Theorem rngonegmn1r 35837

Description: Negation in a ring is the same as right multiplication by -1. (Contributed by Jeff Madsen, 19-Jun-2010.)

Hypotheses

Ref	Expression
ringneg.1	⊢ 𝐺 = (1st ‘𝑅)
ringneg.2	⊢ 𝐻 = (2nd ‘𝑅)
ringneg.3	⊢ 𝑋 = ran 𝐺
ringneg.4	⊢ 𝑁 = (inv‘𝐺)
ringneg.5	⊢ 𝑈 = (GId‘𝐻)

Assertion

Ref	Expression
rngonegmn1r	⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐻(𝑁‘𝑈)))

Proof of Theorem rngonegmn1r

Step	Hyp	Ref	Expression
1		ringneg.3	. . . . . . . . 9 ⊢ 𝑋 = ran 𝐺
2		ringneg.1	. . . . . . . . . 10 ⊢ 𝐺 = (1st ‘𝑅)
3	2	rneqi 5806	. . . . . . . . 9 ⊢ ran 𝐺 = ran (1st ‘𝑅)
4	1, 3	eqtri 2765	. . . . . . . 8 ⊢ 𝑋 = ran (1st ‘𝑅)
5		ringneg.2	. . . . . . . 8 ⊢ 𝐻 = (2nd ‘𝑅)
6		ringneg.5	. . . . . . . 8 ⊢ 𝑈 = (GId‘𝐻)
7	4, 5, 6	rngo1cl 35834	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
8		ringneg.4	. . . . . . . 8 ⊢ 𝑁 = (inv‘𝐺)
9	2, 1, 8	rngonegcl 35822	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋) → (𝑁‘𝑈) ∈ 𝑋)
10	7, 9	mpdan 687	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (𝑁‘𝑈) ∈ 𝑋)
11	10	adantr 484	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝑈) ∈ 𝑋)
12	7	adantr 484	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝑈 ∈ 𝑋)
13	11, 12	jca 515	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝑈) ∈ 𝑋 ∧ 𝑈 ∈ 𝑋))
14	2, 5, 1	rngodi 35799	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋 ∧ 𝑈 ∈ 𝑋)) → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁‘𝑈))𝐺(𝐴𝐻𝑈)))
15	14	3exp2 1356	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ 𝑋 → ((𝑁‘𝑈) ∈ 𝑋 → (𝑈 ∈ 𝑋 → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁‘𝑈))𝐺(𝐴𝐻𝑈))))))
16	15	imp43 431	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ ((𝑁‘𝑈) ∈ 𝑋 ∧ 𝑈 ∈ 𝑋)) → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁‘𝑈))𝐺(𝐴𝐻𝑈)))
17	13, 16	mpdan 687	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁‘𝑈))𝐺(𝐴𝐻𝑈)))
18		eqid 2737	. . . . . . . 8 ⊢ (GId‘𝐺) = (GId‘𝐺)
19	2, 1, 8, 18	rngoaddneg2 35824	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋) → ((𝑁‘𝑈)𝐺𝑈) = (GId‘𝐺))
20	7, 19	mpdan 687	. . . . . 6 ⊢ (𝑅 ∈ RingOps → ((𝑁‘𝑈)𝐺𝑈) = (GId‘𝐺))
21	20	adantr 484	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝑈)𝐺𝑈) = (GId‘𝐺))
22	21	oveq2d 7229	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = (𝐴𝐻(GId‘𝐺)))
23	18, 1, 2, 5	rngorz 35818	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻(GId‘𝐺)) = (GId‘𝐺))
24	22, 23	eqtrd 2777	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = (GId‘𝐺))
25	5, 4, 6	rngoridm 35833	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑈) = 𝐴)
26	25	oveq2d 7229	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻(𝑁‘𝑈))𝐺(𝐴𝐻𝑈)) = ((𝐴𝐻(𝑁‘𝑈))𝐺𝐴))
27	17, 24, 26	3eqtr3rd 2786	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻(𝑁‘𝑈))𝐺𝐴) = (GId‘𝐺))
28	2, 5, 1	rngocl 35796	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋) → (𝐴𝐻(𝑁‘𝑈)) ∈ 𝑋)
29	11, 28	mpd3an3 1464	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻(𝑁‘𝑈)) ∈ 𝑋)
30	2	rngogrpo 35805	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
31	1, 18, 8	grpoinvid2 28610	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐻(𝑁‘𝑈)) ∈ 𝑋) → ((𝑁‘𝐴) = (𝐴𝐻(𝑁‘𝑈)) ↔ ((𝐴𝐻(𝑁‘𝑈))𝐺𝐴) = (GId‘𝐺)))
32	30, 31	syl3an1 1165	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐻(𝑁‘𝑈)) ∈ 𝑋) → ((𝑁‘𝐴) = (𝐴𝐻(𝑁‘𝑈)) ↔ ((𝐴𝐻(𝑁‘𝑈))𝐺𝐴) = (GId‘𝐺)))
33	29, 32	mpd3an3 1464	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) = (𝐴𝐻(𝑁‘𝑈)) ↔ ((𝐴𝐻(𝑁‘𝑈))𝐺𝐴) = (GId‘𝐺)))
34	27, 33	mpbird 260	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐻(𝑁‘𝑈)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ran crn 5552  ‘cfv 6380  (class class class)co 7213  1^st c1st 7759  2^nd c2nd 7760  GrpOpcgr 28570  GIdcgi 28571  invcgn 28572  RingOpscrngo 35789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-1st 7761  df-2nd 7762  df-grpo 28574  df-gid 28575  df-ginv 28576  df-ablo 28626  df-ass 35738  df-exid 35740  df-mgmOLD 35744  df-sgrOLD 35756  df-mndo 35762  df-rngo 35790

This theorem is referenced by:  rngonegrmul  35839