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Theorem rngonegmn1r 34072
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 5507 . . . . . . . . 9 ran 𝐺 = ran (1st𝑅)
41, 3eqtri 2782 . . . . . . . 8 𝑋 = ran (1st𝑅)
5 ringneg.2 . . . . . . . 8 𝐻 = (2nd𝑅)
6 ringneg.5 . . . . . . . 8 𝑈 = (GId‘𝐻)
74, 5, 6rngo1cl 34069 . . . . . . 7 (𝑅 ∈ RingOps → 𝑈𝑋)
8 ringneg.4 . . . . . . . 8 𝑁 = (inv‘𝐺)
92, 1, 8rngonegcl 34057 . . . . . . 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 34034 . . . . . 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 2760 . . . . . . . 8 (GId‘𝐺) = (GId‘𝐺)
192, 1, 8, 18rngoaddneg2 34059 . . . . . . 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 6830 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = (𝐴𝐻(GId‘𝐺)))
2318, 1, 2, 5rngorz 34053 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻(GId‘𝐺)) = (GId‘𝐺))
2422, 23eqtrd 2794 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = (GId‘𝐺))
255, 4, 6rngoridm 34068 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻𝑈) = 𝐴)
2625oveq2d 6830 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝐴𝐻(𝑁𝑈))𝐺(𝐴𝐻𝑈)) = ((𝐴𝐻(𝑁𝑈))𝐺𝐴))
2717, 24, 263eqtr3rd 2803 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝐴𝐻(𝑁𝑈))𝐺𝐴) = (GId‘𝐺))
282, 5, 1rngocl 34031 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋 ∧ (𝑁𝑈) ∈ 𝑋) → (𝐴𝐻(𝑁𝑈)) ∈ 𝑋)
2911, 28mpd3an3 1574 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻(𝑁𝑈)) ∈ 𝑋)
302rngogrpo 34040 . . . 4 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
311, 18, 8grpoinvid2 27713 . . . 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 1574 . 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 1632  wcel 2139  ran crn 5267  cfv 6049  (class class class)co 6814  1st c1st 7332  2nd c2nd 7333  GrpOpcgr 27673  GIdcgi 27674  invcgn 27675  RingOpscrngo 34024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-1st 7334  df-2nd 7335  df-grpo 27677  df-gid 27678  df-ginv 27679  df-ablo 27729  df-ass 33973  df-exid 33975  df-mgmOLD 33979  df-sgrOLD 33991  df-mndo 33997  df-rngo 34025
This theorem is referenced by:  rngonegrmul  34074
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