Theorem idlnegcl 38085

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 2733	. . . 4 ⊢ ran 𝐺 = ran 𝐺
3	1, 2	idlss 38079	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran 𝐺)
4		ssel2 3925	. . . . 5 ⊢ ((𝐼 ⊆ ran 𝐺 ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ ran 𝐺)
5		eqid 2733	. . . . . 6 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
6		idlnegcl.2	. . . . . 6 ⊢ 𝑁 = (inv‘𝐺)
7		eqid 2733	. . . . . 6 ⊢ (GId‘(2nd ‘𝑅)) = (GId‘(2nd ‘𝑅))
8	1, 5, 2, 6, 7	rngonegmn1l 38004	. . . . 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 5883	. . . . . 6 ⊢ ran 𝐺 = ran (1st ‘𝑅)
13	12, 5, 7	rngo1cl 38002	. . . . 5 ⊢ (𝑅 ∈ RingOps → (GId‘(2nd ‘𝑅)) ∈ ran 𝐺)
14	1, 2, 6	rngonegcl 37990	. . . . 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 38083	. . . 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 2833	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) ∈ 𝐼)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ⊆ wss 3898  ran crn 5622  ‘cfv 6488  (class class class)co 7354  1^st c1st 7927  2^nd c2nd 7928  GIdcgi 30474  invcgn 30475  RingOpscrngo 37957  Idlcidl 38070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7311  df-ov 7357  df-1st 7929  df-2nd 7930  df-grpo 30477  df-gid 30478  df-ginv 30479  df-ablo 30529  df-ass 37906  df-exid 37908  df-mgmOLD 37912  df-sgrOLD 37924  df-mndo 37930  df-rngo 37958  df-idl 38073

This theorem is referenced by:  idlsubcl  38086