Theorem isfldidl2 16217

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

Hypotheses

Ref	Expression
isfldidl2.1	|- G = (1st` K)
isfldidl2.2	|- H = (2nd` K)
isfldidl2.3	|- X = ran G
isfldidl2.4	|- Z = (Id` G)

Assertion

Ref	Expression
isfldidl2	|- (K e. Fld <-> (K e. CRing /\ 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 = (Id` G)
5		eqid 1884	. . 3 |- (Id` H) = (Id` H)
6	1, 2, 3, 4, 5	isfldidl 16216	. 2 |- (K e. Fld <-> (K e. CRing /\ (Id` H) =/= Z /\ (Idl` K) = {{Z}, X}))
7		crngrng 16148	. . . . . . 7 |- (K e. CRing -> K e. Ring)
8	1, 2, 3, 4, 5	0ring 16175	. . . . . . . 8 |- (K e. Ring -> (Z = (Id` H) <-> X = {Z}))
9		eqcom 1886	. . . . . . . 8 |- ((Id` H) = Z <-> Z = (Id` H))
10	8, 9	syl5bb 591	. . . . . . 7 |- (K e. Ring -> ((Id` H) = Z <-> X = {Z}))
11	7, 10	syl 12	. . . . . 6 |- (K e. CRing -> ((Id` H) = Z <-> X = {Z}))
12	11	necon3bid 2035	. . . . 5 |- (K e. CRing -> ((Id` H) =/= Z <-> X =/= {Z}))
13	12	anbi1d 679	. . . 4 |- (K e. CRing -> (((Id` H) =/= Z /\ (Idl` K) = {{Z}, X}) <-> (X =/= {Z} /\ (Idl` K) = {{Z}, X})))
14	13	pm5.32i 707	. . 3 |- ((K e. CRing /\ ((Id` H) =/= Z /\ (Idl` K) = {{Z}, X})) <-> (K e. CRing /\ (X =/= {Z} /\ (Idl` K) = {{Z}, X})))
15		3anass 862	. . 3 |- ((K e. CRing /\ (Id` H) =/= Z /\ (Idl` K) = {{Z}, X}) <-> (K e. CRing /\ ((Id` H) =/= Z /\ (Idl` K) = {{Z}, X})))
16		3anass 862	. . 3 |- ((K e. CRing /\ X =/= {Z} /\ (Idl` K) = {{Z}, X}) <-> (K e. CRing /\ (X =/= {Z} /\ (Idl` K) = {{Z}, X})))
17	14, 15, 16	3bitr4i 200	. 2 |- ((K e. CRing /\ (Id` H) =/= Z /\ (Idl` K) = {{Z}, X}) <-> (K e. CRing /\ X =/= {Z} /\ (Idl` K) = {{Z}, X}))
18	6, 17	bitri 190	1 |- (K e. Fld <-> (K e. CRing /\ X =/= {Z} /\ (Idl` K) = {{Z}, X}))

Syntax hints:   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  {csn 3044  {cpr 3045  ran crn 3987  ` cfv 3998  1stc1st 5018  2ndc2nd 5019  Idcgi 9312  Ringcring 9463  Fldcfld 10397  CRingccring 16143  Idlcidl 16155

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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