Theorem idlnegcl 38030

Description: An ideal is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

Ref	Expression
idlnegcl.1	⊢ 𝐺 = (1st ‘𝑅)
idlnegcl.2	⊢ 𝑁 = (inv‘𝐺)

Assertion

Ref	Expression
idlnegcl	⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) ∈ 𝐼)

Proof of Theorem idlnegcl

Step	Hyp	Ref	Expression
1		idlnegcl.1	. . . 4 ⊢ 𝐺 = (1st ‘𝑅)
2		eqid 2736	. . . 4 ⊢ ran 𝐺 = ran 𝐺
3	1, 2	idlss 38024	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran 𝐺)
4		ssel2 3977	. . . . 5 ⊢ ((𝐼 ⊆ ran 𝐺 ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ ran 𝐺)
5		eqid 2736	. . . . . 6 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
6		idlnegcl.2	. . . . . 6 ⊢ 𝑁 = (inv‘𝐺)
7		eqid 2736	. . . . . 6 ⊢ (GId‘(2nd ‘𝑅)) = (GId‘(2nd ‘𝑅))
8	1, 5, 2, 6, 7	rngonegmn1l 37949	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ ran 𝐺) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴))
9	4, 8	sylan2 593	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐼 ⊆ ran 𝐺 ∧ 𝐴 ∈ 𝐼)) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴))
10	9	anassrs 467	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran 𝐺) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴))
11	3, 10	syldanl 602	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴))
12	1	rneqi 5947	. . . . . 6 ⊢ ran 𝐺 = ran (1st ‘𝑅)
13	12, 5, 7	rngo1cl 37947	. . . . 5 ⊢ (𝑅 ∈ RingOps → (GId‘(2nd ‘𝑅)) ∈ ran 𝐺)
14	1, 2, 6	rngonegcl 37935	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (GId‘(2nd ‘𝑅)) ∈ ran 𝐺) → (𝑁‘(GId‘(2nd ‘𝑅))) ∈ ran 𝐺)
15	13, 14	mpdan 687	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑁‘(GId‘(2nd ‘𝑅))) ∈ ran 𝐺)
16	15	ad2antrr 726	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘(GId‘(2nd ‘𝑅))) ∈ ran 𝐺)
17	1, 5, 2	idllmulcl 38028	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ (𝑁‘(GId‘(2nd ‘𝑅))) ∈ ran 𝐺)) → ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴) ∈ 𝐼)
18	17	anassrs 467	. . 3 ⊢ ((((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) ∧ (𝑁‘(GId‘(2nd ‘𝑅))) ∈ ran 𝐺) → ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴) ∈ 𝐼)
19	16, 18	mpdan 687	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴) ∈ 𝐼)
20	11, 19	eqeltrd 2840	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) ∈ 𝐼)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ⊆ wss 3950  ran crn 5685  ‘cfv 6560  (class class class)co 7432  1^st c1st 8013  2^nd c2nd 8014  GIdcgi 30510  invcgn 30511  RingOpscrngo 37902  Idlcidl 38015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-1st 8015  df-2nd 8016  df-grpo 30513  df-gid 30514  df-ginv 30515  df-ablo 30565  df-ass 37851  df-exid 37853  df-mgmOLD 37857  df-sgrOLD 37869  df-mndo 37875  df-rngo 37903  df-idl 38018

This theorem is referenced by:  idlsubcl  38031