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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 5553 . . . . . . . . 9 ran 𝐺 = ran (1st𝑅)
41, 3eqtri 2828 . . . . . . . 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 670 . . . . . 6 (𝑅 ∈ RingOps → (𝑁𝑈) ∈ 𝑋)
1110adantr 468 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑁𝑈) ∈ 𝑋)
127adantr 468 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝑈𝑋)
1311, 12jca 503 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑁𝑈) ∈ 𝑋𝑈𝑋))
142, 5, 1rngodi 34014 . . . . . 6 ((𝑅 ∈ RingOps ∧ (𝐴𝑋 ∧ (𝑁𝑈) ∈ 𝑋𝑈𝑋)) → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁𝑈))𝐺(𝐴𝐻𝑈)))
15143exp2 1456 . . . . 5 (𝑅 ∈ RingOps → (𝐴𝑋 → ((𝑁𝑈) ∈ 𝑋 → (𝑈𝑋 → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁𝑈))𝐺(𝐴𝐻𝑈))))))
1615imp43 416 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ ((𝑁𝑈) ∈ 𝑋𝑈𝑋)) → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁𝑈))𝐺(𝐴𝐻𝑈)))
1713, 16mpdan 670 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁𝑈))𝐺(𝐴𝐻𝑈)))
18 eqid 2806 . . . . . . . 8 (GId‘𝐺) = (GId‘𝐺)
192, 1, 8, 18rngoaddneg2 34039 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑈𝑋) → ((𝑁𝑈)𝐺𝑈) = (GId‘𝐺))
207, 19mpdan 670 . . . . . 6 (𝑅 ∈ RingOps → ((𝑁𝑈)𝐺𝑈) = (GId‘𝐺))
2120adantr 468 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑁𝑈)𝐺𝑈) = (GId‘𝐺))
2221oveq2d 6890 . . . 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 2840 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = (GId‘𝐺))
255, 4, 6rngoridm 34048 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻𝑈) = 𝐴)
2625oveq2d 6890 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝐴𝐻(𝑁𝑈))𝐺(𝐴𝐻𝑈)) = ((𝐴𝐻(𝑁𝑈))𝐺𝐴))
2717, 24, 263eqtr3rd 2849 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝐴𝐻(𝑁𝑈))𝐺𝐴) = (GId‘𝐺))
282, 5, 1rngocl 34011 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋 ∧ (𝑁𝑈) ∈ 𝑋) → (𝐴𝐻(𝑁𝑈)) ∈ 𝑋)
2911, 28mpd3an3 1579 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻(𝑁𝑈)) ∈ 𝑋)
302rngogrpo 34020 . . . 4 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
311, 18, 8grpoinvid2 27712 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋 ∧ (𝐴𝐻(𝑁𝑈)) ∈ 𝑋) → ((𝑁𝐴) = (𝐴𝐻(𝑁𝑈)) ↔ ((𝐴𝐻(𝑁𝑈))𝐺𝐴) = (GId‘𝐺)))
3230, 31syl3an1 1195 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋 ∧ (𝐴𝐻(𝑁𝑈)) ∈ 𝑋) → ((𝑁𝐴) = (𝐴𝐻(𝑁𝑈)) ↔ ((𝐴𝐻(𝑁𝑈))𝐺𝐴) = (GId‘𝐺)))
3329, 32mpd3an3 1579 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑁𝐴) = (𝐴𝐻(𝑁𝑈)) ↔ ((𝐴𝐻(𝑁𝑈))𝐺𝐴) = (GId‘𝐺)))
3427, 33mpbird 248 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑁𝐴) = (𝐴𝐻(𝑁𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2156  ran crn 5312  cfv 6101  (class class class)co 6874  1st c1st 7396  2nd c2nd 7397  GrpOpcgr 27672  GIdcgi 27673  invcgn 27674  RingOpscrngo 34004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6835  df-ov 6877  df-1st 7398  df-2nd 7399  df-grpo 27676  df-gid 27677  df-ginv 27678  df-ablo 27728  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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