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Theorem isfldidl 30383
Description: Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
isfldidl.1  |-  G  =  ( 1st `  K
)
isfldidl.2  |-  H  =  ( 2nd `  K
)
isfldidl.3  |-  X  =  ran  G
isfldidl.4  |-  Z  =  (GId `  G )
isfldidl.5  |-  U  =  (GId `  H )
Assertion
Ref Expression
isfldidl  |-  ( K  e.  Fld  <->  ( K  e. CRingOps 
/\  U  =/=  Z  /\  ( Idl `  K
)  =  { { Z } ,  X }
) )

Proof of Theorem isfldidl
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fldcrng 30319 . . 3  |-  ( K  e.  Fld  ->  K  e. CRingOps )
2 flddivrng 25231 . . . 4  |-  ( K  e.  Fld  ->  K  e. 
DivRingOps )
3 isfldidl.1 . . . . 5  |-  G  =  ( 1st `  K
)
4 isfldidl.2 . . . . 5  |-  H  =  ( 2nd `  K
)
5 isfldidl.3 . . . . 5  |-  X  =  ran  G
6 isfldidl.4 . . . . 5  |-  Z  =  (GId `  G )
7 isfldidl.5 . . . . 5  |-  U  =  (GId `  H )
83, 4, 5, 6, 7dvrunz 25249 . . . 4  |-  ( K  e.  DivRingOps  ->  U  =/=  Z
)
92, 8syl 16 . . 3  |-  ( K  e.  Fld  ->  U  =/=  Z )
103, 4, 5, 6divrngidl 30343 . . . 4  |-  ( K  e.  DivRingOps  ->  ( Idl `  K
)  =  { { Z } ,  X }
)
112, 10syl 16 . . 3  |-  ( K  e.  Fld  ->  ( Idl `  K )  =  { { Z } ,  X } )
121, 9, 113jca 1176 . 2  |-  ( K  e.  Fld  ->  ( K  e. CRingOps  /\  U  =/= 
Z  /\  ( Idl `  K )  =  { { Z } ,  X } ) )
13 crngorngo 30315 . . . . . 6  |-  ( K  e. CRingOps  ->  K  e.  RingOps )
14133ad2ant1 1017 . . . . 5  |-  ( ( K  e. CRingOps  /\  U  =/= 
Z  /\  ( Idl `  K )  =  { { Z } ,  X } )  ->  K  e.  RingOps )
15 simp2 997 . . . . 5  |-  ( ( K  e. CRingOps  /\  U  =/= 
Z  /\  ( Idl `  K )  =  { { Z } ,  X } )  ->  U  =/=  Z )
163rneqi 5235 . . . . . . . . . . . . . . 15  |-  ran  G  =  ran  ( 1st `  K
)
175, 16eqtri 2496 . . . . . . . . . . . . . 14  |-  X  =  ran  ( 1st `  K
)
1817, 4, 7rngo1cl 25245 . . . . . . . . . . . . 13  |-  ( K  e.  RingOps  ->  U  e.  X
)
1913, 18syl 16 . . . . . . . . . . . 12  |-  ( K  e. CRingOps  ->  U  e.  X
)
2019ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( K  e. CRingOps  /\  ( Idl `  K )  =  { { Z } ,  X } )  /\  x  e.  ( X  \  { Z } ) )  ->  U  e.  X )
21 eldif 3491 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( X  \  { Z } )  <->  ( x  e.  X  /\  -.  x  e.  { Z } ) )
22 snssi 4177 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  X  ->  { x }  C_  X )
233, 5igenss 30377 . . . . . . . . . . . . . . . . . . 19  |-  ( ( K  e.  RingOps  /\  {
x }  C_  X
)  ->  { x }  C_  ( K  IdlGen  { x } ) )
2422, 23sylan2 474 . . . . . . . . . . . . . . . . . 18  |-  ( ( K  e.  RingOps  /\  x  e.  X )  ->  { x }  C_  ( K  IdlGen  { x } ) )
25 vex 3121 . . . . . . . . . . . . . . . . . . . . . 22  |-  x  e. 
