Theorem isfldidl 29011

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.

Step	Hyp	Ref	Expression
1		fldcrng 28947	. . 3 |- ( K e. Fld -> K e. CRingOps )
2		flddivrng 24049	. . . 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 )
8	3, 4, 5, 6, 7	dvrunz 24067	. . . 4 |- ( K e. DivRingOps -> U =/= Z )
9	2, 8	syl 16	. . 3 |- ( K e. Fld -> U =/= Z )
10	3, 4, 5, 6	divrngidl 28971	. . . 4 |- ( K e. DivRingOps -> ( Idl ` K ) = { { Z } , X } )
11	2, 10	syl 16	. . 3 |- ( K e. Fld -> ( Idl ` K ) = { { Z } , X } )
12	1, 9, 11	3jca 1168	. 2 |- ( K e. Fld -> ( K e. CRingOps /\ U =/= Z /\ ( Idl ` K ) = { { Z } , X } ) )
13		crngorngo 28943	. . . . . 6 |- ( K e. CRingOps -> K e. RingOps )
14	13	3ad2ant1 1009	. . . . 5 |- ( ( K e. CRingOps /\ U =/= Z /\ ( Idl ` K ) = { { Z } , X } ) -> K e. RingOps )
15		simp2 989	. . . . 5 |- ( ( K e. CRingOps /\ U =/= Z /\ ( Idl ` K ) = { { Z } , X } ) -> U =/= Z )
16	3	rneqi 5169	. . . . . . . . . . . . . . 15 |- ran G = ran ( 1st ` K )
17	5, 16	eqtri 2481	. . . . . . . . . . . . . 14 |- X = ran ( 1st ` K )
18	17, 4, 7	rngo1cl 24063	. . . . . . . . . . . . 13 |- ( K e. RingOps -> U e. X )
19	13, 18	syl 16	. . . . . . . . . . . 12 |- ( K e. CRingOps -> U e. X )
20	19	ad2antrr 725	. . . . . . . . . . 11 |- ( ( ( K e. CRingOps /\ ( Idl ` K ) = { { Z } , X } ) /\ x e. ( X \ { Z } ) ) -> U e. X )
21		eldif 3441	. . . . . . . . . . . . . . . 16 |- ( x e. ( X \ { Z } ) <-> ( x e. X /\ -. x e. { Z } ) )
22		snssi 4120	. . . . . . . . . . . . . . . . . . 19 |- ( x e. X -> { x } C_ X )
23	3, 5	igenss 29005	. . . . . . . . . . . . . . . . . . 19 |- ( ( K e. RingOps /\ { x } C_ X ) -> { x } C_ ( K IdlGen { x } ) )
24	22, 23	sylan2 474	. . . . . . . . . . . . . . . . . 18 |- ( ( K e. RingOps /\ x e. X ) -> { x } C_ ( K IdlGen { x } ) )
25		vex 3075	. . . . . . . . . . . . . . . . . . . . . 22 |- x e. _V
26	25	snss 4102	. . . . . . . . . . . . . . . . . . . . 21 |- ( x e. ( K IdlGen { x } ) <-> { x } C_ ( K IdlGen { x } ) )
27	26	biimpri 206	. . . . . . . . . . . . . . . . . . . 20 |- ( { x } C_ ( K IdlGen { x } ) -> x e. ( K IdlGen { x } ) )
28		eleq2 2525	. . . . . . . . . . . . . . . . . . . 20 |- ( ( K IdlGen { x } ) = { Z } -> ( x e. ( K IdlGen { x } ) <-> x e. { Z } ) )
29	27, 28	syl5ibcom 220	. . . . . . . . . . . . . . . . . . 19 |- ( { x } C_ ( K IdlGen { x } ) -> ( ( K IdlGen { x } ) = { Z } -> x e. { Z } ) )
30	29	con3dimp 441	. . . . . . . . . . . . . . . . . 18 |- ( ( { x } C_ ( K IdlGen { x } ) /\ -. x e. { Z } ) -> -. ( K IdlGen { x } ) = { Z } )
31	24, 30	sylan 471	. . . . . . . . . . . . . . . . 17 |- ( ( ( K e. RingOps /\ x e. X ) /\ -. x e. { Z } ) -> -. ( K IdlGen { x } ) = { Z } )
32	31	anasss 647	. . . . . . . . . . . . . . . 16 |- ( ( K e. RingOps /\ ( x e. X /\ -. x e. { Z } ) ) -> -. ( K IdlGen { x } ) = { Z } )
