Theorem isfldidl 16216

Description: Determine if a ring is a field based on its ideals.

Hypotheses

Ref	Expression
isfldidl.1	|- G = (1st` K)
isfldidl.2	|- H = (2nd` K)
isfldidl.3	|- X = ran G
isfldidl.4	|- Z = (Id` G)
isfldidl.5	|- U = (Id` H)

Assertion

Ref	Expression
isfldidl	|- (K e. Fld <-> (K e. CRing /\ U =/= Z /\ (Idl` K) = {{Z}, X}))

Proof of Theorem isfldidl

Step	Hyp	Ref	Expression
1		fldcrng 16152	. . 3 |- (K e. Fld -> K e. CRing)
2		flddivrng 16113	. . . 4 |- (K e. Fld -> K e. DivRing)
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 = (Id` G)
7		isfldidl.5	. . . . 5 |- U = (Id` H)
8	3, 4, 5, 6, 7	dvrunz 10419	. . . 4 |- (K e. DivRing -> U =/= Z)
9	2, 8	syl 12	. . 3 |- (K e. Fld -> U =/= Z)
10	3, 4, 5, 6	divrngidl 16176	. . . 4 |- (K e. DivRing -> (Idl` K) = {{Z}, X})
11	2, 10	syl 12	. . 3 |- (K e. Fld -> (Idl` K) = {{Z}, X})
12	1, 9, 11	3jca 1050	. 2 |- (K e. Fld -> (K e. CRing /\ U =/= Z /\ (Idl` K) = {{Z}, X}))
13		isfld2 16153	. . 3 |- (K e. Fld <-> (K e. DivRing /\ K e. CRing))
14		crngrng 16148	. . . . . 6 |- (K e. CRing -> K e. Ring)
15	14	3ad2ant1 897	. . . . 5 |- ((K e. CRing /\ U =/= Z /\ (Idl` K) = {{Z}, X}) -> K e. Ring)
16		simp2 877	. . . . 5 |- ((K e. CRing /\ U =/= Z /\ (Idl` K) = {{Z}, X}) -> U =/= Z)
17	3	rneqi 4187	. . . . . . . . . . . . . . 15 |- ran G = ran (1st` K)
18	5, 17	eqtri 1908	. . . . . . . . . . . . . 14 |- X = ran (1st` K)
19	18, 4, 7	ring1cl 10415	. . . . . . . . . . . . 13 |- (K e. Ring -> U e. X)
20	14, 19	syl 12	. . . . . . . . . . . 12 |- (K e. CRing -> U e. X)
21	20	ad2antrr 440	. . . . . . . . . . 11 |- (((K e. CRing /\ (Idl` K) = {{Z}, X}) /\ x e. (X \ {Z})) -> U e. X)
22		eleq2 1958	. . . . . . . . . . . . . . . . . . . . 21 |- ((K IdlGen {x}) = {Z} -> (x e. (K IdlGen {x}) <-> x e. {Z}))
23		visset 2295	. . . . . . . . . . . . . . . . . . . . . . 23 |- x e. _V
24	23	snss 3122	. . . . . . . . . . . . . . . . . . . . . 22 |- (x e. (K IdlGen {x}) <-> {x} C_ (K IdlGen {x}))
25	24	biimpri 169	. . . . . . . . . . . . . . . . . . . . 21 |- ({x} C_ (K IdlGen {x}) -> x e. (K IdlGen {x}))
26	22, 25	syl5cbi 226	. . . . . . . . . . . . . . . . . . . 20 |- ({x} C_ (K IdlGen {x}) -> ((K IdlGen {x}) = {Z} -> x e. {Z}))
27	26	con3d 111	. . . . . . . . . . . . . . . . . . 19 |- ({x} C_ (K IdlGen {x}) -> (-. x e. {Z} -> -. (K IdlGen {x}) = {Z}))
28	27	imp 377	. . . . . . . . . . . . . . . . . 18 |- (({x} C_ (K IdlGen {x}) /\ -. x e. {Z}) -> -. (K IdlGen {x}) = {Z})
29	3, 5	igenss 16210	. . . . . . . . . . . . . . . . . . 19 |- ((K e. Ring /\ {x} C_ X) -> {x} C_ (K IdlGen {x}))
30		snssi 3129	. . . . . . . . . . . . . . . . . . 19 |- (x e. X -> {x} C_ X)
31	29, 30	sylan2 500	. . . . . . . . . . . . . . . . . 18 |- ((K e. Ring /\ x e. X) -> {x} C_ (K IdlGen {x}))
