Theorem rngonegmn1l 39015

Description: Obsolete theorem, use ringnegl 20545 instead. Negation in a ring is the same as left multiplication by -1. (Contributed by Jeff Madsen, 10-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
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 5923	. . . . . . 7 ⊢ ran 𝐺 = ran (1st ‘𝑅)
4	1, 3	eqtri 2786	. . . . . 6 ⊢ 𝑋 = ran (1st ‘𝑅)
5		ringneg.2	. . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅)
6		ringneg.5	. . . . . 6 ⊢ 𝑈 = (GId‘𝐻)
7	4, 5, 6	rngo1cl 39013	. . . . 5 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
8		ringneg.4	. . . . . . 7 ⊢ 𝑁 = (inv‘𝐺)
9	2, 1, 8	rngonegcl 39001	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋) → (𝑁‘𝑈) ∈ 𝑋)
10	7, 9	mpdan 700	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑁‘𝑈) ∈ 𝑋)
11	7, 10	jca 521	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋))
12	2, 5, 1	rngodir 38979	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ (𝑈 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)))
13	12	3exp2 1373	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ 𝑋 → ((𝑁‘𝑈) ∈ 𝑋 → (𝐴 ∈ 𝑋 → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴))))))
14	13	imp42 432	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋)) ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)))
15	14	an32s 665	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ (𝑈 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋)) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)))
16	11, 15	mpidan 702	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)))
17		eqid 2763	. . . . . . . 8 ⊢ (GId‘𝐺) = (GId‘𝐺)
18	2, 1, 8, 17	rngoaddneg1 39002	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋) → (𝑈𝐺(𝑁‘𝑈)) = (GId‘𝐺))
19	7, 18	mpdan 700	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (𝑈𝐺(𝑁‘𝑈)) = (GId‘𝐺))
20	19	adantr 486	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑈𝐺(𝑁‘𝑈)) = (GId‘𝐺))
21	20	oveq1d 7442	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((GId‘𝐺)𝐻𝐴))
22	17, 1, 2, 5	rngolz 38996	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐻𝐴) = (GId‘𝐺))
23	21, 22	eqtrd 2798	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = (GId‘𝐺))
24	5, 4, 6	rngolidm 39011	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑈𝐻𝐴) = 𝐴)
25	24	oveq1d 7442	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)) = (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)))
26	16, 23, 25	3eqtr3rd 2807	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)) = (GId‘𝐺))
27	2, 5, 1	rngocl 38975	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ (𝑁‘𝑈) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋)
28	27	3expa 1136	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ (𝑁‘𝑈) ∈ 𝑋) ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋)
29	28	an32s 665	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ (𝑁‘𝑈) ∈ 𝑋) → ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋)
30	10, 29	mpidan 702	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋)
31	2	rngogrpo 38984	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
32	1, 17, 8	grpoinvid1 31142	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋) → ((𝑁‘𝐴) = ((𝑁‘𝑈)𝐻𝐴) ↔ (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)) = (GId‘𝐺)))
33	31, 32	syl3an1 1181	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋) → ((𝑁‘𝐴) = ((𝑁‘𝑈)𝐻𝐴) ↔ (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)) = (GId‘𝐺)))
34	30, 33	mpd3an3 1491	. 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2146  ran crn 5655  ‘cfv 6546  (class class class)co 7427  1^st c1st 8006  2^nd c2nd 8007  GrpOpcgr 31103  GIdcgi 31104  invcgn 31105  RingOpscrngo 38968

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-1st 8008  df-2nd 8009  df-grpo 31107  df-gid 31108  df-ginv 31109  df-ablo 31159  df-ass 38917  df-exid 38919  df-mgmOLD 38923  df-sgrOLD 38935  df-mndo 38941  df-rngo 38969

This theorem is used by:  rngoneglmul  39017  idlnegcl  39096