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Theorem isfldidl 26693
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 26629 . . 3  |-  ( K  e.  Fld  ->  K  e. CRingOps )
2 flddivrng 21082 . . . 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 21100 . . . 4  |-  ( K  e.  DivRingOps  ->  U  =/=  Z
)
92, 8syl 15 . . 3  |-  ( K  e.  Fld  ->  U  =/=  Z )
103, 4, 5, 6divrngidl 26653 . . . 4  |-  ( K  e.  DivRingOps  ->  ( Idl `  K
)  =  { { Z } ,  X }
)
112, 10syl 15 . . 3  |-  ( K  e.  Fld  ->  ( Idl `  K )  =  { { Z } ,  X } )
121, 9, 113jca 1132 . 2  |-  ( K  e.  Fld  ->  ( K  e. CRingOps  /\  U  =/= 
Z  /\  ( Idl `  K )  =  { { Z } ,  X } ) )
13 crngorngo 26625 . . . . . 6  |-  ( K  e. CRingOps  ->  K  e.  RingOps )
14133ad2ant1 976 . . . . 5  |-  ( ( K  e. CRingOps  /\  U  =/= 
Z  /\  ( Idl `  K )  =  { { Z } ,  X } )  ->  K  e.  RingOps )
15 simp2 956 . . . . 5  |-  ( ( K  e. CRingOps  /\  U  =/= 
Z  /\  ( Idl `  K )  =  { { Z } ,  X } )  ->  U  =/=  Z )
163rneqi 4905 . . . . . . . . . . . . . . 15  |-  ran  G  =  ran  ( 1st `  K
)
175, 16eqtri 2303 . . . . . . . . . . . . . 14  |-  X  =  ran  ( 1st `  K
)
1817, 4, 7rngo1cl 21096 . . . . . . . . . . . . 13  |-  ( K  e.  RingOps  ->  U  e.  X
)
1913, 18syl 15 . . . . . . . . . . . 12  |-  ( K  e. CRingOps  ->  U  e.  X
)
2019ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( K  e. CRingOps  /\  ( Idl `  K )  =  { { Z } ,  X } )  /\  x  e.  ( X  \  { Z } ) )  ->  U  e.  X )
21 eldif 3162 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( X  \  { Z } )  <->  ( x  e.  X  /\  -.  x  e.  { Z } ) )
22 snssi 3759 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  X  ->  { x }  C_  X )
233, 5igenss 26687 . . . . . . . . . . . . . . . . . . 19  |-  ( ( K  e.  RingOps  /\  {
x }  C_  X
)  ->  { x }  C_  ( K  IdlGen  { x } ) )
2422, 23sylan2 460 . . . . . . . . . . . . . . . . . 18  |-  ( ( K  e.  RingOps  /\  x  e.  X )  ->  { x }  C_  ( K  IdlGen  { x } ) )
25 vex 2791 . . . . . . . . . . . . . . . . . . . . . 22  |-  x  e. 
_V
2625snss 3748 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  ( K  IdlGen  { x } )  <->  { x }  C_  ( K  IdlGen  { x } ) )
2726biimpri 197 . . . . . . . . . . . . . . . . . . . 20  |-  ( { x }  C_  ( K  IdlGen  { x }
)  ->  x  e.  ( K  IdlGen  { x } ) )
28 eleq2 2344 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( K  IdlGen  { x }
)  =  { Z }  ->  ( x  e.  ( K  IdlGen  { x } )  <->  x  e.  { Z } ) )
2927, 28syl5ibcom 211 . . . . . . . . . . . . . . . . . . 19  |-  ( { x }  C_  ( K  IdlGen  { x }
)  ->  ( ( K  IdlGen  { x }
)  =  { Z }  ->  x  e.  { Z } ) )
3029con3and 428 . . . . . . . . . . . . . . . . . 18  |-  ( ( { x }  C_  ( K  IdlGen  { x } )  /\  -.  x  e.  { Z } )  ->  -.  ( K  IdlGen  { x } )  =  { Z } )
3124, 30sylan 457 . . . . . . . . . . . . . . . . 17  |-  ( ( ( K  e.  RingOps  /\  x  e.  X )  /\  -.  x  e.  { Z } )  ->  -.  ( K  IdlGen  { x } )  =  { Z } )
3231anasss 628 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  RingOps  /\  (
