Theorem rngonegmn1l 36078

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 5843	. . . . . . 7 ⊢ ran 𝐺 = ran (1st ‘𝑅)
4	1, 3	eqtri 2767	. . . . . 6 ⊢ 𝑋 = ran (1st ‘𝑅)
5		ringneg.2	. . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅)
6		ringneg.5	. . . . . 6 ⊢ 𝑈 = (GId‘𝐻)
7	4, 5, 6	rngo1cl 36076	. . . . 5 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
8		ringneg.4	. . . . . . 7 ⊢ 𝑁 = (inv‘𝐺)
9	2, 1, 8	rngonegcl 36064	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋) → (𝑁‘𝑈) ∈ 𝑋)
10	7, 9	mpdan 683	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑁‘𝑈) ∈ 𝑋)
11	7, 10	jca 511	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋))
12	2, 5, 1	rngodir 36042	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ (𝑈 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)))
13	12	3exp2 1352	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ 𝑋 → ((𝑁‘𝑈) ∈ 𝑋 → (𝐴 ∈ 𝑋 → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴))))))
14	13	imp42 426	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋)) ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)))
15	14	an32s 648	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ (𝑈 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋)) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)))
16	11, 15	mpidan 685	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)))
17		eqid 2739	. . . . . . . 8 ⊢ (GId‘𝐺) = (GId‘𝐺)
18	2, 1, 8, 17	rngoaddneg1 36065	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋) → (𝑈𝐺(𝑁‘𝑈)) = (GId‘𝐺))
19	7, 18	mpdan 683	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (𝑈𝐺(𝑁‘𝑈)) = (GId‘𝐺))
20	19	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑈𝐺(𝑁‘𝑈)) = (GId‘𝐺))
21	20	oveq1d 7283	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((GId‘𝐺)𝐻𝐴))
22	17, 1, 2, 5	rngolz 36059	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐻𝐴) = (GId‘𝐺))
23	21, 22	eqtrd 2779	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = (GId‘𝐺))
24	5, 4, 6	rngolidm 36074	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑈𝐻𝐴) = 𝐴)
25	24	oveq1d 7283	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)) = (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)))
26	16, 23, 25	3eqtr3rd 2788	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)) = (GId‘𝐺))
27	2, 5, 1	rngocl 36038	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ (𝑁‘𝑈) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋)
28	27	3expa 1116	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ (𝑁‘𝑈) ∈ 𝑋) ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋)
29	28	an32s 648	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ (𝑁‘𝑈) ∈ 𝑋) → ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋)
30	10, 29	mpidan 685	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋)
31	2	rngogrpo 36047	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
32	1, 17, 8	grpoinvid1 28869	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋) → ((𝑁‘𝐴) = ((𝑁‘𝑈)𝐻𝐴) ↔ (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)) = (GId‘𝐺)))
33	31, 32	syl3an1 1161	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋) → ((𝑁‘𝐴) = ((𝑁‘𝑈)𝐻𝐴) ↔ (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)) = (GId‘𝐺)))
34	30, 33	mpd3an3 1460	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) = ((𝑁‘𝑈)𝐻𝐴) ↔ (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)) = (GId‘𝐺)))
35	26, 34	mpbird 256	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = ((𝑁‘𝑈)𝐻𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1541   ∈ wcel 2109  ran crn 5589  ‘cfv 6430  (class class class)co 7268  1^st c1st 7815  2^nd c2nd 7816  GrpOpcgr 28830  GIdcgi 28831  invcgn 28832  RingOpscrngo 36031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-riota 7225  df-ov 7271  df-1st 7817  df-2nd 7818  df-grpo 28834  df-gid 28835  df-ginv 28836  df-ablo 28886  df-ass 35980  df-exid 35982  df-mgmOLD 35986  df-sgrOLD 35998  df-mndo 36004  df-rngo 36032

This theorem is referenced by:  rngoneglmul  36080  idlnegcl  36159