Theorem isfldidl 28793

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 28729	. . 3 |- ( K e. Fld -> K e. CRingOps )
2		flddivrng 23837	. . . 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 23855	. . . 4 |- ( K e. DivRingOps -> U =/= Z )
9	2, 8	syl 16	. . 3 |- ( K e. Fld -> U =/= Z )
10	3, 4, 5, 6	divrngidl 28753	. . . 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 1163	. 2 |- ( K e. Fld -> ( K e. CRingOps /\ U =/= Z /\ ( Idl ` K ) = { { Z } , X } ) )
13		crngorngo 28725	. . . . . 6 |- ( K e. CRingOps -> K e. RingOps )
14	13	3ad2ant1 1004	. . . . 5 |- ( ( K e. CRingOps /\ U =/= Z /\ ( Idl ` K ) = { { Z } , X } ) -> K e. RingOps )
15		simp2 984	. . . . 5 |- ( ( K e. CRingOps /\ U =/= Z /\ ( Idl ` K ) = { { Z } , X } ) -> U =/= Z )
16	3	rneqi 5062	. . . . . . . . . . . . . . 15 |- ran G = ran ( 1st ` K )
17	5, 16	eqtri 2461	. . . . . . . . . . . . . 14 |- X = ran ( 1st ` K )
18	17, 4, 7	rngo1cl 23851	. . . . . . . . . . . . 13 |- ( K e. RingOps -> U e. X )
19	13, 18	syl 16	. . . . . . . . . . . 12 |- ( K e. CRingOps -> U e. X )
20	19	ad2antrr 720	. . . . . . . . . . 11 |- ( ( ( K e. CRingOps /\ ( Idl ` K ) = { { Z } , X } ) /\ x e. ( X \ { Z } ) ) -> U e. X )
21		eldif 3335	. . . . . . . . . . . . . . . 16 |- ( x e. ( X \ { Z } ) <-> ( x e. X /\ -. x e. { Z } ) )
22		snssi 4014	. . . . . . . . . . . . . . . . . . 19 |- ( x e. X -> { x } C_ X )
23	3, 5	igenss 28787	. . . . . . . . . . . . . . . . . . 19 |- ( ( K e. RingOps /\ { x } C_ X ) -> { x } C_ ( K IdlGen { x } ) )
24	22, 23	sylan2 471	. . . . . . . . . . . . . . . . . 18 |- ( ( K e. RingOps /\ x e. X ) -> { x } C_ ( K IdlGen { x } ) )
25		vex 2973	. . . . . . . . . . . . . . . . . . . . . 22 |- x e. _V
26	25	snss 3996	. . . . . . . . . . . . . . . . . . . . 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 2502	. . . . . . . . . . . . . . . . . . . 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	con3and 439	. . . . . . . . . . . . . . . . . 18 |- ( ( { x } C_ ( K IdlGen { x } ) /\ -. x e. { Z } ) -> -. ( K IdlGen { x } ) = { Z } )
31	24, 30	sylan 468	. . . . . . . . . . . . . . . . 17 |- ( ( ( K e. RingOps /\ x e. X ) /\ -. x e. { Z } ) -> -. ( K IdlGen { x } ) = { Z } )
32	31	anasss 642	. . . . . . . . . . . . . . . 16 |- ( ( K e. RingOps /\ ( x e. X /\ -. x e. { Z } ) ) -> -. ( K IdlGen { x } ) = { Z } )
33	21, 32	sylan2b 472	. . . . . . . . . . . . . . 15 |- ( ( K e. RingOps /\ x e. ( X \ { Z } ) ) -> -. ( K IdlGen { x } ) = { Z } )
34	33	adantlr 709	. . . . . . . . . . . . . 14 |- ( ( ( K e. RingOps /\ ( Idl ` K ) = { { Z } , X } ) /\ x e. ( X \ { Z } ) ) -> -. ( K IdlGen { x } ) = { Z } )
35		eldifi 3475	. . . . . . . . . . . . . . . . . . . . 21 |- ( x e. ( X \ { Z } ) -> x e. X )
36	35	snssd 4015	. . . . . . . . . . . . . . . . . . . 20 |- ( x e. ( X \ { Z } ) -> { x } C_ X )
37	3, 5	igenidl 28788	. . . . . . . . . . . . . . . . . . . 20 |- ( ( K e. RingOps /\ { x } C_ X ) -> ( K IdlGen { x } ) e. ( Idl ` K ) )