_V
2625snss 4157 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  ( K  IdlGen  { x } )  <->  { x }  C_  ( K  IdlGen  { x } ) )
2726biimpri 206 . . . . . . . . . . . . . . . . . . . 20  |-  ( { x }  C_  ( K  IdlGen  { x }
)  ->  x  e.  ( K  IdlGen  { x } ) )
28 eleq2 2540 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( K  IdlGen  { x }
)  =  { Z }  ->  ( x  e.  ( K  IdlGen  { x } )  <->  x  e.  { Z } ) )
2927, 28syl5ibcom 220 . . . . . . . . . . . . . . . . . . 19  |-  ( { x }  C_  ( K  IdlGen  { x }
)  ->  ( ( K  IdlGen  { x }
)  =  { Z }  ->  x  e.  { Z } ) )
3029con3dimp 441 . . . . . . . . . . . . . . . . . 18  |-  ( ( { x }  C_  ( K  IdlGen  { x } )  /\  -.  x  e.  { Z } )  ->  -.  ( K  IdlGen  { x } )  =  { Z } )
3124, 30sylan 471 . . . . . . . . . . . . . . . . 17  |-  ( ( ( K  e.  RingOps  /\  x  e.  X )  /\  -.  x  e.  { Z } )  ->  -.  ( K  IdlGen  { x } )  =  { Z } )
3231anasss 647 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  RingOps  /\  (
x  e.  X  /\  -.  x  e.  { Z } ) )  ->  -.  ( K  IdlGen  { x } )  =  { Z } )
3321, 32sylan2b 475 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  RingOps  /\  x  e.  ( X  \  { Z } ) )  ->  -.  ( K  IdlGen  { x } )  =  { Z } )
3433adantlr 714 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  RingOps  /\  ( Idl `  K )  =  { { Z } ,  X }
)  /\  x  e.  ( X  \  { Z } ) )  ->  -.  ( K  IdlGen  { x } )  =  { Z } )
35 eldifi 3631 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  ( X  \  { Z } )  ->  x  e.  X )
3635snssd 4178 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  ( X  \  { Z } )  ->  { x }  C_  X )
373, 5igenidl 30378 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( K  e.  RingOps  /\  {
x }  C_  X
)  ->  ( K  IdlGen  { x } )  e.  ( Idl `  K
) )
3836, 37sylan2 474 . . . . . . . . . . . . . . . . . . 19  |-  ( ( K  e.  RingOps  /\  x  e.  ( X  \  { Z } ) )  -> 
( K  IdlGen  { x } )  e.  ( Idl `  K ) )
39 eleq2 2540 . . . . . . . . . . . . . . . . . . 19  |-  ( ( Idl `  K )  =  { { Z } ,  X }  ->  ( ( K  IdlGen  { x } )  e.  ( Idl `  K
)  <->  ( K  IdlGen  { x } )  e. 