x  e.  X  /\  -.  x  e.  { Z } ) )  ->  -.  ( K  IdlGen  { x } )  =  { Z } )
3321, 32sylan2b 461 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  RingOps  /\  x  e.  ( X  \  { Z } ) )  ->  -.  ( K  IdlGen  { x } )  =  { Z } )
3433adantlr 695 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  RingOps  /\  ( Idl `  K )  =  { { Z } ,  X }
)  /\  x  e.  ( X  \  { Z } ) )  ->  -.  ( K  IdlGen  { x } )  =  { Z } )
35 eldifi 3298 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  ( X  \  { Z } )  ->  x  e.  X )
3635snssd 3760 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  ( X  \  { Z } )  ->  { x }  C_  X )
373, 5igenidl 26688 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( K  e.  RingOps  /\  {
x }  C_  X
)  ->  ( K  IdlGen  { x } )  e.  ( Idl `  K
) )
3836, 37sylan2 460 . . . . . . . . . . . . . . . . . . 19  |-  ( ( K  e.  RingOps  /\  x  e.  ( X  \  { Z } ) )  -> 
( K  IdlGen  { x } )  e.  ( Idl `  K ) )
39 eleq2 2344 . . . . . . . . . . . . . . . . . . 19  |-  ( ( Idl `  K )  =  { { Z } ,  X }  ->  ( ( K  IdlGen  { x } )  e.  ( Idl `  K
)  <->  ( K  IdlGen  { x } )  e. 
{ { Z } ,  X } ) )
4038, 39syl5ibcom 211 . . . . . . . . . . . . . . . . . 18  |-  ( ( K  e.  RingOps  /\  x  e.  ( X  \  { Z } ) )  -> 
( ( Idl `  K
)  =  { { Z } ,  X }  ->  ( K  IdlGen  { x } )  e.  { { Z } ,  X } ) )
4140imp 418 . . . . . . . . . . . . . . . . 17  |-  ( ( ( K  e.  RingOps  /\  x  e.  ( X  \  { Z } ) )  /\  ( Idl `  K )  =  { { Z } ,  X } )  ->  ( K  IdlGen  { x }
)  e.  { { Z } ,  X }
)
4241an32s 779 . . . . . . . . . . . . . . . 16  |-  ( ( ( K  e.  RingOps  /\  ( Idl `  K )  =  { { Z } ,  X }
)  /\  x  e.  ( X  \  { Z } ) )  -> 
( K  IdlGen  { x } )  e.  { { Z } ,  X } )
43 ovex 5883 . . . . . . . . . . . . . . . . 17  |-  ( K 
IdlGen  { x } )  e.  _V
4443elpr 3658 . . . . . . . . . . . . . . . 16  |-  ( ( K  IdlGen  { x }
)  e.  { { Z } ,  X }  <->  ( ( K  IdlGen  { x } )  =  { Z }  \/  ( K  IdlGen  { x }
)  =  X ) )
4542, 44sylib 188 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  RingOps  /\  ( Idl `  K )  =  { { Z } ,  X }
)  /\  x  e.  ( X  \  { Z } ) )  -> 
( ( K  IdlGen  { x } )  =  { Z }  \/  ( K  IdlGen  { x } )  =  X ) )
4645ord 366 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  RingOps  /\  ( Idl `  K )  =  { { Z } ,  X }
)  /\  x  e.  ( X  \  { Z } ) )  -> 
( -.  ( K 
IdlGen  { x } )  =  { Z }  ->  ( K  IdlGen  { x } )  =  X ) )
4734, 46mpd 14 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  RingOps  /\  ( Idl `  K )  =  { { Z } ,  X }
)  /\  x  e.  ( X  \  { Z } ) )  -> 
( K  IdlGen  { x } )  =  X )
4813, 47sylanl1 631 . . . . . . . . . . . 12  |-  ( ( ( K  e. CRingOps  /\  ( Idl `  K )  =  { { Z } ,  X } )  /\  x  e.  ( X  \  { Z } ) )  ->  ( K  IdlGen  { x } )  =  X )
493, 4, 5prnc 26692 . . . . . . . . . . . . . 14  |-  ( ( K  e. CRingOps  /\  x  e.  X )  ->  ( K  IdlGen  { x }
)  =  { z  e.  X  |  E. y  e.  X  z  =  ( y H x ) } )
5035, 49sylan2 460 . . . . . . . . . . . . 13  |-  ( ( K  e. CRingOps  /\  x  e.  ( X  \  { Z } ) )  -> 
