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Theorem rngonegmn1r 37929
Description: Negation in a ring is the same as right multiplication by -1. (Contributed by Jeff Madsen, 19-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
rngonegmn1r ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑁𝐴) = (𝐴𝐻(𝑁𝑈)))

Proof of Theorem rngonegmn1r
StepHypRef Expression
1 ringneg.3 . . . . . . . . 9 𝑋 = ran 𝐺
2 ringneg.1 . . . . . . . . . 10 𝐺 = (1st𝑅)
32rneqi 5951 . . . . . . . . 9 ran 𝐺 = ran (1st𝑅)
41, 3eqtri 2763 . . . . . . . 8 𝑋 = ran (1st𝑅)
5 ringneg.2 . . . . . . . 8 𝐻 = (2nd𝑅)
6 ringneg.5 . . . . . . . 8 𝑈 = (GId‘𝐻)
74, 5, 6rngo1cl 37926 . . . . . . 7 (𝑅 ∈ RingOps → 𝑈𝑋)
8 ringneg.4 . . . . . . . 8 𝑁 = (inv‘𝐺)
92, 1, 8rngonegcl 37914 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑈𝑋) → (𝑁𝑈) ∈ 𝑋)
107, 9mpdan 687 . . . . . 6 (𝑅 ∈ RingOps → (𝑁𝑈) ∈ 𝑋)
1110adantr 480 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑁𝑈) ∈ 𝑋)
127adantr 480 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝑈𝑋)
1311, 12jca 511 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑁𝑈) ∈ 𝑋𝑈𝑋))
142, 5, 1rngodi 37891 . . . . . 6 ((𝑅 ∈ RingOps ∧ (𝐴𝑋 ∧ (𝑁𝑈) ∈ 𝑋𝑈𝑋)) → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁𝑈))𝐺(𝐴𝐻𝑈)))
15143exp2 1353 . . . . 5 (𝑅 ∈ RingOps → (𝐴𝑋 → ((𝑁𝑈) ∈ 𝑋 → (𝑈𝑋 → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁𝑈))𝐺(𝐴𝐻𝑈))))))
1615imp43 427 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ ((𝑁𝑈) ∈ 𝑋𝑈𝑋)) → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁𝑈))𝐺(𝐴𝐻𝑈)))
1713, 16mpdan 687 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁𝑈))𝐺(𝐴𝐻𝑈)))
18 eqid 2735 . . . . . . . 8 (GId‘𝐺) = (GId‘𝐺)
192, 1, 8, 18rngoaddneg2 37916 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑈𝑋) → ((𝑁𝑈)𝐺𝑈) = (GId‘𝐺))
207, 19mpdan 687 . . . . . 6 (𝑅 ∈ RingOps → ((𝑁𝑈)𝐺𝑈) = (GId‘𝐺))
2120adantr 480 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑁𝑈)𝐺𝑈) = (GId‘𝐺))
2221oveq2d 7447 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = (𝐴𝐻(GId‘𝐺)))
2318, 1, 2, 5rngorz 37910 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻(GId‘𝐺)) = (GId‘𝐺))
2422, 23eqtrd 2775 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = (GId‘𝐺))
255, 4, 6rngoridm 37925 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻𝑈) = 𝐴)
2625oveq2d 7447 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝐴𝐻(𝑁𝑈))𝐺(𝐴𝐻𝑈)) = ((𝐴𝐻(𝑁𝑈))𝐺𝐴))
2717, 24, 263eqtr3rd 2784 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝐴𝐻(𝑁𝑈))𝐺𝐴) = (GId‘𝐺))
282, 5, 1rngocl 37888 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋 ∧ (𝑁𝑈) ∈ 𝑋) → (𝐴𝐻(𝑁𝑈)) ∈ 𝑋)
2911, 28mpd3an3 1461 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻(𝑁𝑈)) ∈ 𝑋)
302rngogrpo 37897 . . . 4 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
311, 18, 8grpoinvid2 30558 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋 ∧ (𝐴𝐻(𝑁𝑈)) ∈ 𝑋) → ((𝑁𝐴) = (𝐴𝐻(𝑁𝑈)) ↔ ((𝐴𝐻(𝑁𝑈))𝐺𝐴) = (GId‘𝐺)))
3230, 31syl3an1 1162 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋 ∧ (𝐴𝐻(𝑁𝑈)) ∈ 𝑋) → ((𝑁𝐴) = (𝐴𝐻(𝑁𝑈)) ↔ ((𝐴𝐻(𝑁𝑈))𝐺𝐴) = (GId‘𝐺)))
3329, 32mpd3an3 1461 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑁𝐴) = (𝐴𝐻(𝑁𝑈)) ↔ ((𝐴𝐻(𝑁𝑈))𝐺𝐴) = (GId‘𝐺)))
3427, 33mpbird 257 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑁𝐴) = (𝐴𝐻(𝑁𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  ran crn 5690  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  GrpOpcgr 30518  GIdcgi 30519  invcgn 30520  RingOpscrngo 37881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-1st 8013  df-2nd 8014  df-grpo 30522  df-gid 30523  df-ginv 30524  df-ablo 30574  df-ass 37830  df-exid 37832  df-mgmOLD 37836  df-sgrOLD 37848  df-mndo 37854  df-rngo 37882
This theorem is referenced by:  rngonegrmul  37931
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