Theorem rngonegmn1l 33740

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 5352	. . . . . . 7 ⊢ ran 𝐺 = ran (1st ‘𝑅)
4	1, 3	eqtri 2644	. . . . . 6 ⊢ 𝑋 = ran (1st ‘𝑅)
5		ringneg.2	. . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅)
6		ringneg.5	. . . . . 6 ⊢ 𝑈 = (GId‘𝐻)
7	4, 5, 6	rngo1cl 33738	. . . . 5 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
8		ringneg.4	. . . . . . 7 ⊢ 𝑁 = (inv‘𝐺)
9	2, 1, 8	rngonegcl 33726	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋) → (𝑁‘𝑈) ∈ 𝑋)
10	7, 9	mpdan 702	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑁‘𝑈) ∈ 𝑋)
11	7, 10	jca 554	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋))
12	2, 5, 1	rngodir 33704	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ (𝑈 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)))
13	12	3exp2 1285	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ 𝑋 → ((𝑁‘𝑈) ∈ 𝑋 → (𝐴 ∈ 𝑋 → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴))))))
14	13	imp42 620	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋)) ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)))
15	14	an32s 846	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ (𝑈 ∈ 𝑋 ∧ (𝑁‘𝑈) ∈ 𝑋)) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)))
16	11, 15	mpidan 704	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)))
17		eqid 2622	. . . . . . . 8 ⊢ (GId‘𝐺) = (GId‘𝐺)
18	2, 1, 8, 17	rngoaddneg1 33727	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋) → (𝑈𝐺(𝑁‘𝑈)) = (GId‘𝐺))
19	7, 18	mpdan 702	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (𝑈𝐺(𝑁‘𝑈)) = (GId‘𝐺))
20	19	adantr 481	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑈𝐺(𝑁‘𝑈)) = (GId‘𝐺))
21	20	oveq1d 6665	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = ((GId‘𝐺)𝐻𝐴))
22	17, 1, 2, 5	rngolz 33721	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐻𝐴) = (GId‘𝐺))
23	21, 22	eqtrd 2656	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺(𝑁‘𝑈))𝐻𝐴) = (GId‘𝐺))
24	5, 4, 6	rngolidm 33736	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑈𝐻𝐴) = 𝐴)
25	24	oveq1d 6665	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴)𝐺((𝑁‘𝑈)𝐻𝐴)) = (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)))
26	16, 23, 25	3eqtr3rd 2665	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)) = (GId‘𝐺))
27	2, 5, 1	rngocl 33700	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ (𝑁‘𝑈) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋)
28	27	3expa 1265	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ (𝑁‘𝑈) ∈ 𝑋) ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋)
29	28	an32s 846	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ (𝑁‘𝑈) ∈ 𝑋) → ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋)
30	10, 29	mpidan 704	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋)
31	2	rngogrpo 33709	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
32	1, 17, 8	grpoinvid1 27382	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋) → ((𝑁‘𝐴) = ((𝑁‘𝑈)𝐻𝐴) ↔ (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)) = (GId‘𝐺)))
33	31, 32	syl3an1 1359	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ ((𝑁‘𝑈)𝐻𝐴) ∈ 𝑋) → ((𝑁‘𝐴) = ((𝑁‘𝑈)𝐻𝐴) ↔ (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)) = (GId‘𝐺)))
34	30, 33	mpd3an3 1425	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) = ((𝑁‘𝑈)𝐻𝐴) ↔ (𝐴𝐺((𝑁‘𝑈)𝐻𝐴)) = (GId‘𝐺)))
35	26, 34	mpbird 247	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = ((𝑁‘𝑈)𝐻𝐴))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ran crn 5115  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167  GrpOpcgr 27343  GIdcgi 27344  invcgn 27345  RingOpscrngo 33693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-1st 7168  df-2nd 7169  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-ass 33642  df-exid 33644  df-mgmOLD 33648  df-sgrOLD 33660  df-mndo 33666  df-rngo 33694

This theorem is referenced by:  rngoneglmul  33742  idlnegcl  33821