idlnegcl - Mathbox for Jeff Madsen

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Theorem idlnegcl 36890

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

**Hypotheses**
Ref	Expression
idlnegcl.1	⊢ 𝐺 = (1^st ‘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 = (1^st ‘𝑅)
2		eqid 2733	. . . 4 ⊢ ran 𝐺 = ran 𝐺
3	1, 2	idlss 36884	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran 𝐺)
4		ssel2 3978	. . . . 5 ⊢ ((𝐼 ⊆ ran 𝐺 ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ ran 𝐺)
5		eqid 2733	. . . . . 6 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
6		idlnegcl.2	. . . . . 6 ⊢ 𝑁 = (inv‘𝐺)
7		eqid 2733	. . . . . 6 ⊢ (GId‘(2^nd ‘𝑅)) = (GId‘(2^nd ‘𝑅))
8	1, 5, 2, 6, 7	rngonegmn1l 36809	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ ran 𝐺) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2^nd ‘𝑅)))(2^nd ‘𝑅)𝐴))
9	4, 8	sylan2 594	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐼 ⊆ ran 𝐺 ∧ 𝐴 ∈ 𝐼)) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2^nd ‘𝑅)))(2^nd ‘𝑅)𝐴))
10	9	anassrs 469	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran 𝐺) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2^nd ‘𝑅)))(2^nd ‘𝑅)𝐴))
11	3, 10	syldanl 603	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2^nd ‘𝑅)))(2^nd ‘𝑅)𝐴))
12	1	rneqi 5937	. . . . . 6 ⊢ ran 𝐺 = ran (1^st ‘𝑅)
13	12, 5, 7	rngo1cl 36807	. . . . 5 ⊢ (𝑅 ∈ RingOps → (GId‘(2^nd ‘𝑅)) ∈ ran 𝐺)
14	1, 2, 6	rngonegcl 36795	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (GId‘(2^nd ‘𝑅)) ∈ ran 𝐺) → (𝑁‘(GId‘(2^nd ‘𝑅))) ∈ ran 𝐺)
15	13, 14	mpdan 686	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑁‘(GId‘(2^nd ‘𝑅))) ∈ ran 𝐺)
16	15	ad2antrr 725	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘(GId‘(2^nd ‘𝑅))) ∈ ran 𝐺)
17	1, 5, 2	idllmulcl 36888	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ (𝑁‘(GId‘(2^nd ‘𝑅))) ∈ ran 𝐺)) → ((𝑁‘(GId‘(2^nd ‘𝑅)))(2^nd ‘𝑅)𝐴) ∈ 𝐼)
18	17	anassrs 469	. . 3 ⊢ ((((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) ∧ (𝑁‘(GId‘(2^nd ‘𝑅))) ∈ ran 𝐺) → ((𝑁‘(GId‘(2^nd ‘𝑅)))(2^nd ‘𝑅)𝐴) ∈ 𝐼)
19	16, 18	mpdan 686	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → ((𝑁‘(GId‘(2^nd ‘𝑅)))(2^nd ‘𝑅)𝐴) ∈ 𝐼)
20	11, 19	eqeltrd 2834	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) ∈ 𝐼)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3949 ran crn 5678 ‘cfv 6544 (class class class)co 7409 1^st c1st 7973 2^nd c2nd 7974 GIdcgi 29743 invcgn 29744 RingOpscrngo 36762 Idlcidl 36875

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-1st 7975 df-2nd 7976 df-grpo 29746 df-gid 29747 df-ginv 29748 df-ablo 29798 df-ass 36711 df-exid 36713 df-mgmOLD 36717 df-sgrOLD 36729 df-mndo 36735 df-rngo 36763 df-idl 36878

This theorem is referenced by: idlsubcl 36891

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