( K  IdlGen  { x } )  =  {
z  e.  X  |  E. y  e.  X  z  =  ( y H x ) } )
5150adantlr 695 . . . . . . . . . . . 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 2317 . . . . . . . . . . 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 2359 . . . . . . . . . 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 2289 . . . . . . . . . . . 12  |-  ( z  =  U  ->  (
z  =  ( y H x )  <->  U  =  ( y H x ) ) )
5554rexbidv 2564 . . . . . . . . . . 11  |-  ( z  =  U  ->  ( E. y  e.  X  z  =  ( y H x )  <->  E. y  e.  X  U  =  ( y H x ) ) )
5655elrab 2923 . . . . . . . . . 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 188 . . . . . . . . 9  |-  ( ( ( K  e. CRingOps  /\  ( Idl `  K )  =  { { Z } ,  X } )  /\  x  e.  ( X  \  { Z } ) )  ->  ( U  e.  X  /\  E. y  e.  X  U  =  ( y H x ) ) )
5857simprd 449 . . . . . . . 8  |-  ( ( ( K  e. CRingOps  /\  ( Idl `  K )  =  { { Z } ,  X } )  /\  x  e.  ( X  \  { Z } ) )  ->  E. y  e.  X  U  =  ( y H x ) )
59 eqcom 2285 . . . . . . . . 9  |-  ( ( y H x )  =  U  <->  U  =  ( y H x ) )
6059rexbii 2568 . . . . . . . 8  |-  ( E. y  e.  X  ( y H x )  =  U  <->  E. y  e.  X  U  =  ( y H x ) )
6158, 60sylibr 203 . . . . . . 7  |-  ( ( ( K  e. CRingOps  /\  ( Idl `  K )  =  { { Z } ,  X } )  /\  x  e.  ( X  \  { Z } ) )  ->  E. y  e.  X  ( y H x )  =  U )
6261ralrimiva 2626 . . . . . 6  |-  ( ( K  e. CRingOps  /\  ( Idl `  K )  =  { { Z } ,  X } )  ->  A. x  e.  ( X  \  { Z }
) E. y  e.  X  ( y H x )  =  U )
63623adant2 974 . . . . 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 521 . . . 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 26590 . . . 4  |-  ( K  e.  DivRingOps 
<->  ( K  e.  RingOps  /\  ( U  =/=  Z  /\  A. x  e.  ( X  \  { Z } ) E. y  e.  X  ( y H x )  =  U ) ) )
6664, 65sylibr 203 . . 3  |-  ( ( K  e. CRingOps  /\  U  =/= 
Z  /\  ( Idl `  K )  =  { { Z } ,  X } )  ->  K  e. 
DivRingOps )
67 simp1 955 . . 3  |-  ( ( K  e. CRingOps  /\  U  =/= 
Z  /\  ( Idl `  K )  =  { { Z } ,  X } )  ->  K  e. CRingOps )
68 isfld2 26630 . . 3  |-  ( K  e.  Fld  <->  ( K  e. 
DivRingOps 
/\  K  e. CRingOps )
)
6966, 67, 68sylanbrc 645 . 2  |-  ( ( K  e. CRingOps  /\  U  =/= 
Z  /\  ( Idl `  K )  =  { { Z } ,  X } )  ->  K  e.  Fld )
7012, 69impbii 180 1  |-  ( K  e.  Fld  <->  ( K  e. CRingOps 
/\  U  =/=  Z  /\  ( Idl `  K
)  =  { { Z } ,  X }
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547    \ cdif 3149    C_ wss 3152   {csn 3640   {cpr 3641   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121  GIdcgi 20854   RingOpscrngo 21042   DivRingOpscdrng 21072   Fldcfld 21080  CRingOpsccring 26620   Idlcidl 26632    IdlGen cigen 26684
This theorem is referenced by:  isfldidl2  26694
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-int 3863  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-oprab 5862  df-mpt2 5863  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-com2 21078  df-fld 21081  df-crngo 26621  df-idl 26635  df-igen 26685
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