . . . . . 13  |-  ( ( { Z }  C_  i  /\  { Z }  =/=  i )  ->  (
i  \  { Z } )  =/=  (/) )
14 n0 3625 . . . . . . . . . . . . 13  |-  ( ( i  \  { Z } )  =/=  (/)  <->  E. z 
z  e.  ( i 
\  { Z }
) )
1513, 14sylib 190 . . . . . . . . . . . 12  |-  ( ( { Z }  C_  i  /\  { Z }  =/=  i )  ->  E. z 
z  e.  ( i 
\  { Z }
) )
1611, 12, 15syl2anb 467 . . . . . . . . . . 11  |-  ( ( Z  e.  i  /\  i  =/=  { Z }
)  ->  E. z 
z  e.  ( i 
\  { Z }
) )
171, 4idlss 26668 . . . . . . . . . . . . . . . . 17  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  i  C_  X )
18 ssdif 3471 . . . . . . . . . . . . . . . . . 18  |-  ( i 
C_  X  ->  (
i  \  { Z } )  C_  ( X  \  { Z }
) )
1918sselda 3337 . . . . . . . . . . . . . . . . 17  |-  ( ( i  C_  X  /\  z  e.  ( i  \  { Z } ) )  ->  z  e.  ( X  \  { Z } ) )
2017, 19sylan 459 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  z  e.  ( i  \  { Z } ) )  ->  z  e.  ( X  \  { Z } ) )
21 oveq2 6125 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  z  ->  (
y H x )  =  ( y H z ) )
2221eqeq1d 2451 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  z  ->  (
( y H x )  =  (GId `  H )  <->  ( y H z )  =  (GId `  H )
) )
2322rexbidv 2733 . . . . . . . . . . . . . . . . 17  |-  ( x  =  z  ->  ( E. y  e.  ( X  \  { Z }
) ( y H x )  =  (GId
`  H )  <->  E. y  e.  ( X  \  { Z } ) ( y H z )  =  (GId `  H )
) )
2423rspcva 3059 . . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . 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 3458 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  e.  ( i  \  { Z } )  -> 
z  e.  i )
27 eldifi 3458 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  ( X  \  { Z } )  -> 
y  e.  X )
2826, 27anim12i 551 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  e.  ( i 
\  { Z }
)  /\  y  e.  ( X  \  { Z } ) )  -> 
( z  e.  i  /\  y  e.  X
) )
291, 2, 4idllmulcl 26672 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( z  e.  i  /\  y  e.  X
) )  ->  (
y H z )  e.  i )
301, 2, 4, 51idl 26678 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  (
(GId `  H )  e.  i  <->  i  =  X ) )
3130biimpd 200 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R
) )  ->  (
(GId `  H )  e.  i  ->  i  =  X ) )
3231adantr 453 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( z  e.  i  /\  y  e.  X
) )  ->  (
(GId `  H )  e.  i  ->  i  =  X ) )
33 eleq1 2503 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( y H z )  =  (GId `  H
)  ->  ( (
y H z )  e.  i  <->  (GId `  H
)  e.  i ) )
3433imbi1d 310 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( y H z )  =  (GId `  H
)  ->  ( (
( y H z )  e.  i  -> 
i  =  X )  <-> 
( (GId `  H
)  e.  i  -> 
i  =  X ) ) )
3532, 34syl5ibrcom 215 . . . . . . . . . . . . . . . . . . . 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 40 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( z  e.  i  /\  y  e.  X
) )  ->  (
( y H z )  =  (GId `  H )  ->  i  =  X ) )
3728, 36sylan2 462 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  RingOps  /\  i  e.  ( Idl `  R ) )  /\  ( z  e.  ( i  \  { Z } )  /\  y  e.  ( X  \  { Z } ) ) )  ->  ( ( y H z )  =  (GId `  H )  ->  i  =  X ) )
3837anassrs 631 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( R  e.  RingOps 
/\  i  e.  ( Idl `  R ) )  /\  z  e.  ( i  \  { Z } ) )  /\  y  e.  ( X  \  { Z } ) )  ->  ( (
y H z )  =  (GId `  H
)  ->  i  =  X ) )
