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Theorem isfldidl2 26797
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 2296 . . 3  |-  (GId `  H )  =  (GId
`  H )
61, 2, 3, 4, 5isfldidl 26796 . 2  |-  ( K  e.  Fld  <->  ( K  e. CRingOps 
/\  (GId `  H
)  =/=  Z  /\  ( Idl `  K )  =  { { Z } ,  X }
) )
7 crngorngo 26728 . . . . . . 7  |-  ( K  e. CRingOps  ->  K  e.  RingOps )
8 eqcom 2298 . . . . . . . 8  |-  ( (GId
`  H )  =  Z  <->  Z  =  (GId `  H ) )
91, 2, 3, 4, 50rngo 26755 . . . . . . . 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 2494 . . . . 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 1632    e. wcel 1696    =/= wne 2459   {csn 3653   {cpr 3654   ran crn 4706   ` cfv 5271   1stc1st 6136   2ndc2nd 6137  GIdcgi 20870   RingOpscrngo 21058   Fldcfld 21096  CRingOpsccring 26723   Idlcidl 26735
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-grpo 20874  df-gid 20875  df-ginv 20876  df-ablo 20965  df-ass 20996  df-exid 20998  df-mgm 21002  df-sgr 21014  df-mndo 21021  df-rngo 21059  df-drngo 21089  df-com2 21094  df-fld 21097  df-crngo 26724  df-idl 26738  df-igen 26788
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