Theorem rngidl 16172

Description: A ring R is an R ideal.

Hypotheses

Ref	Expression
rngidl.1	|- G = (1st` R)
rngidl.2	|- X = ran G

Assertion

Ref	Expression
rngidl	|- (R e. Ring -> X e. (Idl` R))

Proof of Theorem rngidl

Step	Hyp	Ref	Expression
1		ssid 2634	. . . 4 |- X C_ X
2	1	a1i 8	. . 3 |- (R e. Ring -> X C_ X)
3		rngidl.1	. . . 4 |- G = (1st` R)
4		rngidl.2	. . . 4 |- X = ran G
5		eqid 1884	. . . 4 |- (Id` G) = (Id` G)
6	3, 4, 5	ring0cl 9484	. . 3 |- (R e. Ring -> (Id` G) e. X)
7	3, 4	ringgcl 9477	. . . . . . 7 |- ((R e. Ring /\ x e. X /\ y e. X) -> (xGy) e. X)
8	7	3expa 1067	. . . . . 6 |- (((R e. Ring /\ x e. X) /\ y e. X) -> (xGy) e. X)
9	8	r19.21aiva 2176	. . . . 5 |- ((R e. Ring /\ x e. X) -> A.y e. X (xGy) e. X)
10		eqid 1884	. . . . . . . . . 10 |- (2nd` R) = (2nd` R)
11	3, 10, 4	ringcl 9468	. . . . . . . . 9 |- ((R e. Ring /\ z e. X /\ x e. X) -> (z(2nd` R)x) e. X)
12	11	3com23 1074	. . . . . . . 8 |- ((R e. Ring /\ x e. X /\ z e. X) -> (z(2nd` R)x) e. X)
13	3, 10, 4	ringcl 9468	. . . . . . . 8 |- ((R e. Ring /\ x e. X /\ z e. X) -> (x(2nd` R)z) e. X)
14	12, 13	jca 310	. . . . . . 7 |- ((R e. Ring /\ x e. X /\ z e. X) -> ((z(2nd` R)x) e. X /\ (x(2nd` R)z) e. X))
15	14	3expa 1067	. . . . . 6 |- (((R e. Ring /\ x e. X) /\ z e. X) -> ((z(2nd` R)x) e. X /\ (x(2nd` R)z) e. X))
16	15	r19.21aiva 2176	. . . . 5 |- ((R e. Ring /\ x e. X) -> A.z e. X ((z(2nd` R)x) e. X /\ (x(2nd` R)z) e. X))
17	9, 16	jca 310	. . . 4 |- ((R e. Ring /\ x e. X) -> (A.y e. X (xGy) e. X /\ A.z e. X ((z(2nd` R)x) e. X /\ (x(2nd` R)z) e. X)))
18	17	r19.21aiva 2176	. . 3 |- (R e. Ring -> A.x e. X (A.y e. X (xGy) e. X /\ A.z e. X ((z(2nd` R)x) e. X /\ (x(2nd` R)z) e. X)))
19	2, 6, 18	3jca 1050	. 2 |- (R e. Ring -> (X C_ X /\ (Id` G) e. X /\ A.x e. X (A.y e. X (xGy) e. X /\ A.z e. X ((z(2nd` R)x) e. X /\ (x(2nd` R)z) e. X))))
20	3, 10, 4, 5	isidl 16162	. 2 |- (R e. Ring -> (X e. (Idl` R) <-> (X C_ X /\ (Id` G) e. X /\ A.x e. X (A.y e. X (xGy) e. X /\ A.z e. X ((z(2nd` R)x) e. X /\ (x(2nd` R)z) e. X)))))
21	19, 20	mpbird 213	1 |- (R e. Ring -> X e. (Idl` R))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593  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:  divrngidl 16176  igenval 16209  igenidl 16211

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-abl 9408  df-ring 9464  df-idl 16158

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