32	28, 31	sylan 497	. . . . . . . . . . . . . . . . 17 |- (((K e. Ring /\ x e. X) /\ -. x e. {Z}) -> -. (K IdlGen {x}) = {Z})
33	32	anasss 488	. . . . . . . . . . . . . . . 16 |- ((K e. Ring /\ (x e. X /\ -. x e. {Z})) -> -. (K IdlGen {x}) = {Z})
34		eldif 2609	. . . . . . . . . . . . . . . 16 |- (x e. (X \ {Z}) <-> (x e. X /\ -. x e. {Z}))
35	33, 34	sylan2b 501	. . . . . . . . . . . . . . 15 |- ((K e. Ring /\ x e. (X \ {Z})) -> -. (K IdlGen {x}) = {Z})
36	35	adantlr 429	. . . . . . . . . . . . . 14 |- (((K e. Ring /\ (Idl` K) = {{Z}, X}) /\ x e. (X \ {Z})) -> -. (K IdlGen {x}) = {Z})
37		eleq2 1958	. . . . . . . . . . . . . . . . . . 19 |- ((Idl` K) = {{Z}, X} -> ((K IdlGen {x}) e. (Idl` K) <-> (K IdlGen {x}) e. {{Z}, X}))
38	3, 5	igenidl 16211	. . . . . . . . . . . . . . . . . . . 20 |- ((K e. Ring /\ {x} C_ X) -> (K IdlGen {x}) e. (Idl` K))
39		eldifi 2730	. . . . . . . . . . . . . . . . . . . . 21 |- (x e. (X \ {Z}) -> x e. X)
40	39	snssd 3130	. . . . . . . . . . . . . . . . . . . 20 |- (x e. (X \ {Z}) -> {x} C_ X)
41	38, 40	sylan2 500	. . . . . . . . . . . . . . . . . . 19 |- ((K e. Ring /\ x e. (X \ {Z})) -> (K IdlGen {x}) e. (Idl` K))
42	37, 41	syl5cbi 226	. . . . . . . . . . . . . . . . . 18 |- ((K e. Ring /\ x e. (X \ {Z})) -> ((Idl` K) = {{Z}, X} -> (K IdlGen {x}) e. {{Z}, X}))
43	42	imp 377	. . . . . . . . . . . . . . . . 17 |- (((K e. Ring /\ x e. (X \ {Z})) /\ (Idl` K) = {{Z}, X}) -> (K IdlGen {x}) e. {{Z}, X})
44	43	an1rs 547	. . . . . . . . . . . . . . . 16 |- (((K e. Ring /\ (Idl` K) = {{Z}, X}) /\ x e. (X \ {Z})) -> (K IdlGen {x}) e. {{Z}, X})
45		oprex 4907	. . . . . . . . . . . . . . . . 17 |- (K IdlGen {x}) e. _V
46	45	elpr 3061	. . . . . . . . . . . . . . . 16 |- ((K IdlGen {x}) e. {{Z}, X} <-> ((K IdlGen {x}) = {Z} \/ (K IdlGen {x}) = X))
47	44, 46	sylib 215	. . . . . . . . . . . . . . 15 |- (((K e. Ring /\ (Idl` K) = {{Z}, X}) /\ x e. (X \ {Z})) -> ((K IdlGen {x}) = {Z} \/ (K IdlGen {x}) = X))
48	47	ord 249	. . . . . . . . . . . . . 14 |- (((K e. Ring /\ (Idl` K) = {{Z}, X}) /\ x e. (X \ {Z})) -> (-. (K IdlGen {x}) = {Z} -> (K IdlGen {x}) = X))
49	36, 48	mpd 29	. . . . . . . . . . . . 13 |- (((K e. Ring /\ (Idl` K) = {{Z}, X}) /\ x e. (X \ {Z})) -> (K IdlGen {x}) = X)
50	49, 14	sylanl1 509	. . . . . . . . . . . 12 |- (((K e. CRing /\ (Idl` K) = {{Z}, X}) /\ x e. (X \ {Z})) -> (K IdlGen {x}) = X)
51	3, 4, 5	prnc 16215	. . . . . . . . . . . . . 14 |- ((K e. CRing /\ x e. X) -> (K IdlGen {x}) = {z e. X | E.y e. X z = (yHx)})
52	51, 39	sylan2 500	. . . . . . . . . . . . 13 |- ((K e. CRing /\ x e. (X \ {Z})) -> (K IdlGen {x}) = {z e. X | E.y e. X z = (yHx)})
53	52	adantlr 429	. . . . . . . . . . . 12 |- (((K e. CRing /\ (Idl` K) = {{Z}, X}) /\ x e. (X \ {Z})) -> (K IdlGen {x}) = {z e. X | E.y e. X z = (yHx)})