33	21, 32	sylan2b 475	. . . . . . . . . . . . . . 15 |- ( ( K e. RingOps /\ x e. ( X \ { Z } ) ) -> -. ( K IdlGen { x } ) = { Z } )
34	33	adantlr 714	. . . . . . . . . . . . . 14 |- ( ( ( K e. RingOps /\ ( Idl ` K ) = { { Z } , X } ) /\ x e. ( X \ { Z } ) ) -> -. ( K IdlGen { x } ) = { Z } )
35		eldifi 3581	. . . . . . . . . . . . . . . . . . . . 21 |- ( x e. ( X \ { Z } ) -> x e. X )
36	35	snssd 4121	. . . . . . . . . . . . . . . . . . . 20 |- ( x e. ( X \ { Z } ) -> { x } C_ X )
37	3, 5	igenidl 29006	. . . . . . . . . . . . . . . . . . . 20 |- ( ( K e. RingOps /\ { x } C_ X ) -> ( K IdlGen { x } ) e. ( Idl ` K ) )
38	36, 37	sylan2 474	. . . . . . . . . . . . . . . . . . 19 |- ( ( K e. RingOps /\ x e. ( X \ { Z } ) ) -> ( K IdlGen { x } ) e. ( Idl ` K ) )
39		eleq2 2525	. . . . . . . . . . . . . . . . . . 19 |- ( ( Idl ` K ) = { { Z } , X } -> ( ( K IdlGen { x } ) e. ( Idl ` K ) <-> ( K IdlGen { x } ) e. { { Z } , X } ) )
40	38, 39	syl5ibcom 220	. . . . . . . . . . . . . . . . . 18 |- ( ( K e. RingOps /\ x e. ( X \ { Z } ) ) -> ( ( Idl ` K ) = { { Z } , X } -> ( K IdlGen { x } ) e. { { Z } , X } ) )
41	40	imp 429	. . . . . . . . . . . . . . . . 17 |- ( ( ( K e. RingOps /\ x e. ( X \ { Z } ) ) /\ ( Idl ` K ) = { { Z } , X } ) -> ( K IdlGen { x } ) e. { { Z } , X } )
42	41	an32s 802	. . . . . . . . . . . . . . . 16 |- ( ( ( K e. RingOps /\ ( Idl ` K ) = { { Z } , X } ) /\ x e. ( X \ { Z } ) ) -> ( K IdlGen { x } ) e. { { Z } , X } )
43		ovex 6220	. . . . . . . . . . . . . . . . 17 |- ( K IdlGen { x } ) e. _V
44	43	elpr 3998	. . . . . . . . . . . . . . . 16 |- ( ( K IdlGen { x } ) e. { { Z } , X } <-> ( ( K IdlGen { x } ) = { Z } \/ ( K IdlGen { x } ) = X ) )
45	42, 44	sylib 196	. . . . . . . . . . . . . . 15 |- ( ( ( K e. RingOps /\ ( Idl ` K ) = { { Z } , X } ) /\ x e. ( X \ { Z } ) ) -> ( ( K IdlGen { x } ) = { Z } \/ ( K IdlGen { x } ) = X ) )
46	45	ord 377	. . . . . . . . . . . . . 14 |- ( ( ( K e. RingOps /\ ( Idl ` K ) = { { Z } , X } ) /\ x e. ( X \ { Z } ) ) -> ( -. ( K IdlGen { x } ) = { Z } -> ( K IdlGen { x } ) = X ) )
47	34, 46	mpd 15	. . . . . . . . . . . . 13 |- ( ( ( K e. RingOps /\ ( Idl ` K ) = { { Z } , X } ) /\ x e. ( X \ { Z } ) ) -> ( K IdlGen { x } ) = X )
48	13, 47	sylanl1 650	. . . . . . . . . . . 12 |- ( ( ( K e. CRingOps /\ ( Idl ` K ) = { { Z } , X } ) /\ x e. ( X \ { Z } ) ) -> ( K IdlGen { x } ) = X )
49	3, 4, 5	prnc 29010	. . . . . . . . . . . . . 14 |- ( ( K e. CRingOps /\ x e. X ) -> ( K IdlGen { x } ) = { z e. X | E. y e. X z = ( y H x ) } )
50	35, 49	sylan2 474	. . . . . . . . . . . . 13 |- ( ( K e. CRingOps /\ x e. ( X \ { Z } ) ) -> ( K IdlGen { x } ) = { z e. X | E. y e. X z = ( y H x ) } )
51	50	adantlr 714	. . . . . . . . . . . 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 ) } )
52	48, 51	eqtr3d 2495	. . . . . . . . . . 11 |- ( ( ( K e. CRingOps /\ ( Idl ` K ) = { { Z } , X } ) /\ x e. ( X \ { Z } ) ) -> X = { z e. X | E. y e. X z = ( y H x ) } )