38	36, 37	sylan2 471	. . . . . . . . . . . . . . . . . . 19 |- ( ( K e. RingOps /\ x e. ( X \ { Z } ) ) -> ( K IdlGen { x } ) e. ( Idl ` K ) )
39		eleq2 2502	. . . . . . . . . . . . . . . . . . 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 797	. . . . . . . . . . . . . . . 16 |- ( ( ( K e. RingOps /\ ( Idl ` K ) = { { Z } , X } ) /\ x e. ( X \ { Z } ) ) -> ( K IdlGen { x } ) e. { { Z } , X } )
43		ovex 6115	. . . . . . . . . . . . . . . . 17 |- ( K IdlGen { x } ) e. _V
44	43	elpr 3892	. . . . . . . . . . . . . . . 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 645	. . . . . . . . . . . 12 |- ( ( ( K e. CRingOps /\ ( Idl ` K ) = { { Z } , X } ) /\ x e. ( X \ { Z } ) ) -> ( K IdlGen { x } ) = X )
49	3, 4, 5	prnc 28792	. . . . . . . . . . . . . 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 471	. . . . . . . . . . . . 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 709	. . . . . . . . . . . 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 2475	. . . . . . . . . . 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 2517	. . . . . . . . . 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 2447	. . . . . . . . . . . 12 |- ( z = U -> ( z = ( y H x ) <-> U = ( y H x ) ) )
55	54	rexbidv 2734	. . . . . . . . . . 11 |- ( z = U -> ( E. y e. X z = ( y H x ) <-> E. y e. X U = ( y H x ) ) )
56	55	elrab 3114	. . . . . . . . . 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 460	. . . . . . . 8 |- ( ( ( K e. CRingOps /\ ( Idl ` K ) = { { Z } , X } ) /\ x e. ( X \ { Z } ) ) -> E. y e. X U = ( y H x ) )
59		eqcom 2443	. . . . . . . . 9 |- ( ( y H x ) = U <-> U = ( y H x ) )
60	59	rexbii 2738	. . . . . . . 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 2797	. . . . . 6 |- ( ( K e. CRingOps /\ ( Idl ` K ) = { { Z } , X } ) -> A. x e. ( X \ { Z } ) E. y e. X ( y H x ) = U )
63	62	3adant2 1002	. . . . 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 532	. . . 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 28690	. . . 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 983	. . 3 |- ( ( K e. CRingOps /\ U =/= Z /\ ( Idl ` K ) = { { Z } , X } ) -> K e. CRingOps )
68		isfld2 28730	. . 3 |- ( K e. Fld <-> ( K e. DivRingOps /\ K e. CRingOps ) )
69	66, 67, 68	sylanbrc 659	. 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 960   = wceq 1364   e. wcel 1761   =/= wne 2604   A.wral 2713   E.wrex 2714   {crab 2717   \ cdif 3322   C_ wss 3325   {csn 3874   {cpr 3876   ran crn 4837   ` cfv 5415  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575  GIdcgi 23609   RingOpscrngo 23797   DivRingOpscdrng 23827   Fldcfld 23835  CRingOpsccring 28720   Idlcidl 28732   IdlGen cigen 28784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-1o 6916  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-grpo 23613  df-gid 23614  df-ginv 23615  df-ablo 23704  df-ass 23735  df-exid 23737  df-mgm 23741  df-sgr 23753  df-mndo 23760  df-rngo 23798  df-drngo 23828  df-com2 23833  df-fld 23836  df-crngo 28721  df-idl 28735  df-igen 28785

This theorem is referenced by:  isfldidl2  28794