{ { Z } ,  X } ) )
4038, 39syl5ibcom 220 . . . . . . . . . . . . . . . . . 18  |-  ( ( K  e.  RingOps  /\  x  e.  ( X  \  { Z } ) )  -> 
( ( Idl `  K
)  =  { { Z } ,  X }  ->  ( K  IdlGen  { x } )  e.  { { Z } ,  X } ) )
4140imp 429 . . . . . . . . . . . . . . . . 17  |-  ( ( ( K  e.  RingOps  /\  x  e.  ( X  \  { Z } ) )  /\  ( Idl `  K )  =  { { Z } ,  X } )  ->  ( K  IdlGen  { x }
)  e.  { { Z } ,  X }
)
4241an32s 802 . . . . . . . . . . . . . . . 16  |-  ( ( ( K  e.  RingOps  /\  ( Idl `  K )  =  { { Z } ,  X }
)  /\  x  e.  ( X  \  { Z } ) )  -> 
( K  IdlGen  { x } )  e.  { { Z } ,  X } )
43 ovex 6320 . . . . . . . . . . . . . . . . 17  |-  ( K 
IdlGen  { x } )  e.  _V
4443elpr 4051 . . . . . . . . . . . . . . . 16  |-  ( ( K  IdlGen  { x }
)  e.  { { Z } ,  X }  <->  ( ( K  IdlGen  { x } )  =  { Z }  \/  ( K  IdlGen  { x }
)  =  X ) )
4542, 44sylib 196 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  RingOps  /\  ( Idl `  K )  =  { { Z } ,  X }
)  /\  x  e.  ( X  \  { Z } ) )  -> 
( ( K  IdlGen  { x } )  =  { Z }  \/  ( K  IdlGen  { x } )  =  X ) )
4645ord 377 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  RingOps  /\  ( Idl `  K )  =  { { Z } ,  X }
)  /\  x  e.  ( X  \  { Z } ) )  -> 
( -.  ( K 
IdlGen  { x } )  =  { Z }  ->  ( K  IdlGen  { x } )  =  X ) )
4734, 46mpd 15 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  RingOps  /\  ( Idl `  K )  =  { { Z } ,  X }
)  /\  x  e.  ( X  \  { Z } ) )  -> 
( K  IdlGen  { x } )  =  X )
4813, 47sylanl1 650 . . . . . . . . . . . 12  |-  ( ( ( K  e. CRingOps  /\  ( Idl `  K )  =  { { Z } ,  X } )  /\  x  e.  ( X  \  { Z } ) )  ->  ( K  IdlGen  { x } )  =  X )
493, 4, 5prnc 30382 . . . . . . . . . . . . . 14  |-  ( ( K  e. CRingOps  /\  x  e.  X )  ->  ( K  IdlGen  { x }
)  =  { z  e.  X  |  E. y  e.  X  z  =  ( y H x ) } )
5035, 49sylan2 474 . . . . . . . . . . . . 13  |-  ( ( K  e. CRingOps  /\  x  e.  ( X  \  { Z } ) )  -> 
( K  IdlGen  { x } )  =  {
z  e.  X  |  E. y  e.  X  z  =  ( y H x ) } )
5150adantlr 714 . . . . . . . . . . . 12  |-  ( ( ( K  e. CRingOps  /\  ( Idl `  K )  =  { { Z } ,  X } )  /\  x  e.  ( X  \  { Z } ) )  ->  ( K  IdlGen  { x } )  =  { z  e.  X  |  E. y  e.  X  z  =  ( y H x ) } )
5248, 51eqtr3d 2510 . . . . . . . . . . 11  |-  ( ( ( K  e. CRingOps  /\  ( Idl `  K )  =  { { Z } ,  X } )  /\  x  e.  ( X  \  { Z } ) )  ->  X  =  { z  e.  X  |  E. y  e.  X  z  =  ( y H x ) } )
5320, 52eleqtrd 2557 . . . . . . . . . 10  |-  ( ( ( K  e. CRingOps  /\  ( Idl `  K )  =  { { Z } ,  X } )  /\  x  e.  ( X  \  { Z } ) )  ->  U  e.  { z  e.  X  |  E. y  e.  X  z  =  ( y H x ) } )
54 eqeq1 2471 . . . . . . . . . . . 12  |-  ( z  =  U  ->  (
z  =  ( y H x )  <->  U  =  ( y H x ) ) )
5554rexbidv 2978 . . . . . . . . . . 11  |-  ( z  =  U  ->  ( E. y  e.  X  z  =  ( y H x )  <->  E. y  e.  X  U  =  ( y H x ) ) )
