Theorem smprngpr 16200

Description: A simple ring (one whose only ideals are 0 and R) is a prime ring.

Hypotheses

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

Assertion

Ref	Expression
smprngpr	|- ((R e. Ring /\ U =/= Z /\ (Idl` R) = {{Z}, X}) -> R e. PrRing)

Proof of Theorem smprngpr

Step	Hyp	Ref	Expression
1		smprngpr.1	. . 3 |- G = (1st` R)
2		smprngpr.4	. . 3 |- Z = (Id` G)
3	1, 2	isprrng 16198	. 2 |- (R e. PrRing <-> (R e. Ring /\ {Z} e. (PrIdl` R)))
4		simp1 876	. 2 |- ((R e. Ring /\ U =/= Z /\ (Idl` R) = {{Z}, X}) -> R e. Ring)
5	1, 2	0idl 16173	. . . . 5 |- (R e. Ring -> {Z} e. (Idl` R))
6	5	3ad2ant1 897	. . . 4 |- ((R e. Ring /\ U =/= Z /\ (Idl` R) = {{Z}, X}) -> {Z} e. (Idl` R))
7		smprngpr.2	. . . . . . . . 9 |- H = (2nd` R)
8		smprngpr.3	. . . . . . . . 9 |- X = ran G
9		smprngpr.5	. . . . . . . . 9 |- U = (Id` H)
10	1, 7, 8, 2, 9	0ring 16175	. . . . . . . 8 |- (R e. Ring -> (Z = U <-> X = {Z}))
11		eqcom 1886	. . . . . . . 8 |- (U = Z <-> Z = U)
12		eqcom 1886	. . . . . . . 8 |- ({Z} = X <-> X = {Z})
13	10, 11, 12	3bitr4g 614	. . . . . . 7 |- (R e. Ring -> (U = Z <-> {Z} = X))
14	13	necon3bid 2035	. . . . . 6 |- (R e. Ring -> (U =/= Z <-> {Z} =/= X))
15	14	biimpa 460	. . . . 5 |- ((R e. Ring /\ U =/= Z) -> {Z} =/= X)
16	15	3adant3 896	. . . 4 |- ((R e. Ring /\ U =/= Z /\ (Idl` R) = {{Z}, X}) -> {Z} =/= X)
17		df-pr 3050	. . . . . . . . 9 |- {{Z}, X} = ({{Z}} u. {X})
18	17	eqeq2i 1894	. . . . . . . 8 |- ((Idl` R) = {{Z}, X} <-> (Idl` R) = ({{Z}} u. {X}))
19		eleq2 1958	. . . . . . . . . 10 |- ((Idl` R) = ({{Z}} u. {X}) -> (i e. (Idl` R) <-> i e. ({{Z}} u. {X})))
20		eleq2 1958	. . . . . . . . . 10 |- ((Idl` R) = ({{Z}} u. {X}) -> (j e. (Idl` R) <-> j e. ({{Z}} u. {X})))
21	19, 20	anbi12d 690	. . . . . . . . 9 |- ((Idl` R) = ({{Z}} u. {X}) -> ((i e. (Idl` R) /\ j e. (Idl` R)) <-> (i e. ({{Z}} u. {X}) /\ j e. ({{Z}} u. {X}))))
22		elun 2741	. . . . . . . . . . 11 |- (i e. ({{Z}} u. {X}) <-> (i e. {{Z}} \/ i e. {X}))
23		elsn 3058	. . . . . . . . . . . 12 |- (i e. {{Z}} <-> i = {Z})
24		elsn 3058	. . . . . . . . . . . 12 |- (i e. {X} <-> i = X)
25	23, 24	orbi12i 277	. . . . . . . . . . 11 |- ((i e. {{Z}} \/ i e. {X}) <-> (i = {Z} \/ i = X))
26	22, 25	bitri 190	. . . . . . . . . 10 |- (i e. ({{Z}} u. {X}) <-> (i = {Z} \/ i = X))
27		elun 2741	. . . . . . . . . . 11 |- (j e. ({{Z}} u. {X}) <-> (j e. {{Z}} \/ j e. {X}))
28		elsn 3058	. . . . . . . . . . . 12 |- (j e. {{Z}} <-> j = {Z})
29		elsn 3058	. . . . . . . . . . . 12 |- (j e. {X} <-> j = X)