3938rexlimdva 2837 . . . . . . . . . . . . . . . 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 420 . . . . . . . . . . . . . . 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 458 . . . . . . . . . . . . . 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 781 . . . . . . . . . . . . 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 425 . . . . . . . . . . . 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 1648 . . . . . . . . . . 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 31 . . . . . . . . . 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 658 . . . . . . . . 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 781 . . . . . . . 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 2695 . . . . . . . 8  |-  ( ( i  =  { Z }  \/  i  =  X )  <->  ( i  =/=  { Z }  ->  i  =  X ) )
4947, 48sylibr 205 . . . . . . 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 425 . . . . . 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 26677 . . . . . . . . 9  |-  ( R  e.  RingOps  ->  { Z }  e.  ( Idl `  R
) )
52 eleq1 2503 . . . . . . . . 9  |-  ( i  =  { Z }  ->  ( i  e.  ( Idl `  R )  <->  { Z }  e.  ( Idl `  R ) ) )
5351, 52syl5ibrcom 215 . . . . . . . 8  |-  ( R  e.  RingOps  ->  ( i  =  { Z }  ->  i  e.  ( Idl `  R
) ) )
541, 4rngoidl 26676 . . . . . . . . 9  |-  ( R  e.  RingOps  ->  X  e.  ( Idl `  R ) )
55 eleq1 2503 . . . . . . . . 9  |-  ( i  =  X  ->  (
i  e.  ( Idl `  R )  <->  X  e.  ( Idl `  R ) ) )
5654, 55syl5ibrcom 215 . . . . . . . 8  |-  ( R  e.  RingOps  ->  ( i  =  X  ->  i  e.  ( Idl `  R ) ) )
5753, 56jaod 371 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( ( i  =  { Z }  \/  i  =  X
)  ->  i  e.  ( Idl `  R ) ) )
5857adantr 453 . . . . . 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 185 . . . . 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 2968 . . . . . 6  |-  i  e. 
_V
6160elpr 3861 . . . . 5  |-  ( i  e.  { { Z } ,  X }  <->  ( i  =  { Z }  \/  i  =  X ) )
6259, 61syl6bbr 256 . . . 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 2441 . . 3  |-  ( ( R  e.  RingOps  /\  A. x  e.  ( X  \  { Z } ) E. y  e.  ( X  \  { Z } ) ( y H x )  =  (GId `  H )
)  ->  ( Idl `  R )  =  { { Z } ,  X } )
6463adantrl 698 . 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 189 1  |-  ( R  e.  DivRingOps  ->  ( Idl `  R
)  =  { { Z } ,  X }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360   E.wex 1551    = wceq 1654    e. wcel 1728    =/= wne 2606   A.wral 2712   E.wrex 2713   _Vcvv 2965    \ cdif 3306    C_ wss 3309   (/)c0 3616   {csn 3843   {cpr 3844   ran crn 4914   ` cfv 5489  (class class class)co 6117   1stc1st 6383   2ndc2nd 6384  GIdcgi 21813   RingOpscrngo 22001   DivRingOpscdrng 22031   Idlcidl 26659
This theorem is referenced by:  divrngpr  26705  isfldidl  26720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-tr 4334  df-eprel 4529  df-id 4533  df-po 4538  df-so 4539  df-fr 4576  df-we 4578  df-ord 4619  df-on 4620  df-lim 4621  df-suc 4622  df-om 4881  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-1st 6385  df-2nd 6386  df-riota 6585  df-1o 6760  df-er 6941  df-en 7146  df-dom 7147  df-sdom 7148  df-fin 7149  df-grpo 21817  df-gid 21818  df-ginv 21819  df-ablo 21908  df-ass 21939  df-exid 21941  df-mgm 21945  df-sgr 21957  df-mndo 21964  df-rngo 22002  df-drngo 22032  df-idl 26662
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