54	50, 53	eqtr3d 1927	. . . . . . . . . . 11 |- (((K e. CRing /\ (Idl` K) = {{Z}, X}) /\ x e. (X \ {Z})) -> X = {z e. X | E.y e. X z = (yHx)})
55	21, 54	eleqtrd 1973	. . . . . . . . . 10 |- (((K e. CRing /\ (Idl` K) = {{Z}, X}) /\ x e. (X \ {Z})) -> U e. {z e. X | E.y e. X z = (yHx)})
56		eqeq1 1890	. . . . . . . . . . . 12 |- (z = U -> (z = (yHx) <-> U = (yHx)))
57	56	rexbidv 2124	. . . . . . . . . . 11 |- (z = U -> (E.y e. X z = (yHx) <-> E.y e. X U = (yHx)))
58	57	elrab 2414	. . . . . . . . . 10 |- (U e. {z e. X | E.y e. X z = (yHx)} <-> (U e. X /\ E.y e. X U = (yHx)))
59	55, 58	sylib 215	. . . . . . . . 9 |- (((K e. CRing /\ (Idl` K) = {{Z}, X}) /\ x e. (X \ {Z})) -> (U e. X /\ E.y e. X U = (yHx)))
60	59	simprd 352	. . . . . . . 8 |- (((K e. CRing /\ (Idl` K) = {{Z}, X}) /\ x e. (X \ {Z})) -> E.y e. X U = (yHx))
61		eqcom 1886	. . . . . . . . 9 |- ((yHx) = U <-> U = (yHx))
62	61	rexbii 2128	. . . . . . . 8 |- (E.y e. X (yHx) = U <-> E.y e. X U = (yHx))
63	60, 62	sylibr 217	. . . . . . 7 |- (((K e. CRing /\ (Idl` K) = {{Z}, X}) /\ x e. (X \ {Z})) -> E.y e. X (yHx) = U)
64	63	r19.21aiva 2176	. . . . . 6 |- ((K e. CRing /\ (Idl` K) = {{Z}, X}) -> A.x e. (X \ {Z})E.y e. X (yHx) = U)
65	64	3adant2 895	. . . . 5 |- ((K e. CRing /\ U =/= Z /\ (Idl` K) = {{Z}, X}) -> A.x e. (X \ {Z})E.y e. X (yHx) = U)
66	15, 16, 65	jca32 312	. . . 4 |- ((K e. CRing /\ U =/= Z /\ (Idl` K) = {{Z}, X}) -> (K e. Ring /\ (U =/= Z /\ A.x e. (X \ {Z})E.y e. X (yHx) = U)))
67	3, 4, 6, 5, 7	isdivrng3 16112	. . . 4 |- (K e. DivRing <-> (K e. Ring /\ (U =/= Z /\ A.x e. (X \ {Z})E.y e. X (yHx) = U)))
68	66, 67	sylibr 217	. . 3 |- ((K e. CRing /\ U =/= Z /\ (Idl` K) = {{Z}, X}) -> K e. DivRing)
69		simp1 876	. . 3 |- ((K e. CRing /\ U =/= Z /\ (Idl` K) = {{Z}, X}) -> K e. CRing)
70	13, 68, 69	sylanbrc 527	. 2 |- ((K e. CRing /\ U =/= Z /\ (Idl` K) = {{Z}, X}) -> K e. Fld)
71	12, 70	impbii 174	1 |- (K e. Fld <-> (K e. CRing /\ U =/= Z /\ (Idl` K) = {{Z}, X}))

Syntax hints:  -. wn 2   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106  {crab 2108   \ cdif 2590   C_ wss 2593  {csn 3044  {cpr 3045  ran crn 3987  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Idcgi 9312  Ringcring 9463  DivRingcdrng 9491  Fldcfld 10397  CRingccring 16143  Idlcidl 16155   IdlGen cigen 16207

This theorem is referenced by:  isfldidl2 16217

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-1o 5177  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-grp 9316  df-gid 9317  df-ginv 9318  df-abl 9408  df-ring 9464  df-drng 9492  df-ass 10360  df-exid 10362  df-mgm 10366  df-sgr 10378  df-mnd 10385  df-com2 10395  df-fld 10398  df-cring 16144  df-idl 16158  df-igen 16208

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