30	28, 29	orbi12i 277	. . . . . . . . . . 11 |- ((j e. {{Z}} \/ j e. {X}) <-> (j = {Z} \/ j = X))
31	27, 30	bitri 190	. . . . . . . . . 10 |- (j e. ({{Z}} u. {X}) <-> (j = {Z} \/ j = X))
32	26, 31	anbi12i 540	. . . . . . . . 9 |- ((i e. ({{Z}} u. {X}) /\ j e. ({{Z}} u. {X})) <-> ((i = {Z} \/ i = X) /\ (j = {Z} \/ j = X)))
33	21, 32	syl6bb 595	. . . . . . . 8 |- ((Idl` R) = ({{Z}} u. {X}) -> ((i e. (Idl` R) /\ j e. (Idl` R)) <-> ((i = {Z} \/ i = X) /\ (j = {Z} \/ j = X))))
34	18, 33	sylbi 216	. . . . . . 7 |- ((Idl` R) = {{Z}, X} -> ((i e. (Idl` R) /\ j e. (Idl` R)) <-> ((i = {Z} \/ i = X) /\ (j = {Z} \/ j = X))))
35	34	3ad2ant3 899	. . . . . 6 |- ((R e. Ring /\ U =/= Z /\ (Idl` R) = {{Z}, X}) -> ((i e. (Idl` R) /\ j e. (Idl` R)) <-> ((i = {Z} \/ i = X) /\ (j = {Z} \/ j = X))))
36		eqimss 2665	. . . . . . . . . . . 12 |- (i = {Z} -> i C_ {Z})
37	36	orcd 294	. . . . . . . . . . 11 |- (i = {Z} -> (i C_ {Z} \/ j C_ {Z}))
38	37	adantr 425	. . . . . . . . . 10 |- ((i = {Z} /\ j = {Z}) -> (i C_ {Z} \/ j C_ {Z}))
39	38	a1d 15	. . . . . . . . 9 |- ((i = {Z} /\ j = {Z}) -> (A.x e. i A.y e. j (xHy) e. {Z} -> (i C_ {Z} \/ j C_ {Z})))
40	39	a1i 8	. . . . . . . 8 |- ((R e. Ring /\ U =/= Z) -> ((i = {Z} /\ j = {Z}) -> (A.x e. i A.y e. j (xHy) e. {Z} -> (i C_ {Z} \/ j C_ {Z}))))
41		eqimss 2665	. . . . . . . . . . . 12 |- (j = {Z} -> j C_ {Z})
42	41	olcd 295	. . . . . . . . . . 11 |- (j = {Z} -> (i C_ {Z} \/ j C_ {Z}))
43	42	adantl 424	. . . . . . . . . 10 |- ((i = X /\ j = {Z}) -> (i C_ {Z} \/ j C_ {Z}))
44	43	a1d 15	. . . . . . . . 9 |- ((i = X /\ j = {Z}) -> (A.x e. i A.y e. j (xHy) e. {Z} -> (i C_ {Z} \/ j C_ {Z})))
45	44	a1i 8	. . . . . . . 8 |- ((R e. Ring /\ U =/= Z) -> ((i = X /\ j = {Z}) -> (A.x e. i A.y e. j (xHy) e. {Z} -> (i C_ {Z} \/ j C_ {Z}))))
46	37	adantr 425	. . . . . . . . . 10 |- ((i = {Z} /\ j = X) -> (i C_ {Z} \/ j C_ {Z}))
47	46	a1d 15	. . . . . . . . 9 |- ((i = {Z} /\ j = X) -> (A.x e. i A.y e. j (xHy) e. {Z} -> (i C_ {Z} \/ j C_ {Z})))
48	47	a1i 8	. . . . . . . 8 |- ((R e. Ring /\ U =/= Z) -> ((i = {Z} /\ j = X) -> (A.x e. i A.y e. j (xHy) e. {Z} -> (i C_ {Z} \/ j C_ {Z}))))
49		raleq 2266	. . . . . . . . . . 11 |- (i = X -> (A.x e. i A.y e. j (xHy) e. {Z} <-> A.x e. X A.y e. j (xHy) e. {Z}))
50		raleq 2266	. . . . . . . . . . . 12 |- (j = X -> (A.y e. j (xHy) e. {Z} <-> A.y e. X (xHy) e. {Z}))
51	50	ralbidv 2123	. . . . . . . . . . 11 |- (j = X -> (A.x e. X A.y e. j (xHy) e. {Z} <-> A.x e. X A.y e. X (xHy) e. {Z}))
52	49, 51	sylan9bb 599	. . . . . . . . . 10 |- ((i = X /\ j = X) -> (A.x e. i A.y e. j (xHy) e. {Z} <-> A.x e. X A.y e. X (xHy) e. {Z}))
