Theorem 0idl 16173

Description: The set containing only 0 is an ideal.

Hypotheses

Ref	Expression
0idl.1	|- G = (1st` R)
0idl.2	|- Z = (Id` G)

Assertion

Ref	Expression
0idl	|- (R e. Ring -> {Z} e. (Idl` R))

Proof of Theorem 0idl

Step	Hyp	Ref	Expression
1		0idl.1	. . . . 5 |- G = (1st` R)
2		eqid 1884	. . . . 5 |- ran G = ran G
3		0idl.2	. . . . 5 |- Z = (Id` G)
4	1, 2, 3	ring0cl 9484	. . . 4 |- (R e. Ring -> Z e. ran G)
5	4	snssd 3130	. . 3 |- (R e. Ring -> {Z} C_ ran G)
6		fvex 4689	. . . . . 6 |- (Id` G) e. _V
7	3, 6	eqeltri 1967	. . . . 5 |- Z e. _V
8	7	snid 3069	. . . 4 |- Z e. {Z}
9	8	a1i 8	. . 3 |- (R e. Ring -> Z e. {Z})
10		opreq1 4889	. . . . . . . . 9 |- (x = Z -> (xGy) = (ZGy))
11	10	eleq1d 1963	. . . . . . . 8 |- (x = Z -> ((xGy) e. {Z} <-> (ZGy) e. {Z}))
12	11	ralbidv 2123	. . . . . . 7 |- (x = Z -> (A.y e. {Z} (xGy) e. {Z} <-> A.y e. {Z} (ZGy) e. {Z}))
13		opreq2 4890	. . . . . . . . . 10 |- (x = Z -> (z(2nd` R)x) = (z(2nd` R)Z))
14	13	eleq1d 1963	. . . . . . . . 9 |- (x = Z -> ((z(2nd` R)x) e. {Z} <-> (z(2nd` R)Z) e. {Z}))
15		opreq1 4889	. . . . . . . . . 10 |- (x = Z -> (x(2nd` R)z) = (Z(2nd` R)z))
16	15	eleq1d 1963	. . . . . . . . 9 |- (x = Z -> ((x(2nd` R)z) e. {Z} <-> (Z(2nd` R)z) e. {Z}))
17	14, 16	anbi12d 690	. . . . . . . 8 |- (x = Z -> (((z(2nd` R)x) e. {Z} /\ (x(2nd` R)z) e. {Z}) <-> ((z(2nd` R)Z) e. {Z} /\ (Z(2nd` R)z) e. {Z})))
18	17	ralbidv 2123	. . . . . . 7 |- (x = Z -> (A.z e. ran G((z(2nd` R)x) e. {Z} /\ (x(2nd` R)z) e. {Z}) <-> A.z e. ran G((z(2nd` R)Z) e. {Z} /\ (Z(2nd` R)z) e. {Z})))
19	12, 18	anbi12d 690	. . . . . 6 |- (x = Z -> ((A.y e. {Z} (xGy) e. {Z} /\ A.z e. ran G((z(2nd` R)x) e. {Z} /\ (x(2nd` R)z) e. {Z})) <-> (A.y e. {Z} (ZGy) e. {Z} /\ A.z e. ran G((z(2nd` R)Z) e. {Z} /\ (Z(2nd` R)z) e. {Z}))))
20		opreq2 4890	. . . . . . . . . . 11 |- (y = Z -> (ZGy) = (ZGZ))
21	20	eleq1d 1963	. . . . . . . . . 10 |- (y = Z -> ((ZGy) e. {Z} <-> (ZGZ) e. {Z}))
22	1, 2, 3	ring0rid 9485	. . . . . . . . . . . 12 |- ((R e. Ring /\ Z e. ran G) -> (ZGZ) = Z)
23	4, 22	mpdan 768	. . . . . . . . . . 11 |- (R e. Ring -> (ZGZ) = Z)
24		oprex 4907	. . . . . . . . . . . 12 |- (ZGZ) e. _V
25	24	elsnc 3065	. . . . . . . . . . 11 |- ((ZGZ) e. {Z} <-> (ZGZ) = Z)
26	23, 25	sylibr 217	. . . . . . . . . 10 |- (R e. Ring -> (ZGZ) e. {Z})
27	21, 26	syl5cbir 228	. . . . . . . . 9 |- (R e. Ring -> (y = Z -> (ZGy) e. {Z}))
28		elsn 3058	. . . . . . . . 9 |- (y e. {Z} <-> y = Z)
29	27, 28	syl5ib 223	. . . . . . . 8 |- (R e. Ring -> (y e. {Z} -> (ZGy) e. {Z}))
