Theorem isfldidl2 33868

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

Step	Hyp	Ref	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 2622	. . 3 |- (GId ` H ) = (GId ` H )
6	1, 2, 3, 4, 5	isfldidl 33867	. 2 |- ( K e. Fld <-> ( K e. CRingOps /\ (GId ` H ) =/= Z /\ ( Idl ` K ) = { { Z } , X } ) )
7		crngorngo 33799	. . . . . . 7 |- ( K e. CRingOps -> K e. RingOps )
8		eqcom 2629	. . . . . . . 8 |- ( (GId ` H ) = Z <-> Z = (GId ` H ) )
9	1, 2, 3, 4, 5	0rngo 33826	. . . . . . . 8 |- ( K e. RingOps -> ( Z = (GId ` H ) <-> X = { Z } ) )
10	8, 9	syl5bb 272	. . . . . . 7 |- ( K e. RingOps -> ( (GId ` H ) = Z <-> X = { Z } ) )
11	7, 10	syl 17	. . . . . 6 |- ( K e. CRingOps -> ( (GId ` H ) = Z <-> X = { Z } ) )
12	11	necon3bid 2838	. . . . 5 |- ( K e. CRingOps -> ( (GId ` H ) =/= Z <-> X =/= { Z } ) )
13	12	anbi1d 741	. . . 4 |- ( K e. CRingOps -> ( ( (GId ` H ) =/= Z /\ ( Idl ` K ) = { { Z } , X } ) <-> ( X =/= { Z } /\ ( Idl ` K ) = { { Z } , X } ) ) )
14	13	pm5.32i 669	. . 3 |- ( ( K e. CRingOps /\ ( (GId ` H ) =/= Z /\ ( Idl ` K ) = { { Z } , X } ) ) <-> ( K e. CRingOps /\ ( X =/= { Z } /\ ( Idl ` K ) = { { Z } , X } ) ) )
15		3anass 1042	. . 3 |- ( ( K e. CRingOps /\ (GId ` H ) =/= Z /\ ( Idl ` K ) = { { Z } , X } ) <-> ( K e. CRingOps /\ ( (GId ` H ) =/= Z /\ ( Idl ` K ) = { { Z } , X } ) ) )
16		3anass 1042	. . 3 |- ( ( K e. CRingOps /\ X =/= { Z } /\ ( Idl ` K ) = { { Z } , X } ) <-> ( K e. CRingOps /\ ( X =/= { Z } /\ ( Idl ` K ) = { { Z } , X } ) ) )
17	14, 15, 16	3bitr4i 292	. 2 |- ( ( K e. CRingOps /\ (GId ` H ) =/= Z /\ ( Idl ` K ) = { { Z } , X } ) <-> ( K e. CRingOps /\ X =/= { Z } /\ ( Idl ` K ) = { { Z } , X } ) )
18	6, 17	bitri 264	1 |- ( K e. Fld <-> ( K e. CRingOps /\ X =/= { Z } /\ ( Idl ` K ) = { { Z } , X } ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {csn 4177   {cpr 4179   ran crn 5115   ` cfv 5888   1stc1st 7166   2ndc2nd 7167  GIdcgi 27344   RingOpscrngo 33693   Fldcfld 33790  CRingOpsccring 33792   Idlcidl 33806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-ass 33642  df-exid 33644  df-mgmOLD 33648  df-sgrOLD 33660  df-mndo 33666  df-rngo 33694  df-drngo 33748  df-com2 33789  df-fld 33791  df-crngo 33793  df-idl 33809  df-igen 33859

This theorem is referenced by: (None)