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Theorem isfldidl2 26680
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 2437 . . 3  |-  (GId `  H )  =  (GId
`  H )
61, 2, 3, 4, 5isfldidl 26679 . 2  |-  ( K  e.  Fld  <->  ( K  e. CRingOps 
/\  (GId `  H
)  =/=  Z  /\  ( Idl `  K )  =  { { Z } ,  X }
) )
7 crngorngo 26611 . . . . . . 7  |-  ( K  e. CRingOps  ->  K  e.  RingOps )
8 eqcom 2439 . . . . . . . 8  |-  ( (GId
`  H )  =  Z  <->  Z  =  (GId `  H ) )
91, 2, 3, 4, 50rngo 26638 . . . . . . . 8  |-  ( K  e.  RingOps  ->  ( Z  =  (GId `  H )  <->  X  =  { Z }
) )
108, 9syl5bb 250 . . . . . . 7  |-  ( K  e.  RingOps  ->  ( (GId `  H )  =  Z  <-> 
X  =  { Z } ) )
117, 10syl 16 . . . . . 6  |-  ( K  e. CRingOps  ->  ( (GId `  H )  =  Z  <-> 
X  =  { Z } ) )
1211necon3bid 2637 . . . . 5  |-  ( K  e. CRingOps  ->  ( (GId `  H )  =/=  Z  <->  X  =/=  { Z }
) )
1312anbi1d 687 . . . 4  |-  ( K  e. CRingOps  ->  ( ( (GId
`  H )  =/= 
Z  /\  ( Idl `  K )  =  { { Z } ,  X } )  <->  ( X  =/=  { Z }  /\  ( Idl `  K )  =  { { Z } ,  X }
) ) )
1413pm5.32i 620 . . 3  |-  ( ( K  e. CRingOps  /\  (
(GId `  H )  =/=  Z  /\  ( Idl `  K )  =  { { Z } ,  X } ) )  <->  ( K  e. CRingOps 
/\  ( X  =/= 
{ Z }  /\  ( Idl `  K )  =  { { Z } ,  X }
) ) )
15 3anass 941 . . 3  |-  ( ( K  e. CRingOps  /\  (GId `  H )  =/=  Z  /\  ( Idl `  K
)  =  { { Z } ,  X }
)  <->  ( K  e. CRingOps  /\  ( (GId `  H
)  =/=  Z  /\  ( Idl `  K )  =  { { Z } ,  X }
) ) )
16 3anass 941 . . 3  |-  ( ( K  e. CRingOps  /\  X  =/= 
{ Z }  /\  ( Idl `  K )  =  { { Z } ,  X }
)  <->  ( K  e. CRingOps  /\  ( X  =/=  { Z }  /\  ( Idl `  K )  =  { { Z } ,  X } ) ) )
1714, 15, 163bitr4i 270 . 2  |-  ( ( K  e. CRingOps  /\  (GId `  H )  =/=  Z  /\  ( Idl `  K
)  =  { { Z } ,  X }
)  <->  ( K  e. CRingOps  /\  X  =/=  { Z }  /\  ( Idl `  K
)  =  { { Z } ,  X }
) )
186, 17bitri 242 1  |-  ( K  e.  Fld  <->  ( K  e. CRingOps 
/\  X  =/=  { Z }  /\  ( Idl `  K )  =  { { Z } ,  X } ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   {csn 3815   {cpr 3816   ran crn 4880   ` cfv 5455   1stc1st 6348   2ndc2nd 6349  GIdcgi 21776   RingOpscrngo 21964   Fldcfld 22002  CRingOpsccring 26606   Idlcidl 26618
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-1o 6725  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-grpo 21780  df-gid 21781  df-ginv 21782  df-ablo 21871  df-ass 21902  df-exid 21904  df-mgm 21908  df-sgr 21920  df-mndo 21927  df-rngo 21965  df-drngo 21995  df-com2 22000  df-fld 22003  df-crngo 26607  df-idl 26621  df-igen 26671
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