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Theorem rngonegmn1r 34052
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 5505 . . . . . . . . 9 ran 𝐺 = ran (1st𝑅)
41, 3eqtri 2780 . . . . . . . 8 𝑋 = ran (1st𝑅)
5 ringneg.2 . . . . . . . 8 𝐻 = (2nd𝑅)
6 ringneg.5 . . . . . . . 8 𝑈 = (GId‘𝐻)
74, 5, 6rngo1cl 34049 . . . . . . 7 (𝑅 ∈ RingOps → 𝑈𝑋)
8 ringneg.4 . . . . . . . 8 𝑁 = (inv‘𝐺)
92, 1, 8rngonegcl 34037 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑈𝑋) → (𝑁𝑈) ∈ 𝑋)
107, 9mpdan 705 . . . . . 6 (𝑅 ∈ RingOps → (𝑁𝑈) ∈ 𝑋)
1110adantr 472 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑁𝑈) ∈ 𝑋)
127adantr 472 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝑈𝑋)
1311, 12jca 555 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑁𝑈) ∈ 𝑋𝑈𝑋))
142, 5, 1rngodi 34014 . . . . . 6 ((𝑅 ∈ RingOps ∧ (𝐴𝑋 ∧ (𝑁𝑈) ∈ 𝑋𝑈𝑋)) → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁𝑈))𝐺(𝐴𝐻𝑈)))
15143exp2 1448 . . . . 5 (𝑅 ∈ RingOps → (𝐴𝑋 → ((𝑁𝑈) ∈ 𝑋 → (𝑈𝑋 → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁𝑈))𝐺(𝐴𝐻𝑈))))))
1615imp43 622 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ ((𝑁𝑈) ∈ 𝑋𝑈𝑋)) → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁𝑈))𝐺(𝐴𝐻𝑈)))
1713, 16mpdan 705 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁𝑈))𝐺(𝐴𝐻𝑈)))
18 eqid 2758 . . . . . . . 8 (GId‘𝐺) = (GId‘𝐺)
192, 1, 8, 18rngoaddneg2 34039 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑈𝑋) → ((𝑁𝑈)𝐺𝑈) = (GId‘𝐺))
207, 19mpdan 705 . . . . . 6 (𝑅 ∈ RingOps → ((𝑁𝑈)𝐺𝑈) = (GId‘𝐺))
2120adantr 472 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑁𝑈)𝐺𝑈) = (GId‘𝐺))
2221oveq2d 6827 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = (𝐴𝐻(GId‘𝐺)))
2318, 1, 2, 5rngorz 34033 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻(GId‘𝐺)) = (GId‘𝐺))
2422, 23eqtrd 2792 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = (GId‘𝐺))
255, 4, 6rngoridm 34048 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻𝑈) = 𝐴)
2625oveq2d 6827 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝐴𝐻(𝑁𝑈))𝐺(𝐴𝐻𝑈)) = ((𝐴𝐻(𝑁𝑈))𝐺𝐴))
2717, 24, 263eqtr3rd 2801 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝐴𝐻(𝑁𝑈))𝐺𝐴) = (GId‘𝐺))
282, 5, 1rngocl 34011 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋 ∧ (𝑁𝑈) ∈ 𝑋) → (𝐴𝐻(𝑁𝑈)) ∈ 𝑋)
2911, 28mpd3an3 1572 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻(𝑁𝑈)) ∈ 𝑋)
302rngogrpo 34020 . . . 4 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
311, 18, 8grpoinvid2 27690 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋 ∧ (𝐴𝐻(𝑁𝑈)) ∈ 𝑋) → ((𝑁𝐴) = (𝐴𝐻(𝑁𝑈)) ↔ ((𝐴𝐻(𝑁𝑈))𝐺𝐴) = (GId‘𝐺)))
3230, 31syl3an1 1167 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋 ∧ (𝐴𝐻(𝑁𝑈)) ∈ 𝑋) → ((𝑁𝐴) = (𝐴𝐻(𝑁𝑈)) ↔ ((𝐴𝐻(𝑁𝑈))𝐺𝐴) = (GId‘𝐺)))
3329, 32mpd3an3 1572 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑁𝐴) = (𝐴𝐻(𝑁𝑈)) ↔ ((𝐴𝐻(𝑁𝑈))𝐺𝐴) = (GId‘𝐺)))
3427, 33mpbird 247 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑁𝐴) = (𝐴𝐻(𝑁𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1630  wcel 2137  ran crn 5265  cfv 6047  (class class class)co 6811  1st c1st 7329  2nd c2nd 7330  GrpOpcgr 27650  GIdcgi 27651  invcgn 27652  RingOpscrngo 34004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-riota 6772  df-ov 6814  df-1st 7331  df-2nd 7332  df-grpo 27654  df-gid 27655  df-ginv 27656  df-ablo 27706  df-ass 33953  df-exid 33955  df-mgmOLD 33959  df-sgrOLD 33971  df-mndo 33977  df-rngo 34005
This theorem is referenced by:  rngonegrmul  34054
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