53	20, 52	eleqtrd 2542	. . . . . . . . . 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 2456	. . . . . . . . . . . 12 |- ( z = U -> ( z = ( y H x ) <-> U = ( y H x ) ) )
55	54	rexbidv 2857	. . . . . . . . . . 11 |- ( z = U -> ( E. y e. X z = ( y H x ) <-> E. y e. X U = ( y H x ) ) )
56	55	elrab 3218	. . . . . . . . . 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 ) ) )
57	53, 56	sylib 196	. . . . . . . . 9 |- ( ( ( K e. CRingOps /\ ( Idl ` K ) = { { Z } , X } ) /\ x e. ( X \ { Z } ) ) -> ( U e. X /\ E. y e. X U = ( y H x ) ) )
58	57	simprd 463	. . . . . . . 8 |- ( ( ( K e. CRingOps /\ ( Idl ` K ) = { { Z } , X } ) /\ x e. ( X \ { Z } ) ) -> E. y e. X U = ( y H x ) )
59		eqcom 2461	. . . . . . . . 9 |- ( ( y H x ) = U <-> U = ( y H x ) )
60	59	rexbii 2861	. . . . . . . 8 |- ( E. y e. X ( y H x ) = U <-> E. y e. X U = ( y H x ) )
61	58, 60	sylibr 212	. . . . . . 7 |- ( ( ( K e. CRingOps /\ ( Idl ` K ) = { { Z } , X } ) /\ x e. ( X \ { Z } ) ) -> E. y e. X ( y H x ) = U )
62	61	ralrimiva 2827	. . . . . 6 |- ( ( K e. CRingOps /\ ( Idl ` K ) = { { Z } , X } ) -> A. x e. ( X \ { Z } ) E. y e. X ( y H x ) = U )
63	62	3adant2 1007	. . . . 5 |- ( ( K e. CRingOps /\ U =/= Z /\ ( Idl ` K ) = { { Z } , X } ) -> A. x e. ( X \ { Z } ) E. y e. X ( y H x ) = U )
64	14, 15, 63	jca32 535	. . . 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 ) ) )
65	3, 4, 6, 5, 7	isdrngo3 28908	. . . 4 |- ( K e. DivRingOps <-> ( K e. RingOps /\ ( U =/= Z /\ A. x e. ( X \ { Z } ) E. y e. X ( y H x ) = U ) ) )
66	64, 65	sylibr 212	. . 3 |- ( ( K e. CRingOps /\ U =/= Z /\ ( Idl ` K ) = { { Z } , X } ) -> K e. DivRingOps )
67		simp1 988	. . 3 |- ( ( K e. CRingOps /\ U =/= Z /\ ( Idl ` K ) = { { Z } , X } ) -> K e. CRingOps )
68		isfld2 28948	. . 3 |- ( K e. Fld <-> ( K e. DivRingOps /\ K e. CRingOps ) )
69	66, 67, 68	sylanbrc 664	. 2 |- ( ( K e. CRingOps /\ U =/= Z /\ ( Idl ` K ) = { { Z } , X } ) -> K e. Fld )
70	12, 69	impbii 188	1 |- ( K e. Fld <-> ( K e. CRingOps /\ U =/= Z /\ ( Idl ` K ) = { { Z } , X } ) )

Syntax hints:   -. wn 3   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2645   A.wral 2796   E.wrex 2797   {crab 2800   \ cdif 3428   C_ wss 3431   {csn 3980   {cpr 3982   ran crn 4944   ` cfv 5521  (class class class)co 6195   1stc1st 6680   2ndc2nd 6681  GIdcgi 23821   RingOpscrngo 24009   DivRingOpscdrng 24039   Fldcfld 24047  CRingOpsccring 28938   Idlcidl 28950   IdlGen cigen 29002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-1o 7025  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-grpo 23825  df-gid 23826  df-ginv 23827  df-ablo 23916  df-ass 23947  df-exid 23949  df-mgm 23953  df-sgr 23965  df-mndo 23972  df-rngo 24010  df-drngo 24040  df-com2 24045  df-fld 24048  df-crngo 28939  df-idl 28953  df-igen 29003

This theorem is referenced by:  isfldidl2  29012