Theorem idlnegcl 36107

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 2738	. . . 4 ⊢ ran 𝐺 = ran 𝐺
3	1, 2	idlss 36101	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran 𝐺)
4		ssel2 3912	. . . . 5 ⊢ ((𝐼 ⊆ ran 𝐺 ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ ran 𝐺)
5		eqid 2738	. . . . . 6 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
6		idlnegcl.2	. . . . . 6 ⊢ 𝑁 = (inv‘𝐺)
7		eqid 2738	. . . . . 6 ⊢ (GId‘(2nd ‘𝑅)) = (GId‘(2nd ‘𝑅))
8	1, 5, 2, 6, 7	rngonegmn1l 36026	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ ran 𝐺) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴))
9	4, 8	sylan2 592	. . . 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 601	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴))
12	1	rneqi 5835	. . . . . 6 ⊢ ran 𝐺 = ran (1st ‘𝑅)
13	12, 5, 7	rngo1cl 36024	. . . . 5 ⊢ (𝑅 ∈ RingOps → (GId‘(2nd ‘𝑅)) ∈ ran 𝐺)
14	1, 2, 6	rngonegcl 36012	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (GId‘(2nd ‘𝑅)) ∈ ran 𝐺) → (𝑁‘(GId‘(2nd ‘𝑅))) ∈ ran 𝐺)
15	13, 14	mpdan 683	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑁‘(GId‘(2nd ‘𝑅))) ∈ ran 𝐺)
16	15	ad2antrr 722	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘(GId‘(2nd ‘𝑅))) ∈ ran 𝐺)
17	1, 5, 2	idllmulcl 36105	. . . 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 683	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴) ∈ 𝐼)
20	11, 19	eqeltrd 2839	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) ∈ 𝐼)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ⊆ wss 3883  ran crn 5581  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803  GIdcgi 28753  invcgn 28754  RingOpscrngo 35979  Idlcidl 36092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-1st 7804  df-2nd 7805  df-grpo 28756  df-gid 28757  df-ginv 28758  df-ablo 28808  df-ass 35928  df-exid 35930  df-mgmOLD 35934  df-sgrOLD 35946  df-mndo 35952  df-rngo 35980  df-idl 36095

This theorem is referenced by:  idlsubcl  36108