Theorem rngonegmn1r 39142

Description: Obsolete theorem, use ringnegr 20585 instead. Negation in a ring is the same as right multiplication by -1. (Contributed by Jeff Madsen, 19-Jun-2010.) (New usage is discouraged.) (Proof modification is discouraged.)

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 5931	. . . . . . . . 9 ⊢ ran 𝐺 = ran (1st ‘𝑅)
4	1, 3	eqtri 2786	. . . . . . . 8 ⊢ 𝑋 = ran (1st ‘𝑅)
5		ringneg.2	. . . . . . . 8 ⊢ 𝐻 = (2nd ‘𝑅)
6		ringneg.5	. . . . . . . 8 ⊢ 𝑈 = (GId‘𝐻)
7	4, 5, 6	rngo1cl 39139	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
8		ringneg.4	. . . . . . . 8 ⊢ 𝑁 = (inv‘𝐺)
9	2, 1, 8	rngonegcl 39127	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋) → (𝑁‘𝑈) ∈ 𝑋)
10	7, 9	mpdan 700	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (𝑁‘𝑈) ∈ 𝑋)
11	10	adantr 486	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝑈) ∈ 𝑋)
12	7	adantr 486	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝑈 ∈ 𝑋)
13	11, 12	jca 521	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝑈) ∈ 𝑋 ∧ 𝑈 ∈ 𝑋))
14	2, 5, 1	rngodi 39104	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋 ∧ 𝑈 ∈ 𝑋)) → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁‘𝑈))𝐺(𝐴𝐻𝑈)))
15	14	3exp2 1373	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ 𝑋 → ((𝑁‘𝑈) ∈ 𝑋 → (𝑈 ∈ 𝑋 → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁‘𝑈))𝐺(𝐴𝐻𝑈))))))
16	15	imp43 433	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ ((𝑁‘𝑈) ∈ 𝑋 ∧ 𝑈 ∈ 𝑋)) → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁‘𝑈))𝐺(𝐴𝐻𝑈)))
17	13, 16	mpdan 700	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁‘𝑈))𝐺(𝐴𝐻𝑈)))
18		eqid 2763	. . . . . . . 8 ⊢ (GId‘𝐺) = (GId‘𝐺)
19	2, 1, 8, 18	rngoaddneg2 39129	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋) → ((𝑁‘𝑈)𝐺𝑈) = (GId‘𝐺))
20	7, 19	mpdan 700	. . . . . 6 ⊢ (𝑅 ∈ RingOps → ((𝑁‘𝑈)𝐺𝑈) = (GId‘𝐺))
21	20	adantr 486	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝑈)𝐺𝑈) = (GId‘𝐺))
22	21	oveq2d 7467	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = (𝐴𝐻(GId‘𝐺)))
23	18, 1, 2, 5	rngorz 39123	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻(GId‘𝐺)) = (GId‘𝐺))
24	22, 23	eqtrd 2798	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻((𝑁‘𝑈)𝐺𝑈)) = (GId‘𝐺))
25	5, 4, 6	rngoridm 39138	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑈) = 𝐴)
26	25	oveq2d 7467	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻(𝑁‘𝑈))𝐺(𝐴𝐻𝑈)) = ((𝐴𝐻(𝑁‘𝑈))𝐺𝐴))
27	17, 24, 26	3eqtr3rd 2807	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻(𝑁‘𝑈))𝐺𝐴) = (GId‘𝐺))
28	2, 5, 1	rngocl 39101	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋) → (𝐴𝐻(𝑁‘𝑈)) ∈ 𝑋)
29	11, 28	mpd3an3 1492	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻(𝑁‘𝑈)) ∈ 𝑋)
30	2	rngogrpo 39110	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
31	1, 18, 8	grpoinvid2 31188	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐻(𝑁‘𝑈)) ∈ 𝑋) → ((𝑁‘𝐴) = (𝐴𝐻(𝑁‘𝑈)) ↔ ((𝐴𝐻(𝑁‘𝑈))𝐺𝐴) = (GId‘𝐺)))
32	30, 31	syl3an1 1181	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐻(𝑁‘𝑈)) ∈ 𝑋) → ((𝑁‘𝐴) = (𝐴𝐻(𝑁‘𝑈)) ↔ ((𝐴𝐻(𝑁‘𝑈))𝐺𝐴) = (GId‘𝐺)))
33	29, 32	mpd3an3 1492	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) = (𝐴𝐻(𝑁‘𝑈)) ↔ ((𝐴𝐻(𝑁‘𝑈))𝐺𝐴) = (GId‘𝐺)))
34	27, 33	mpbird 260	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐻(𝑁‘𝑈)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ran crn 5659  ‘cfv 6566  (class class class)co 7451  1^st c1st 8033  2^nd c2nd 8034  GrpOpcgr 31148  GIdcgi 31149  invcgn 31150  RingOpscrngo 39094

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-1st 8035  df-2nd 8036  df-grpo 31152  df-gid 31153  df-ginv 31154  df-ablo 31204  df-ass 39043  df-exid 39045  df-mgmOLD 39049  df-sgrOLD 39061  df-mndo 39067  df-rngo 39095

This theorem is used by:  rngonegrmul  39144