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Theorem isfldidl2 26694
Description: Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 6-Jan-2011.)
Hypotheses
Ref Expression
isfldidl2.1  |-  G  =  ( 1st `  K
)
isfldidl2.2  |-  H  =  ( 2nd `  K
)
isfldidl2.3  |-  X  =  ran  G
isfldidl2.4  |-  Z  =  (GId `  G )
Assertion
Ref Expression
isfldidl2  |-  ( K  e.  Fld  <->  ( K  e. CRingOps 
/\  X  =/=  { Z }  /\  ( Idl `  K )  =  { { Z } ,  X } ) )

Proof of Theorem isfldidl2
StepHypRef Expression
1 isfldidl2.1 . . 3  |-  G  =  ( 1st `  K
)
2 isfldidl2.2 . . 3  |-  H  =  ( 2nd `  K
)
3 isfldidl2.3 . . 3  |-  X  =  ran  G
4 isfldidl2.4 . . 3  |-  Z  =  (GId `  G )
5 eqid 2283 . . 3  |-  (GId `  H )  =  (GId
`  H )
61, 2, 3, 4, 5isfldidl 26693 . 2  |-  ( K  e.  Fld  <->  ( K  e. CRingOps 
/\  (GId `  H
)  =/=  Z  /\  ( Idl `  K )  =  { { Z } ,  X }
) )
7 crngorngo 26625 . . . . . . 7  |-  ( K  e. CRingOps  ->  K  e.  RingOps )
8 eqcom 2285 . . . . . . . 8  |-  ( (GId
`  H )  =  Z  <->  Z  =  (GId `  H ) )
91, 2, 3, 4, 50rngo 26652 . . . . . . . 8  |-  ( K  e.  RingOps  ->  ( Z  =  (GId `  H )  <->  X  =  { Z }
) )
108, 9syl5bb 248 . . . . . . 7  |-  ( K  e.  RingOps  ->  ( (GId `  H )  =  Z  <-> 
X  =  { Z } ) )
117, 10syl 15 . . . . . 6  |-  ( K  e. CRingOps  ->  ( (GId `  H )  =  Z  <-> 
X  =  { Z } ) )
1211necon3bid 2481 . . . . 5  |-  ( K  e. CRingOps  ->  ( (GId `  H )  =/=  Z  <->  X  =/=  { Z }
) )
1312anbi1d 685 . . . 4  |-  ( K  e. CRingOps  ->  ( ( (GId
`  H )  =/= 
Z  /\  ( Idl `  K )  =  { { Z } ,  X } )  <->  ( X  =/=  { Z }  /\  ( Idl `  K )  =  { { Z } ,  X }
) ) )
1413pm5.32i 618 . . 3  |-  ( ( K  e. CRingOps  /\  (
(GId `  H )  =/=  Z  /\  ( Idl `  K )  =  { { Z } ,  X } ) )  <->  ( K  e. CRingOps 
/\  ( X  =/= 
{ Z }  /\  ( Idl `  K )  =  { { Z } ,  X }
) ) )
15 3anass 938 . . 3  |-  ( ( K  e. CRingOps  /\  (GId `  H )  =/=  Z  /\  ( Idl `  K
)  =  { { Z } ,  X }
)  <->  ( K  e. CRingOps  /\  ( (GId `  H
)  =/=  Z  /\  ( Idl `  K )  =  { { Z } ,  X }
) ) )
16 3anass 938 . . 3  |-  ( ( K  e. CRingOps  /\  X  =/= 
{ Z }  /\  ( Idl `  K )  =  { { Z } ,  X }
)  <->  ( K  e. CRingOps  /\  ( X  =/=  { Z }  /\  ( Idl `  K )  =  { { Z } ,  X } ) ) )
1714, 15, 163bitr4i 268 . 2  |-  ( ( K  e. CRingOps  /\  (GId `  H )  =/=  Z  /\  ( Idl `  K
)  =  { { Z } ,  X }
)  <->  ( K  e. CRingOps  /\  X  =/=  { Z }  /\  ( Idl `  K
)  =  { { Z } ,  X }
) )
186, 17bitri 240 1  |-  ( K  e.  Fld  <->  ( K  e. CRingOps 
/\  X  =/=  { Z }  /\  ( Idl `  K )  =  { { Z } ,  X } ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   {csn 3640   {cpr 3641   ran crn 4690   ` cfv 5255   1stc1st 6120   2ndc2nd 6121  GIdcgi 20854   RingOpscrngo 21042   Fldcfld 21080  CRingOpsccring 26620   Idlcidl 26632
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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