Theorem rngonegmn1r 36027

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 5835	. . . . . . . . 9 ⊢ ran 𝐺 = ran (1st ‘𝑅)
4	1, 3	eqtri 2766	. . . . . . . 8 ⊢ 𝑋 = ran (1st ‘𝑅)
5		ringneg.2	. . . . . . . 8 ⊢ 𝐻 = (2nd ‘𝑅)
6		ringneg.5	. . . . . . . 8 ⊢ 𝑈 = (GId‘𝐻)
7	4, 5, 6	rngo1cl 36024	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
8		ringneg.4	. . . . . . . 8 ⊢ 𝑁 = (inv‘𝐺)
9	2, 1, 8	rngonegcl 36012	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋) → (𝑁‘𝑈) ∈ 𝑋)
10	7, 9	mpdan 683	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (𝑁‘𝑈) ∈ 𝑋)
11	10	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝑈) ∈ 𝑋)
12	7	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝑈 ∈ 𝑋)
13	11, 12	jca 511	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝑈) ∈ 𝑋 ∧ 𝑈 ∈ 𝑋))
14	2, 5, 1	rngodi 35989	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋 ∧ 𝑈 ∈ 𝑋)) → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁‘𝑈))𝐺(𝐴𝐻𝑈)))
15	14	3exp2 1352	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ 𝑋 → ((𝑁‘𝑈) ∈ 𝑋 → (𝑈 ∈ 𝑋 → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁‘𝑈))𝐺(𝐴𝐻𝑈))))))
16	15	imp43 427	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ ((𝑁‘𝑈) ∈ 𝑋 ∧ 𝑈 ∈ 𝑋)) → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁‘𝑈))𝐺(𝐴𝐻𝑈)))
17	13, 16	mpdan 683	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁‘𝑈))𝐺(𝐴𝐻𝑈)))
18		eqid 2738	. . . . . . . 8 ⊢ (GId‘𝐺) = (GId‘𝐺)
19	2, 1, 8, 18	rngoaddneg2 36014	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋) → ((𝑁‘𝑈)𝐺𝑈) = (GId‘𝐺))
20	7, 19	mpdan 683	. . . . . 6 ⊢ (𝑅 ∈ RingOps → ((𝑁‘𝑈)𝐺𝑈) = (GId‘𝐺))
21	20	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝑈)𝐺𝑈) = (GId‘𝐺))
22	21	oveq2d 7271	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = (𝐴𝐻(GId‘𝐺)))
23	18, 1, 2, 5	rngorz 36008	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻(GId‘𝐺)) = (GId‘𝐺))
24	22, 23	eqtrd 2778	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = (GId‘𝐺))
25	5, 4, 6	rngoridm 36023	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑈) = 𝐴)
26	25	oveq2d 7271	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻(𝑁‘𝑈))𝐺(𝐴𝐻𝑈)) = ((𝐴𝐻(𝑁‘𝑈))𝐺𝐴))
27	17, 24, 26	3eqtr3rd 2787	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻(𝑁‘𝑈))𝐺𝐴) = (GId‘𝐺))
28	2, 5, 1	rngocl 35986	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋) → (𝐴𝐻(𝑁‘𝑈)) ∈ 𝑋)
29	11, 28	mpd3an3 1460	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻(𝑁‘𝑈)) ∈ 𝑋)
30	2	rngogrpo 35995	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
31	1, 18, 8	grpoinvid2 28792	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐻(𝑁‘𝑈)) ∈ 𝑋) → ((𝑁‘𝐴) = (𝐴𝐻(𝑁‘𝑈)) ↔ ((𝐴𝐻(𝑁‘𝑈))𝐺𝐴) = (GId‘𝐺)))
32	30, 31	syl3an1 1161	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐻(𝑁‘𝑈)) ∈ 𝑋) → ((𝑁‘𝐴) = (𝐴𝐻(𝑁‘𝑈)) ↔ ((𝐴𝐻(𝑁‘𝑈))𝐺𝐴) = (GId‘𝐺)))
33	29, 32	mpd3an3 1460	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) = (𝐴𝐻(𝑁‘𝑈)) ↔ ((𝐴𝐻(𝑁‘𝑈))𝐺𝐴) = (GId‘𝐺)))
34	27, 33	mpbird 256	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐻(𝑁‘𝑈)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ran crn 5581  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803  GrpOpcgr 28752  GIdcgi 28753  invcgn 28754  RingOpscrngo 35979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-1st 7804  df-2nd 7805  df-grpo 28756  df-gid 28757  df-ginv 28758  df-ablo 28808  df-ass 35928  df-exid 35930  df-mgmOLD 35934  df-sgrOLD 35946  df-mndo 35952  df-rngo 35980

This theorem is referenced by:  rngonegrmul  36029