idlnegcl - Mathbox for Jeff Madsen

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Theorem idlnegcl 37528

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 2728	. . . 4 ⊢ ran 𝐺 = ran 𝐺
3	1, 2	idlss 37522	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran 𝐺)
4		ssel2 3977	. . . . 5 ⊢ ((𝐼 ⊆ ran 𝐺 ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ ran 𝐺)
5		eqid 2728	. . . . . 6 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
6		idlnegcl.2	. . . . . 6 ⊢ 𝑁 = (inv‘𝐺)
7		eqid 2728	. . . . . 6 ⊢ (GId‘(2^nd ‘𝑅)) = (GId‘(2^nd ‘𝑅))
8	1, 5, 2, 6, 7	rngonegmn1l 37447	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ ran 𝐺) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2^nd ‘𝑅)))(2^nd ‘𝑅)𝐴))
9	4, 8	sylan2 591	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐼 ⊆ ran 𝐺 ∧ 𝐴 ∈ 𝐼)) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2^nd ‘𝑅)))(2^nd ‘𝑅)𝐴))
10	9	anassrs 466	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran 𝐺) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2^nd ‘𝑅)))(2^nd ‘𝑅)𝐴))
11	3, 10	syldanl 600	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2^nd ‘𝑅)))(2^nd ‘𝑅)𝐴))
12	1	rneqi 5943	. . . . . 6 ⊢ ran 𝐺 = ran (1^st ‘𝑅)
13	12, 5, 7	rngo1cl 37445	. . . . 5 ⊢ (𝑅 ∈ RingOps → (GId‘(2^nd ‘𝑅)) ∈ ran 𝐺)
14	1, 2, 6	rngonegcl 37433	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (GId‘(2^nd ‘𝑅)) ∈ ran 𝐺) → (𝑁‘(GId‘(2^nd ‘𝑅))) ∈ ran 𝐺)
15	13, 14	mpdan 685	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑁‘(GId‘(2^nd ‘𝑅))) ∈ ran 𝐺)
16	15	ad2antrr 724	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘(GId‘(2^nd ‘𝑅))) ∈ ran 𝐺)
17	1, 5, 2	idllmulcl 37526	. . . 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 466	. . 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 685	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → ((𝑁‘(GId‘(2^nd ‘𝑅)))(2^nd ‘𝑅)𝐴) ∈ 𝐼)
20	11, 19	eqeltrd 2829	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) ∈ 𝐼)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⊆ wss 3949 ran crn 5683 ‘cfv 6553 (class class class)co 7426 1^st c1st 7997 2^nd c2nd 7998 GIdcgi 30320 invcgn 30321 RingOpscrngo 37400 Idlcidl 37513

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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-1st 7999 df-2nd 8000 df-grpo 30323 df-gid 30324 df-ginv 30325 df-ablo 30375 df-ass 37349 df-exid 37351 df-mgmOLD 37355 df-sgrOLD 37367 df-mndo 37373 df-rngo 37401 df-idl 37516

This theorem is referenced by: idlsubcl 37529

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