5655elrab 3266 . . . . . . . . . 10  |-  ( U  e.  { z  e.  X  |  E. y  e.  X  z  =  ( y H x ) }  <->  ( U  e.  X  /\  E. y  e.  X  U  =  ( y H x ) ) )
5753, 56sylib 196 . . . . . . . . 9  |-  ( ( ( K  e. CRingOps  /\  ( Idl `  K )  =  { { Z } ,  X } )  /\  x  e.  ( X  \  { Z } ) )  ->  ( U  e.  X  /\  E. y  e.  X  U  =  ( y H x ) ) )
5857simprd 463 . . . . . . . 8  |-  ( ( ( K  e. CRingOps  /\  ( Idl `  K )  =  { { Z } ,  X } )  /\  x  e.  ( X  \  { Z } ) )  ->  E. y  e.  X  U  =  ( y H x ) )
59 eqcom 2476 . . . . . . . . 9  |-  ( ( y H x )  =  U  <->  U  =  ( y H x ) )
6059rexbii 2969 . . . . . . . 8  |-  ( E. y  e.  X  ( y H x )  =  U  <->  E. y  e.  X  U  =  ( y H x ) )
6158, 60sylibr 212 . . . . . . 7  |-  ( ( ( K  e. CRingOps  /\  ( Idl `  K )  =  { { Z } ,  X } )  /\  x  e.  ( X  \  { Z } ) )  ->  E. y  e.  X  ( y H x )  =  U )
6261ralrimiva 2881 . . . . . 6  |-  ( ( K  e. CRingOps  /\  ( Idl `  K )  =  { { Z } ,  X } )  ->  A. x  e.  ( X  \  { Z }
) E. y  e.  X  ( y H x )  =  U )
63623adant2 1015 . . . . 5  |-  ( ( K  e. CRingOps  /\  U  =/= 
Z  /\  ( Idl `  K )  =  { { Z } ,  X } )  ->  A. x  e.  ( X  \  { Z } ) E. y  e.  X  ( y H x )  =  U )
6414, 15, 63jca32 535 . . . 4  |-  ( ( K  e. CRingOps  /\  U  =/= 
Z  /\  ( Idl `  K )  =  { { Z } ,  X } )  ->  ( K  e.  RingOps  /\  ( U  =/=  Z  /\  A. x  e.  ( X  \  { Z } ) E. y  e.  X  ( y H x )  =  U ) ) )
653, 4, 6, 5, 7isdrngo3 30280 . . . 4  |-  ( K  e.  DivRingOps 
<->  ( K  e.  RingOps  /\  ( U  =/=  Z  /\  A. x  e.  ( X  \  { Z } ) E. y  e.  X  ( y H x )  =  U ) ) )
6664, 65sylibr 212 . . 3  |-  ( ( K  e. CRingOps  /\  U  =/= 
Z  /\  ( Idl `  K )  =  { { Z } ,  X } )  ->  K  e. 
DivRingOps )
67 simp1 996 . . 3  |-  ( ( K  e. CRingOps  /\  U  =/= 
Z  /\  ( Idl `  K )  =  { { Z } ,  X } )  ->  K  e. CRingOps )
68 isfld2 30320 . . 3  |-  ( K  e.  Fld  <->  ( K  e. 
DivRingOps 
/\  K  e. CRingOps )
)
6966, 67, 68sylanbrc 664 . 2  |-  ( ( K  e. CRingOps  /\  U  =/= 
Z  /\  ( Idl `  K )  =  { { Z } ,  X } )  ->  K  e.  Fld )
7012, 69impbii 188 1  |-  ( K  e.  Fld  <->  ( K  e. CRingOps 
/\  U  =/=  Z  /\  ( Idl `  K
)  =  { { Z } ,  X }
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818   {crab 2821    \ cdif 3478    C_ wss 3481   {csn 4033   {cpr 4035   ran crn 5006   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794  GIdcgi 25003   RingOpscrngo 25191   DivRingOpscdrng 25221   Fldcfld 25229  CRingOpsccring 30310   Idlcidl 30322    IdlGen cigen 30374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-1o 7142  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-grpo 25007  df-gid 25008  df-ginv 25009  df-ablo 25098  df-ass 25129  df-exid 25131  df-mgmOLD 25135  df-sgrOLD 25147  df-mndo 25154  df-rngo 25192  df-drngo 25222  df-com2 25227  df-fld 25230  df-crngo 30311  df-idl 30325  df-igen 30375
This theorem is referenced by:  isfldidl2  30384
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