idlnegcl - Mathbox for Jeff Madsen

	Mathbox for Jeff Madsen		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > idlnegcl		Structured version Visualization version GIF version

Theorem idlnegcl 37403

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 2726	. . . 4 ⊢ ran 𝐺 = ran 𝐺
3	1, 2	idlss 37397	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran 𝐺)
4		ssel2 3972	. . . . 5 ⊢ ((𝐼 ⊆ ran 𝐺 ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ ran 𝐺)
5		eqid 2726	. . . . . 6 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
6		idlnegcl.2	. . . . . 6 ⊢ 𝑁 = (inv‘𝐺)
7		eqid 2726	. . . . . 6 ⊢ (GId‘(2^nd ‘𝑅)) = (GId‘(2^nd ‘𝑅))
8	1, 5, 2, 6, 7	rngonegmn1l 37322	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ ran 𝐺) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2^nd ‘𝑅)))(2^nd ‘𝑅)𝐴))
9	4, 8	sylan2 592	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐼 ⊆ ran 𝐺 ∧ 𝐴 ∈ 𝐼)) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2^nd ‘𝑅)))(2^nd ‘𝑅)𝐴))
10	9	anassrs 467	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran 𝐺) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2^nd ‘𝑅)))(2^nd ‘𝑅)𝐴))
11	3, 10	syldanl 601	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2^nd ‘𝑅)))(2^nd ‘𝑅)𝐴))
12	1	rneqi 5930	. . . . . 6 ⊢ ran 𝐺 = ran (1^st ‘𝑅)
13	12, 5, 7	rngo1cl 37320	. . . . 5 ⊢ (𝑅 ∈ RingOps → (GId‘(2^nd ‘𝑅)) ∈ ran 𝐺)
14	1, 2, 6	rngonegcl 37308	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (GId‘(2^nd ‘𝑅)) ∈ ran 𝐺) → (𝑁‘(GId‘(2^nd ‘𝑅))) ∈ ran 𝐺)
15	13, 14	mpdan 684	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑁‘(GId‘(2^nd ‘𝑅))) ∈ ran 𝐺)
16	15	ad2antrr 723	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘(GId‘(2^nd ‘𝑅))) ∈ ran 𝐺)
17	1, 5, 2	idllmulcl 37401	. . . 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 467	. . 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 684	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → ((𝑁‘(GId‘(2^nd ‘𝑅)))(2^nd ‘𝑅)𝐴) ∈ 𝐼)
20	11, 19	eqeltrd 2827	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) ∈ 𝐼)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ⊆ wss 3943 ran crn 5670 ‘cfv 6537 (class class class)co 7405 1^st c1st 7972 2^nd c2nd 7973 GIdcgi 30252 invcgn 30253 RingOpscrngo 37275 Idlcidl 37388

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-1st 7974 df-2nd 7975 df-grpo 30255 df-gid 30256 df-ginv 30257 df-ablo 30307 df-ass 37224 df-exid 37226 df-mgmOLD 37230 df-sgrOLD 37242 df-mndo 37248 df-rngo 37276 df-idl 37391

This theorem is referenced by: idlsubcl 37404

W3C validator