Theorem rngonegmn1l 38479

Description: Negation in a ring is the same as left multiplication by -1. (Contributed by Jeff Madsen, 10-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
rngonegmn1l	⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = ((𝑁‘𝑈)𝐻𝐴))

Proof of Theorem rngonegmn1l

Step	Hyp	Ref	Expression
1		ringneg.3	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
2		ringneg.1	. . . . . . . 8 ⊢ 𝐺 = (1st ‘𝑅)
3	2	rneqi 5928	. . . . . . 7 ⊢ ran 𝐺 = ran (1st ‘𝑅)
4	1, 3	eqtri 2792	. . . . . 6 ⊢ 𝑋 = ran (1st ‘𝑅)
5		ringneg.2	. . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅)
6		ringneg.5	. . . . . 6 ⊢ 𝑈 = (GId‘𝐻)
7	4, 5, 6	rngo1cl 38477	. . . . 5 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
8		ringneg.4	. . . . . . 7 ⊢ 𝑁 = (inv‘𝐺)
9	2, 1, 8	rngonegcl 38465	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋) → (𝑁‘𝑈) ∈ 𝑋)
10	7, 9	mpdan 699	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑁‘𝑈) ∈ 𝑋)
11	7, 10	jca 520	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋))
12	2, 5, 1	rngodir 38443	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ (𝑈 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)))
13	12	3exp2 1371	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ 𝑋 → ((𝑁‘𝑈) ∈ 𝑋 → (𝐴 ∈ 𝑋 → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴))))))
14	13	imp42 431	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋)) ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)))
15	14	an32s 664	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ (𝑈 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋)) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)))
16	11, 15	mpidan 701	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)))
17		eqid 2769	. . . . . . . 8 ⊢ (GId‘𝐺) = (GId‘𝐺)
18	2, 1, 8, 17	rngoaddneg1 38466	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋) → (𝑈𝐺(𝑁‘𝑈)) = (GId‘𝐺))
19	7, 18	mpdan 699	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (𝑈𝐺(𝑁‘𝑈)) = (GId‘𝐺))
20	19	adantr 485	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑈𝐺(𝑁‘𝑈)) = (GId‘𝐺))
21	20	oveq1d 7426	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((GId‘𝐺)𝐻𝐴))
22	17, 1, 2, 5	rngolz 38460	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐻𝐴) = (GId‘𝐺))
23	21, 22	eqtrd 2804	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = (GId‘𝐺))
24	5, 4, 6	rngolidm 38475	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑈𝐻𝐴) = 𝐴)
25	24	oveq1d 7426	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)) = (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)))
26	16, 23, 25	3eqtr3rd 2813	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)) = (GId‘𝐺))
27	2, 5, 1	rngocl 38439	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ (𝑁‘𝑈) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋)
28	27	3expa 1134	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ (𝑁‘𝑈) ∈ 𝑋) ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋)
29	28	an32s 664	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ (𝑁‘𝑈) ∈ 𝑋) → ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋)
30	10, 29	mpidan 701	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋)
31	2	rngogrpo 38448	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
32	1, 17, 8	grpoinvid1 30820	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋) → ((𝑁‘𝐴) = ((𝑁‘𝑈)𝐻𝐴) ↔ (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)) = (GId‘𝐺)))
33	31, 32	syl3an1 1179	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋) → ((𝑁‘𝐴) = ((𝑁‘𝑈)𝐻𝐴) ↔ (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)) = (GId‘𝐺)))
34	30, 33	mpd3an3 1488	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) = ((𝑁‘𝑈)𝐻𝐴) ↔ (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)) = (GId‘𝐺)))
35	26, 34	mpbird 260	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = ((𝑁‘𝑈)𝐻𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ran crn 5663  ‘cfv 6537  (class class class)co 7411  1^st c1st 7983  2^nd c2nd 7984  GrpOpcgr 30781  GIdcgi 30782  invcgn 30783  RingOpscrngo 38432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-1st 7985  df-2nd 7986  df-grpo 30785  df-gid 30786  df-ginv 30787  df-ablo 30837  df-ass 38381  df-exid 38383  df-mgmOLD 38387  df-sgrOLD 38399  df-mndo 38405  df-rngo 38433

This theorem is referenced by:  rngoneglmul  38481  idlnegcl  38560