53	52	imbi1d 675	. . . . . . . . 9 |- ((i = X /\ j = X) -> ((A.x e. i A.y e. j (xHy) e. {Z} -> (i C_ {Z} \/ j C_ {Z})) <-> (A.x e. X A.y e. X (xHy) e. {Z} -> (i C_ {Z} \/ j C_ {Z}))))
54	1	rneqi 4187	. . . . . . . . . . . . . . 15 |- ran G = ran (1st` R)
55	8, 54	eqtri 1908	. . . . . . . . . . . . . 14 |- X = ran (1st` R)
56	55, 7, 9	ring1cl 10415	. . . . . . . . . . . . 13 |- (R e. Ring -> U e. X)
57	56	adantr 425	. . . . . . . . . . . 12 |- ((R e. Ring /\ U =/= Z) -> U e. X)
58	7, 55, 9	ringlidm 10410	. . . . . . . . . . . . . . . . 17 |- ((R e. Ring /\ U e. X) -> (UHU) = U)
59	56, 58	mpdan 768	. . . . . . . . . . . . . . . 16 |- (R e. Ring -> (UHU) = U)
60	59	eleq1d 1963	. . . . . . . . . . . . . . 15 |- (R e. Ring -> ((UHU) e. {Z} <-> U e. {Z}))
61		fvex 4689	. . . . . . . . . . . . . . . . 17 |- (Id` H) e. _V
62	9, 61	eqeltri 1967	. . . . . . . . . . . . . . . 16 |- U e. _V
63	62	elsnc 3065	. . . . . . . . . . . . . . 15 |- (U e. {Z} <-> U = Z)
64	60, 63	syl6bb 595	. . . . . . . . . . . . . 14 |- (R e. Ring -> ((UHU) e. {Z} <-> U = Z))
65	64	necon3bbid 2034	. . . . . . . . . . . . 13 |- (R e. Ring -> (-. (UHU) e. {Z} <-> U =/= Z))
66	65	biimpar 461	. . . . . . . . . . . 12 |- ((R e. Ring /\ U =/= Z) -> -. (UHU) e. {Z})
67		opreq1 4889	. . . . . . . . . . . . . . 15 |- (x = U -> (xHy) = (UHy))
68	67	eleq1d 1963	. . . . . . . . . . . . . 14 |- (x = U -> ((xHy) e. {Z} <-> (UHy) e. {Z}))
69	68	notbid 673	. . . . . . . . . . . . 13 |- (x = U -> (-. (xHy) e. {Z} <-> -. (UHy) e. {Z}))
70		opreq2 4890	. . . . . . . . . . . . . . 15 |- (y = U -> (UHy) = (UHU))
71	70	eleq1d 1963	. . . . . . . . . . . . . 14 |- (y = U -> ((UHy) e. {Z} <-> (UHU) e. {Z}))
72	71	notbid 673	. . . . . . . . . . . . 13 |- (y = U -> (-. (UHy) e. {Z} <-> -. (UHU) e. {Z}))
73	69, 72	rcla42ev 2385	. . . . . . . . . . . 12 |- ((U e. X /\ U e. X /\ -. (UHU) e. {Z}) -> E.x e. X E.y e. X -. (xHy) e. {Z})
74	57, 57, 66, 73	syl111anc 1100	. . . . . . . . . . 11 |- ((R e. Ring /\ U =/= Z) -> E.x e. X E.y e. X -. (xHy) e. {Z})
75		rexnal 2114	. . . . . . . . . . . . 13 |- (E.y e. X -. (xHy) e. {Z} <-> -. A.y e. X (xHy) e. {Z})
76	75	rexbii 2128	. . . . . . . . . . . 12 |- (E.x e. X E.y e. X -. (xHy) e. {Z} <-> E.x e. X -. A.y e. X (xHy) e. {Z})
77		rexnal 2114	. . . . . . . . . . . 12 |- (E.x e. X -. A.y e. X (xHy) e. {Z} <-> -. A.x e. X A.y e. X (xHy) e. {Z})
78	76, 77	bitri 190	. . . . . . . . . . 11 |- (E.x e. X E.y e. X -. (xHy) e. {Z} <-> -. A.x e. X A.y e. X (xHy) e. {Z})
79	74, 78	sylib 215	. . . . . . . . . 10 |- ((R e. Ring /\ U =/= Z) -> -. A.x e. X A.y e. X (xHy) e. {Z})
