Theorem idlnegcl 16170

Description: An ideal is closed under negation.

Hypotheses

Ref	Expression
idlnegcl.1	|- G = (1st` R)
idlnegcl.2	|- N = (inv` G)

Assertion

Ref	Expression
idlnegcl	|- (((R e. Ring /\ I e. (Idl` R)) /\ A e. I) -> (N` A) e. I)

Proof of Theorem idlnegcl

Step	Hyp	Ref	Expression
1		idlnegcl.1	. . . 4 |- G = (1st` R)
2		eqid 1884	. . . 4 |- ran G = ran G
3	1, 2	idlss 16164	. . 3 |- ((R e. Ring /\ I e. (Idl` R)) -> I C_ ran G)
4		eqid 1884	. . . . . 6 |- (2nd` R) = (2nd` R)
5		idlnegcl.2	. . . . . 6 |- N = (inv` G)
6		eqid 1884	. . . . . 6 |- (Id` (2nd` R)) = (Id` (2nd` R))
7	1, 4, 2, 5, 6	ringnegmn1l 16102	. . . . 5 |- ((R e. Ring /\ A e. ran G) -> (N` A) = ((N` (Id` (2nd` R)))(2nd` R)A))
8		ssel2 2616	. . . . 5 |- ((I C_ ran G /\ A e. I) -> A e. ran G)
9	7, 8	sylan2 500	. . . 4 |- ((R e. Ring /\ (I C_ ran G /\ A e. I)) -> (N` A) = ((N` (Id` (2nd` R)))(2nd` R)A))
10	9	anassrs 489	. . 3 |- (((R e. Ring /\ I C_ ran G) /\ A e. I) -> (N` A) = ((N` (Id` (2nd` R)))(2nd` R)A))
11	3, 10	syldanl 15649	. 2 |- (((R e. Ring /\ I e. (Idl` R)) /\ A e. I) -> (N` A) = ((N` (Id` (2nd` R)))(2nd` R)A))
12	1	rneqi 4187	. . . . . 6 |- ran G = ran (1st` R)
13	12, 4, 6	ring1cl 10415	. . . . 5 |- (R e. Ring -> (Id` (2nd` R)) e. ran G)
14	1, 2, 5	ringnegcl 16098	. . . . 5 |- ((R e. Ring /\ (Id` (2nd` R)) e. ran G) -> (N` (Id` (2nd` R))) e. ran G)
15	13, 14	mpdan 768	. . . 4 |- (R e. Ring -> (N` (Id` (2nd` R))) e. ran G)
16	15	ad2antrr 440	. . 3 |- (((R e. Ring /\ I e. (Idl` R)) /\ A e. I) -> (N` (Id` (2nd` R))) e. ran G)
17	1, 4, 2	idllmulcl 16168	. . . 4 |- (((R e. Ring /\ I e. (Idl` R)) /\ (A e. I /\ (N` (Id` (2nd` R))) e. ran G)) -> ((N` (Id` (2nd` R)))(2nd` R)A) e. I)
18	17	anassrs 489	. . 3 |- ((((R e. Ring /\ I e. (Idl` R)) /\ A e. I) /\ (N` (Id` (2nd` R))) e. ran G) -> ((N` (Id` (2nd` R)))(2nd` R)A) e. I)
19	16, 18	mpdan 768	. 2 |- (((R e. Ring /\ I e. (Idl` R)) /\ A e. I) -> ((N` (Id` (2nd` R)))(2nd` R)A) e. I)
20	11, 19	eqeltrd 1971	1 |- (((R e. Ring /\ I e. (Idl` R)) /\ A e. I) -> (N` A) e. I)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   C_ wss 2593  ran crn 3987  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Idcgi 9312  invcgn 9313  Ringcring 9463  Idlcidl 16155

This theorem is referenced by:  idlsubcl 16171

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

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