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Theorem rngonegmn1r 34752
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 5689 . . . . . . . . 9 ran 𝐺 = ran (1st𝑅)
41, 3eqtri 2819 . . . . . . . 8 𝑋 = ran (1st𝑅)
5 ringneg.2 . . . . . . . 8 𝐻 = (2nd𝑅)
6 ringneg.5 . . . . . . . 8 𝑈 = (GId‘𝐻)
74, 5, 6rngo1cl 34749 . . . . . . 7 (𝑅 ∈ RingOps → 𝑈𝑋)
8 ringneg.4 . . . . . . . 8 𝑁 = (inv‘𝐺)
92, 1, 8rngonegcl 34737 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑈𝑋) → (𝑁𝑈) ∈ 𝑋)
107, 9mpdan 683 . . . . . 6 (𝑅 ∈ RingOps → (𝑁𝑈) ∈ 𝑋)
1110adantr 481 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑁𝑈) ∈ 𝑋)
127adantr 481 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → 𝑈𝑋)
1311, 12jca 512 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑁𝑈) ∈ 𝑋𝑈𝑋))
142, 5, 1rngodi 34714 . . . . . 6 ((𝑅 ∈ RingOps ∧ (𝐴𝑋 ∧ (𝑁𝑈) ∈ 𝑋𝑈𝑋)) → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁𝑈))𝐺(𝐴𝐻𝑈)))
15143exp2 1347 . . . . 5 (𝑅 ∈ RingOps → (𝐴𝑋 → ((𝑁𝑈) ∈ 𝑋 → (𝑈𝑋 → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁𝑈))𝐺(𝐴𝐻𝑈))))))
1615imp43 428 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ ((𝑁𝑈) ∈ 𝑋𝑈𝑋)) → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁𝑈))𝐺(𝐴𝐻𝑈)))
1713, 16mpdan 683 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = ((𝐴𝐻(𝑁𝑈))𝐺(𝐴𝐻𝑈)))
18 eqid 2795 . . . . . . . 8 (GId‘𝐺) = (GId‘𝐺)
192, 1, 8, 18rngoaddneg2 34739 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑈𝑋) → ((𝑁𝑈)𝐺𝑈) = (GId‘𝐺))
207, 19mpdan 683 . . . . . 6 (𝑅 ∈ RingOps → ((𝑁𝑈)𝐺𝑈) = (GId‘𝐺))
2120adantr 481 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑁𝑈)𝐺𝑈) = (GId‘𝐺))
2221oveq2d 7032 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = (𝐴𝐻(GId‘𝐺)))
2318, 1, 2, 5rngorz 34733 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻(GId‘𝐺)) = (GId‘𝐺))
2422, 23eqtrd 2831 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻((𝑁𝑈)𝐺𝑈)) = (GId‘𝐺))
255, 4, 6rngoridm 34748 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻𝑈) = 𝐴)
2625oveq2d 7032 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝐴𝐻(𝑁𝑈))𝐺(𝐴𝐻𝑈)) = ((𝐴𝐻(𝑁𝑈))𝐺𝐴))
2717, 24, 263eqtr3rd 2840 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝐴𝐻(𝑁𝑈))𝐺𝐴) = (GId‘𝐺))
282, 5, 1rngocl 34711 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋 ∧ (𝑁𝑈) ∈ 𝑋) → (𝐴𝐻(𝑁𝑈)) ∈ 𝑋)
2911, 28mpd3an3 1454 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻(𝑁𝑈)) ∈ 𝑋)
302rngogrpo 34720 . . . 4 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
311, 18, 8grpoinvid2 27997 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋 ∧ (𝐴𝐻(𝑁𝑈)) ∈ 𝑋) → ((𝑁𝐴) = (𝐴𝐻(𝑁𝑈)) ↔ ((𝐴𝐻(𝑁𝑈))𝐺𝐴) = (GId‘𝐺)))
3230, 31syl3an1 1156 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋 ∧ (𝐴𝐻(𝑁𝑈)) ∈ 𝑋) → ((𝑁𝐴) = (𝐴𝐻(𝑁𝑈)) ↔ ((𝐴𝐻(𝑁𝑈))𝐺𝐴) = (GId‘𝐺)))
3329, 32mpd3an3 1454 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑁𝐴) = (𝐴𝐻(𝑁𝑈)) ↔ ((𝐴𝐻(𝑁𝑈))𝐺𝐴) = (GId‘𝐺)))
3427, 33mpbird 258 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑁𝐴) = (𝐴𝐻(𝑁𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  ran crn 5444  cfv 6225  (class class class)co 7016  1st c1st 7543  2nd c2nd 7544  GrpOpcgr 27957  GIdcgi 27958  invcgn 27959  RingOpscrngo 34704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-1st 7545  df-2nd 7546  df-grpo 27961  df-gid 27962  df-ginv 27963  df-ablo 28013  df-ass 34653  df-exid 34655  df-mgmOLD 34659  df-sgrOLD 34671  df-mndo 34677  df-rngo 34705
This theorem is referenced by:  rngonegrmul  34754
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