30	29	r19.21aiv 2175	. . . . . . 7 |- (R e. Ring -> A.y e. {Z} (ZGy) e. {Z})
31		eqid 1884	. . . . . . . . . . 11 |- (2nd` R) = (2nd` R)
32	3, 2, 1, 31	ringrz 9488	. . . . . . . . . 10 |- ((R e. Ring /\ z e. ran G) -> (z(2nd` R)Z) = Z)
33		oprex 4907	. . . . . . . . . . 11 |- (z(2nd` R)Z) e. _V
34	33	elsnc 3065	. . . . . . . . . 10 |- ((z(2nd` R)Z) e. {Z} <-> (z(2nd` R)Z) = Z)
35	32, 34	sylibr 217	. . . . . . . . 9 |- ((R e. Ring /\ z e. ran G) -> (z(2nd` R)Z) e. {Z})
36	3, 2, 1, 31	ringlz 9487	. . . . . . . . . 10 |- ((R e. Ring /\ z e. ran G) -> (Z(2nd` R)z) = Z)
37		oprex 4907	. . . . . . . . . . 11 |- (Z(2nd` R)z) e. _V
38	37	elsnc 3065	. . . . . . . . . 10 |- ((Z(2nd` R)z) e. {Z} <-> (Z(2nd` R)z) = Z)
39	36, 38	sylibr 217	. . . . . . . . 9 |- ((R e. Ring /\ z e. ran G) -> (Z(2nd` R)z) e. {Z})
40	35, 39	jca 310	. . . . . . . 8 |- ((R e. Ring /\ z e. ran G) -> ((z(2nd` R)Z) e. {Z} /\ (Z(2nd` R)z) e. {Z}))
41	40	r19.21aiva 2176	. . . . . . 7 |- (R e. Ring -> A.z e. ran G((z(2nd` R)Z) e. {Z} /\ (Z(2nd` R)z) e. {Z}))
42	30, 41	jca 310	. . . . . 6 |- (R e. Ring -> (A.y e. {Z} (ZGy) e. {Z} /\ A.z e. ran G((z(2nd` R)Z) e. {Z} /\ (Z(2nd` R)z) e. {Z})))
43	19, 42	syl5cbir 228	. . . . 5 |- (R e. Ring -> (x = Z -> (A.y e. {Z} (xGy) e. {Z} /\ A.z e. ran G((z(2nd` R)x) e. {Z} /\ (x(2nd` R)z) e. {Z}))))
44		elsn 3058	. . . . 5 |- (x e. {Z} <-> x = Z)
45	43, 44	syl5ib 223	. . . 4 |- (R e. Ring -> (x e. {Z} -> (A.y e. {Z} (xGy) e. {Z} /\ A.z e. ran G((z(2nd` R)x) e. {Z} /\ (x(2nd` R)z) e. {Z}))))
46	45	r19.21aiv 2175	. . 3 |- (R e. Ring -> A.x e. {Z} (A.y e. {Z} (xGy) e. {Z} /\ A.z e. ran G((z(2nd` R)x) e. {Z} /\ (x(2nd` R)z) e. {Z})))
47	5, 9, 46	3jca 1050	. 2 |- (R e. Ring -> ({Z} C_ ran G /\ Z e. {Z} /\ A.x e. {Z} (A.y e. {Z} (xGy) e. {Z} /\ A.z e. ran G((z(2nd` R)x) e. {Z} /\ (x(2nd` R)z) e. {Z}))))
48	1, 31, 2, 3	isidl 16162	. 2 |- (R e. Ring -> ({Z} e. (Idl` R) <-> ({Z} C_ ran G /\ Z e. {Z} /\ A.x e. {Z} (A.y e. {Z} (xGy) e. {Z} /\ A.z e. ran G((z(2nd` R)x) e. {Z} /\ (x(2nd` R)z) e. {Z})))))
49	47, 48	mpbird 213	1 |- (R e. Ring -> {Z} e. (Idl` R))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593  {csn 3044  ran crn 3987  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Idcgi 9312  Ringcring 9463  Idlcidl 16155

This theorem is referenced by:  0ring 16175  divrngidl 16176  smprngpr 16200  isdmn3 16222

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-idl 16158

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