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Theorem rngonegmn1l 33411
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 = (1st𝑅)
ringneg.2 𝐻 = (2nd𝑅)
ringneg.3 𝑋 = ran 𝐺
ringneg.4 𝑁 = (inv‘𝐺)
ringneg.5 𝑈 = (GId‘𝐻)
Assertion
Ref Expression
rngonegmn1l ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑁𝐴) = ((𝑁𝑈)𝐻𝐴))

Proof of Theorem rngonegmn1l
StepHypRef Expression
1 ringneg.3 . . . . . . 7 𝑋 = ran 𝐺
2 ringneg.1 . . . . . . . 8 𝐺 = (1st𝑅)
32rneqi 5322 . . . . . . 7 ran 𝐺 = ran (1st𝑅)
41, 3eqtri 2643 . . . . . 6 𝑋 = ran (1st𝑅)
5 ringneg.2 . . . . . 6 𝐻 = (2nd𝑅)
6 ringneg.5 . . . . . 6 𝑈 = (GId‘𝐻)
74, 5, 6rngo1cl 33409 . . . . 5 (𝑅 ∈ RingOps → 𝑈𝑋)
8 ringneg.4 . . . . . . 7 𝑁 = (inv‘𝐺)
92, 1, 8rngonegcl 33397 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑈𝑋) → (𝑁𝑈) ∈ 𝑋)
107, 9mpdan 701 . . . . 5 (𝑅 ∈ RingOps → (𝑁𝑈) ∈ 𝑋)
117, 10jca 554 . . . 4 (𝑅 ∈ RingOps → (𝑈𝑋 ∧ (𝑁𝑈) ∈ 𝑋))
122, 5, 1rngodir 33375 . . . . . . 7 ((𝑅 ∈ RingOps ∧ (𝑈𝑋 ∧ (𝑁𝑈) ∈ 𝑋𝐴𝑋)) → ((𝑈𝐺(𝑁𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁𝑈)𝐻𝐴)))
13123exp2 1282 . . . . . 6 (𝑅 ∈ RingOps → (𝑈𝑋 → ((𝑁𝑈) ∈ 𝑋 → (𝐴𝑋 → ((𝑈𝐺(𝑁𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁𝑈)𝐻𝐴))))))
1413imp42 619 . . . . 5 (((𝑅 ∈ RingOps ∧ (𝑈𝑋 ∧ (𝑁𝑈) ∈ 𝑋)) ∧ 𝐴𝑋) → ((𝑈𝐺(𝑁𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁𝑈)𝐻𝐴)))
1514an32s 845 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ (𝑈𝑋 ∧ (𝑁𝑈) ∈ 𝑋)) → ((𝑈𝐺(𝑁𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁𝑈)𝐻𝐴)))
1611, 15mpidan 703 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑈𝐺(𝑁𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁𝑈)𝐻𝐴)))
17 eqid 2621 . . . . . . . 8 (GId‘𝐺) = (GId‘𝐺)
182, 1, 8, 17rngoaddneg1 33398 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑈𝑋) → (𝑈𝐺(𝑁𝑈)) = (GId‘𝐺))
197, 18mpdan 701 . . . . . 6 (𝑅 ∈ RingOps → (𝑈𝐺(𝑁𝑈)) = (GId‘𝐺))
2019adantr 481 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑈𝐺(𝑁𝑈)) = (GId‘𝐺))
2120oveq1d 6630 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑈𝐺(𝑁𝑈))𝐻𝐴) = ((GId‘𝐺)𝐻𝐴))
2217, 1, 2, 5rngolz 33392 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((GId‘𝐺)𝐻𝐴) = (GId‘𝐺))
2321, 22eqtrd 2655 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑈𝐺(𝑁𝑈))𝐻𝐴) = (GId‘𝐺))
245, 4, 6rngolidm 33407 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑈𝐻𝐴) = 𝐴)
2524oveq1d 6630 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑈𝐻𝐴)𝐺((𝑁𝑈)𝐻𝐴)) = (𝐴𝐺((𝑁𝑈)𝐻𝐴)))
2616, 23, 253eqtr3rd 2664 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐺((𝑁𝑈)𝐻𝐴)) = (GId‘𝐺))
272, 5, 1rngocl 33371 . . . . . 6 ((𝑅 ∈ RingOps ∧ (𝑁𝑈) ∈ 𝑋𝐴𝑋) → ((𝑁𝑈)𝐻𝐴) ∈ 𝑋)
28273expa 1262 . . . . 5 (((𝑅 ∈ RingOps ∧ (𝑁𝑈) ∈ 𝑋) ∧ 𝐴𝑋) → ((𝑁𝑈)𝐻𝐴) ∈ 𝑋)
2928an32s 845 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ (𝑁𝑈) ∈ 𝑋) → ((𝑁𝑈)𝐻𝐴) ∈ 𝑋)
3010, 29mpidan 703 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑁𝑈)𝐻𝐴) ∈ 𝑋)
312rngogrpo 33380 . . . 4 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
321, 17, 8grpoinvid1 27270 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋 ∧ ((𝑁𝑈)𝐻𝐴) ∈ 𝑋) → ((𝑁𝐴) = ((𝑁𝑈)𝐻𝐴) ↔ (𝐴𝐺((𝑁𝑈)𝐻𝐴)) = (GId‘𝐺)))
3331, 32syl3an1 1356 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋 ∧ ((𝑁𝑈)𝐻𝐴) ∈ 𝑋) → ((𝑁𝐴) = ((𝑁𝑈)𝐻𝐴) ↔ (𝐴𝐺((𝑁𝑈)𝐻𝐴)) = (GId‘𝐺)))
3430, 33mpd3an3 1422 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑁𝐴) = ((𝑁𝑈)𝐻𝐴) ↔ (𝐴𝐺((𝑁𝑈)𝐻𝐴)) = (GId‘𝐺)))
3526, 34mpbird 247 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑁𝐴) = ((𝑁𝑈)𝐻𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  ran crn 5085  cfv 5857  (class class class)co 6615  1st c1st 7126  2nd c2nd 7127  GrpOpcgr 27231  GIdcgi 27232  invcgn 27233  RingOpscrngo 33364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-1st 7128  df-2nd 7129  df-grpo 27235  df-gid 27236  df-ginv 27237  df-ablo 27287  df-ass 33313  df-exid 33315  df-mgmOLD 33319  df-sgrOLD 33331  df-mndo 33337  df-rngo 33365
This theorem is referenced by:  rngoneglmul  33413  idlnegcl  33492
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