Theorem rngonegmn1r 37902

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 5962	. . . . . . . . 9 ⊢ ran 𝐺 = ran (1st ‘𝑅)
4	1, 3	eqtri 2768	. . . . . . . 8 ⊢ 𝑋 = ran (1st ‘𝑅)
5		ringneg.2	. . . . . . . 8 ⊢ 𝐻 = (2nd ‘𝑅)
6		ringneg.5	. . . . . . . 8 ⊢ 𝑈 = (GId‘𝐻)
7	4, 5, 6	rngo1cl 37899	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
8		ringneg.4	. . . . . . . 8 ⊢ 𝑁 = (inv‘𝐺)
9	2, 1, 8	rngonegcl 37887	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋) → (𝑁‘𝑈) ∈ 𝑋)
10	7, 9	mpdan 686	. . . . . 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 37864	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋 ∧ 𝑈 ∈ 𝑋)) → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁‘𝑈))𝐺(𝐴𝐻𝑈)))
15	14	3exp2 1354	. . . . 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 686	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁‘𝑈))𝐺(𝐴𝐻𝑈)))
18		eqid 2740	. . . . . . . 8 ⊢ (GId‘𝐺) = (GId‘𝐺)
19	2, 1, 8, 18	rngoaddneg2 37889	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋) → ((𝑁‘𝑈)𝐺𝑈) = (GId‘𝐺))
20	7, 19	mpdan 686	. . . . . 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 7464	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = (𝐴𝐻(GId‘𝐺)))
23	18, 1, 2, 5	rngorz 37883	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻(GId‘𝐺)) = (GId‘𝐺))
24	22, 23	eqtrd 2780	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = (GId‘𝐺))
25	5, 4, 6	rngoridm 37898	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑈) = 𝐴)
26	25	oveq2d 7464	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻(𝑁‘𝑈))𝐺(𝐴𝐻𝑈)) = ((𝐴𝐻(𝑁‘𝑈))𝐺𝐴))
27	17, 24, 26	3eqtr3rd 2789	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻(𝑁‘𝑈))𝐺𝐴) = (GId‘𝐺))
28	2, 5, 1	rngocl 37861	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋) → (𝐴𝐻(𝑁‘𝑈)) ∈ 𝑋)
29	11, 28	mpd3an3 1462	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻(𝑁‘𝑈)) ∈ 𝑋)
30	2	rngogrpo 37870	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
31	1, 18, 8	grpoinvid2 30561	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐻(𝑁‘𝑈)) ∈ 𝑋) → ((𝑁‘𝐴) = (𝐴𝐻(𝑁‘𝑈)) ↔ ((𝐴𝐻(𝑁‘𝑈))𝐺𝐴) = (GId‘𝐺)))
32	30, 31	syl3an1 1163	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐻(𝑁‘𝑈)) ∈ 𝑋) → ((𝑁‘𝐴) = (𝐴𝐻(𝑁‘𝑈)) ↔ ((𝐴𝐻(𝑁‘𝑈))𝐺𝐴) = (GId‘𝐺)))
33	29, 32	mpd3an3 1462	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) = (𝐴𝐻(𝑁‘𝑈)) ↔ ((𝐴𝐻(𝑁‘𝑈))𝐺𝐴) = (GId‘𝐺)))
34	27, 33	mpbird 257	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐻(𝑁‘𝑈)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ran crn 5701  ‘cfv 6573  (class class class)co 7448  1^st c1st 8028  2^nd c2nd 8029  GrpOpcgr 30521  GIdcgi 30522  invcgn 30523  RingOpscrngo 37854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-1st 8030  df-2nd 8031  df-grpo 30525  df-gid 30526  df-ginv 30527  df-ablo 30577  df-ass 37803  df-exid 37805  df-mgmOLD 37809  df-sgrOLD 37821  df-mndo 37827  df-rngo 37855

This theorem is referenced by:  rngonegrmul  37904