80	79	pm2.21d 94	. . . . . . . . 9 |- ((R e. Ring /\ U =/= Z) -> (A.x e. X A.y e. X (xHy) e. {Z} -> (i C_ {Z} \/ j C_ {Z})))
81	53, 80	syl5cbir 228	. . . . . . . 8 |- ((R e. Ring /\ U =/= Z) -> ((i = X /\ j = X) -> (A.x e. i A.y e. j (xHy) e. {Z} -> (i C_ {Z} \/ j C_ {Z}))))
82	40, 45, 48, 81	ccased 830	. . . . . . 7 |- ((R e. Ring /\ U =/= Z) -> (((i = {Z} \/ i = X) /\ (j = {Z} \/ j = X)) -> (A.x e. i A.y e. j (xHy) e. {Z} -> (i C_ {Z} \/ j C_ {Z}))))
83	82	3adant3 896	. . . . . 6 |- ((R e. Ring /\ U =/= Z /\ (Idl` R) = {{Z}, X}) -> (((i = {Z} \/ i = X) /\ (j = {Z} \/ j = X)) -> (A.x e. i A.y e. j (xHy) e. {Z} -> (i C_ {Z} \/ j C_ {Z}))))
84	35, 83	sylbid 220	. . . . 5 |- ((R e. Ring /\ U =/= Z /\ (Idl` R) = {{Z}, X}) -> ((i e. (Idl` R) /\ j e. (Idl` R)) -> (A.x e. i A.y e. j (xHy) e. {Z} -> (i C_ {Z} \/ j C_ {Z}))))
85	84	r19.21aivv 2183	. . . 4 |- ((R e. Ring /\ U =/= Z /\ (Idl` R) = {{Z}, X}) -> A.i e. (Idl` R)A.j e. (Idl` R)(A.x e. i A.y e. j (xHy) e. {Z} -> (i C_ {Z} \/ j C_ {Z})))
86	6, 16, 85	3jca 1050	. . 3 |- ((R e. Ring /\ U =/= Z /\ (Idl` R) = {{Z}, X}) -> ({Z} e. (Idl` R) /\ {Z} =/= X /\ A.i e. (Idl` R)A.j e. (Idl` R)(A.x e. i A.y e. j (xHy) e. {Z} -> (i C_ {Z} \/ j C_ {Z}))))
87	1, 7, 8	ispridl 16182	. . . 4 |- (R e. Ring -> ({Z} e. (PrIdl` R) <-> ({Z} e. (Idl` R) /\ {Z} =/= X /\ A.i e. (Idl` R)A.j e. (Idl` R)(A.x e. i A.y e. j (xHy) e. {Z} -> (i C_ {Z} \/ j C_ {Z})))))
88	87	3ad2ant1 897	. . 3 |- ((R e. Ring /\ U =/= Z /\ (Idl` R) = {{Z}, X}) -> ({Z} e. (PrIdl` R) <-> ({Z} e. (Idl` R) /\ {Z} =/= X /\ A.i e. (Idl` R)A.j e. (Idl` R)(A.x e. i A.y e. j (xHy) e. {Z} -> (i C_ {Z} \/ j C_ {Z})))))
89	86, 88	mpbird 213	. 2 |- ((R e. Ring /\ U =/= Z /\ (Idl` R) = {{Z}, X}) -> {Z} e. (PrIdl` R))
90	3, 4, 89	sylanbrc 527	1 |- ((R e. Ring /\ U =/= Z /\ (Idl` R) = {{Z}, X}) -> R e. PrRing)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106  _Vcvv 2292   u. cun 2591   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  Idlcidl 16155  PrIdlcpridl 16156  PrRingcprrng 16194

This theorem is referenced by:  divrngpr 16201

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-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-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  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-grp 9316  df-gid 9317  df-ginv 9318  df-abl 9408  df-ring 9464  df-ass 10360  df-exid 10362  df-mgm 10366  df-sgr 10378  df-mnd 10385  df-idl 16158  df-pridl 16159  df-prrng 16196

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