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Theorem rngonegmn1r 36805
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 = (1st β€˜π‘…)
ringneg.2 𝐻 = (2nd β€˜π‘…)
ringneg.3 𝑋 = ran 𝐺
ringneg.4 𝑁 = (invβ€˜πΊ)
ringneg.5 π‘ˆ = (GIdβ€˜π»)
Assertion
Ref Expression
rngonegmn1r ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) = (𝐴𝐻(π‘β€˜π‘ˆ)))
Proof of Theorem rngonegmn1r
StepHypRef Expression
1 ringneg.3 . . . . . . . . 9 𝑋 = ran 𝐺
2 ringneg.1 . . . . . . . . . 10 𝐺 = (1st β€˜π‘…)
32rneqi 5936 . . . . . . . . 9 ran 𝐺 = ran (1st β€˜π‘…)
41, 3eqtri 2760 . . . . . . . 8 𝑋 = ran (1st β€˜π‘…)
5 ringneg.2 . . . . . . . 8 𝐻 = (2nd β€˜π‘…)
6 ringneg.5 . . . . . . . 8 π‘ˆ = (GIdβ€˜π»)
74, 5, 6rngo1cl 36802 . . . . . . 7 (𝑅 ∈ RingOps β†’ π‘ˆ ∈ 𝑋)
8 ringneg.4 . . . . . . . 8 𝑁 = (invβ€˜πΊ)
92, 1, 8rngonegcl 36790 . . . . . . 7 ((𝑅 ∈ RingOps ∧ π‘ˆ ∈ 𝑋) β†’ (π‘β€˜π‘ˆ) ∈ 𝑋)
107, 9mpdan 685 . . . . . 6 (𝑅 ∈ RingOps β†’ (π‘β€˜π‘ˆ) ∈ 𝑋)
1110adantr 481 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π‘ˆ) ