Theorem divrngidl 16176

Description: The only ideals in a division ring are the zero ideal and the unit ideal.

Hypotheses

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

Assertion

Ref	Expression
divrngidl	|- (R e. DivRing -> (Idl` R) = {{Z}, X})

Proof of Theorem divrngidl

Step	Hyp	Ref	Expression
1		divrngidl.1	. . 3 |- G = (1st` R)
2		divrngidl.2	. . 3 |- H = (2nd` R)
3		divrngidl.4	. . 3 |- Z = (Id` G)
4		divrngidl.3	. . 3 |- X = ran G
5		eqid 1884	. . 3 |- (Id` H) = (Id` H)
6	1, 2, 3, 4, 5	isdivrng2 16111	. 2 |- (R e. DivRing <-> (R e. Ring /\ ((Id` H) =/= Z /\ A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = (Id` H))))
7	1, 3	idl0cl 16166	. . . . . . . . . . 11 |- ((R e. Ring /\ i e. (Idl` R)) -> Z e. i)
8	7	adantr 425	. . . . . . . . . 10 |- (((R e. Ring /\ i e. (Idl` R)) /\ A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = (Id` H)) -> Z e. i)
9		opreq2 4890	. . . . . . . . . . . . . . . . . . 19 |- (x = z -> (yHx) = (yHz))
10	9	eqeq1d 1892	. . . . . . . . . . . . . . . . . 18 |- (x = z -> ((yHx) = (Id` H) <-> (yHz) = (Id` H)))
11	10	rexbidv 2124	. . . . . . . . . . . . . . . . 17 |- (x = z -> (E.y e. (X \ {Z})(yHx) = (Id` H) <-> E.y e. (X \ {Z})(yHz) = (Id` H)))
12	11	rcla4va 2378	. . . . . . . . . . . . . . . 16 |- ((z e. (X \ {Z}) /\ A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = (Id` H)) -> E.y e. (X \ {Z})(yHz) = (Id` H))
13		ssel2 2616	. . . . . . . . . . . . . . . . . 18 |- (((i \ {Z}) C_ (X \ {Z}) /\ z e. (i \ {Z})) -> z e. (X \ {Z}))
14		ssdif 2740	. . . . . . . . . . . . . . . . . 18 |- (i C_ X -> (i \ {Z}) C_ (X \ {Z}))
15	13, 14	sylan 497	. . . . . . . . . . . . . . . . 17 |- ((i C_ X /\ z e. (i \ {Z})) -> z e. (X \ {Z}))
16	1, 4	idlss 16164	. . . . . . . . . . . . . . . . 17 |- ((R e. Ring /\ i e. (Idl` R)) -> i C_ X)
17	15, 16	sylan 497	. . . . . . . . . . . . . . . 16 |- (((R e. Ring /\ i e. (Idl` R)) /\ z e. (i \ {Z})) -> z e. (X \ {Z}))
18	12, 17	sylan 497	. . . . . . . . . . . . . . 15 |- ((((R e. Ring /\ i e. (Idl` R)) /\ z e. (i \ {Z})) /\ A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = (Id` H)) -> E.y e. (X \ {Z})(yHz) = (Id` H))
19	1, 2, 4	idllmulcl 16168	. . . . . . . . . . . . . . . . . . . 20 |- (((R e. Ring /\ i e. (Idl` R)) /\ (z e. i /\ y e. X)) -> (yHz) e. i)
20		eleq1 1957	. . . . . . . . . . . . . . . . . . . . . 22 |- ((yHz) = (Id` H) -> ((yHz) e. i <-> (Id` H) e. i))
21	20	imbi1d 675	. . . . . . . . . . . . . . . . . . . . 21 |- ((yHz) = (Id` H) -> (((yHz) e. i -> i = X) <-> ((Id` H) e. i -> i = X)))
22	1, 2, 4, 5	1idl 16174	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((R e. Ring /\ i e. (Idl` R)) -> ((Id` H) e. i <-> i = X))
23	22	biimpd 170	. . . . . . . . . . . . . . . . . . . . . 22 |- ((R e. Ring /\ i e. (Idl` R)) -> ((Id` H) e. i -> i = X))
24	23	adantr 425	. . . . . . . . . . . . . . . . . . . . 21 |- (((R e. Ring /\ i e. (Idl` R)) /\ (z e. i /\ y e. X)) -> ((Id` H) e. i -> i = X))
25	21, 24	syl5cbir 228	. . . . . . . . . . . . . . . . . . . 20 |- (((R e. Ring /\ i e. (Idl` R)) /\ (z e. i /\ y e. X)) -> ((yHz) = (Id` H) -> ((yHz) e. i -> i = X)))
26	19, 25	mpid 58	. . . . . . . . . . . . . . . . . . 19 |- (((R e. Ring /\ i e. (Idl` R)) /\ (z e. i /\ y e. X)) -> ((yHz) = (Id` H) -> i = X))
27		eldifi 2730	. . . . . . . . . . . . . . . . . . . 20 |- (z e. (i \ {Z}) -> z e. i)
28		eldifi 2730	. . . . . . . . . . . . . . . . . . . 20 |- (y e. (X \ {Z}) -> y e. X)
29	27, 28	anim12i 360	. . . . . . . . . . . . . . . . . . 19 |- ((z e. (i \ {Z}) /\ y e. (X \ {Z})) -> (z e. i /\ y e. X))
30	26, 29	sylan2 500	. . . . . . . . . . . . . . . . . 18 |- (((R e. Ring /\ i e. (Idl` R)) /\ (z e. (i \ {Z}) /\ y e. (X \ {Z}))) -> ((yHz) = (Id` H) -> i = X))
31	30	anassrs 489	. . . . . . . . . . . . . . . . 17 |- ((((R e. Ring /\ i e. (Idl` R)) /\ z e. (i \ {Z})) /\ y e. (X \ {Z})) -> ((yHz) = (Id` H) -> i = X))
32	31	r19.23adva 2216	. . . . . . . . . . . . . . . 16 |- (((R e. Ring /\ i e. (Idl` R)) /\ z e. (i \ {Z})) -> (E.y e. (X \ {Z})(yHz) = (Id` H) -> i = X))
33	32	imp 377	. . . . . . . . . . . . . . 15 |- ((((R e. Ring /\ i e. (Idl` R)) /\ z e. (i \ {Z})) /\ E.y e. (X \ {Z})(yHz) = (Id` H)) -> i = X)
34	18, 33	syldan 516	. . . . . . . . . . . . . 14 |- ((((R e. Ring /\ i e. (Idl` R)) /\ z e. (i \ {Z})) /\ A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = (Id` H)) -> i = X)
35	34	an1rs 547	. . . . . . . . . . . . 13 |- ((((R e. Ring /\ i e. (Idl` R)) /\ A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = (Id` H)) /\ z e. (i \ {Z})) -> i = X)
36	35	ex 402	. . . . . . . . . . . 12 |- (((R e. Ring /\ i e. (Idl` R)) /\ A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = (Id` H)) -> (z e. (i \ {Z}) -> i = X))
37	36	19.23adv 1584	. . . . . . . . . . 11 |- (((R e. Ring /\ i e. (Idl` R)) /\ A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = (Id` H)) -> (E.z z e. (i \ {Z}) -> i = X))
38		pssdifn0 2936	. . . . . . . . . . . . 13 |- (({Z} C_ i /\ {Z} =/= i) -> (i \ {Z}) =/= (/))
39		n0 2884	. . . . . . . . . . . . 13 |- ((i \ {Z}) =/= (/) <-> E.z z e. (i \ {Z}))
40	38, 39	sylib 215	. . . . . . . . . . . 12 |- (({Z} C_ i /\ {Z} =/= i) -> E.z z e. (i \ {Z}))
41		fvex 4689	. . . . . . . . . . . . . 14 |- (Id` G) e. _V
42	3, 41	eqeltri 1967	. . . . . . . . . . . . 13 |- Z e. _V
43	42	snss 3122	. . . . . . . . . . . 12 |- (Z e. i <-> {Z} C_ i)
44		necom 2094	. . . . . . . . . . . 12 |- (i =/= {Z} <-> {Z} =/= i)
45	40, 43, 44	syl2anb 504	. . . . . . . . . . 11 |- ((Z e. i /\ i =/= {Z}) -> E.z z e. (i \ {Z}))
46	37, 45	syl5 20	. . . . . . . . . 10 |- (((R e. Ring /\ i e. (Idl` R)) /\ A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = (Id` H)) -> ((Z e. i /\ i =/= {Z}) -> i = X))
47	8, 46	mpand 765	. . . . . . . . 9 |- (((R e. Ring /\ i e. (Idl` R)) /\ A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = (Id` H)) -> (i =/= {Z} -> i = X))
48	47	an1rs 547	. . . . . . . 8 |- (((R e. Ring /\ A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = (Id` H)) /\ i e. (Idl` R)) -> (i =/= {Z} -> i = X))
49		neor 2096	. . . . . . . 8 |- ((i = {Z} \/ i = X) <-> (i =/= {Z} -> i = X))
50	48, 49	sylibr 217	. . . . . . 7 |- (((R e. Ring /\ A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = (Id` H)) /\ i e. (Idl` R)) -> (i = {Z} \/ i = X))
51	50	ex 402	. . . . . 6 |- ((R e. Ring /\ A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = (Id` H)) -> (i e. (Idl` R) -> (i = {Z} \/ i = X)))
52		eleq1 1957	. . . . . . . . 9 |- (i = {Z} -> (i e. (Idl` R) <-> {Z} e. (Idl` R)))
53	1, 3	0idl 16173	. . . . . . . . 9 |- (R e. Ring -> {Z} e. (Idl` R))
54	52, 53	syl5cbir 228	. . . . . . . 8 |- (R e. Ring -> (i = {Z} -> i e. (Idl` R)))
55		eleq1 1957	. . . . . . . . 9 |- (i = X -> (i e. (Idl` R) <-> X e. (Idl` R)))
56	1, 4	rngidl 16172	. . . . . . . . 9 |- (R e. Ring -> X e. (Idl` R))
57	55, 56	syl5cbir 228	. . . . . . . 8 |- (R e. Ring -> (i = X -> i e. (Idl` R)))
58	54, 57	jaod 469	. . . . . . 7 |- (R e. Ring -> ((i = {Z} \/ i = X) -> i e. (Idl` R)))
59	58	adantr 425	. . . . . 6 |- ((R e. Ring /\ A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = (Id` H)) -> ((i = {Z} \/ i = X) -> i e. (Idl` R)))
60	51, 59	impbid 574	. . . . 5 |- ((R e. Ring /\ A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = (Id` H)) -> (i e. (Idl` R) <-> (i = {Z} \/ i = X)))
61		visset 2295	. . . . . 6 |- i e. _V
62	61	elpr 3061	. . . . 5 |- (i e. {{Z}, X} <-> (i = {Z} \/ i = X))
63	60, 62	syl6bbr 597	. . . 4 |- ((R e. Ring /\ A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = (Id` H)) -> (i e. (Idl` R) <-> i e. {{Z}, X}))
64	63	eqrdv 1882	. . 3 |- ((R e. Ring /\ A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = (Id` H)) -> (Idl` R) = {{Z}, X})
65	64	adantrl 430	. 2 |- ((R e. Ring /\ ((Id` H) =/= Z /\ A.x e. (X \ {Z})E.y e. (X \ {Z})(yHx) = (Id` H))) -> (Idl` R) = {{Z}, X})
66	6, 65	sylbi 216	1 |- (R e. DivRing -> (Idl` R) = {{Z}, X})

Syntax hints:   -> wi 3   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  A.wral 2105  E.wrex 2106  _Vcvv 2292   \ cdif 2590   C_ wss 2593  (/)c0 2875  {csn 3044  {cpr 3045  ran crn 3987  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Idcgi 9312  Ringcring 9463  DivRingcdrng 9491  Idlcidl 16155

This theorem is referenced by:  divrngpr 16201  isfldidl 16216

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-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-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-idl 16158

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