Theorem rngonegmn1l 37273

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 5936	. . . . . . 7 ⊢ ran 𝐺 = ran (1st ‘𝑅)
4	1, 3	eqtri 2759	. . . . . 6 ⊢ 𝑋 = ran (1st ‘𝑅)
5		ringneg.2	. . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅)
6		ringneg.5	. . . . . 6 ⊢ 𝑈 = (GId‘𝐻)
7	4, 5, 6	rngo1cl 37271	. . . . 5 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
8		ringneg.4	. . . . . . 7 ⊢ 𝑁 = (inv‘𝐺)
9	2, 1, 8	rngonegcl 37259	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋) → (𝑁‘𝑈) ∈ 𝑋)
10	7, 9	mpdan 684	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑁‘𝑈) ∈ 𝑋)
11	7, 10	jca 511	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋))
12	2, 5, 1	rngodir 37237	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ (𝑈 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)))
13	12	3exp2 1353	. . . . . 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 649	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ (𝑈 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋)) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)))
16	11, 15	mpidan 686	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)))
17		eqid 2731	. . . . . . . 8 ⊢ (GId‘𝐺) = (GId‘𝐺)
18	2, 1, 8, 17	rngoaddneg1 37260	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋) → (𝑈𝐺(𝑁‘𝑈)) = (GId‘𝐺))
19	7, 18	mpdan 684	. . . . . 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 7427	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((GId‘𝐺)𝐻𝐴))
22	17, 1, 2, 5	rngolz 37254	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐻𝐴) = (GId‘𝐺))
23	21, 22	eqtrd 2771	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = (GId‘𝐺))
24	5, 4, 6	rngolidm 37269	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑈𝐻𝐴) = 𝐴)
25	24	oveq1d 7427	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)) = (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)))
26	16, 23, 25	3eqtr3rd 2780	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)) = (GId‘𝐺))
27	2, 5, 1	rngocl 37233	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ (𝑁‘𝑈) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋)
28	27	3expa 1117	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ (𝑁‘𝑈) ∈ 𝑋) ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋)
29	28	an32s 649	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ (𝑁‘𝑈) ∈ 𝑋) → ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋)
30	10, 29	mpidan 686	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋)
31	2	rngogrpo 37242	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
32	1, 17, 8	grpoinvid1 30214	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋) → ((𝑁‘𝐴) = ((𝑁‘𝑈)𝐻𝐴) ↔ (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)) = (GId‘𝐺)))
33	31, 32	syl3an1 1162	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋) → ((𝑁‘𝐴) = ((𝑁‘𝑈)𝐻𝐴) ↔ (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)) = (GId‘𝐺)))
34	30, 33	mpd3an3 1461	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) = ((𝑁‘𝑈)𝐻𝐴) ↔ (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)) = (GId‘𝐺)))
35	26, 34	mpbird 257	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = ((𝑁‘𝑈)𝐻𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ran crn 5677  ‘cfv 6543  (class class class)co 7412  1^st c1st 7977  2^nd c2nd 7978  GrpOpcgr 30175  GIdcgi 30176  invcgn 30177  RingOpscrngo 37226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-1st 7979  df-2nd 7980  df-grpo 30179  df-gid 30180  df-ginv 30181  df-ablo 30231  df-ass 37175  df-exid 37177  df-mgmOLD 37181  df-sgrOLD 37193  df-mndo 37199  df-rngo 37227

This theorem is referenced by:  rngoneglmul  37275  idlnegcl  37354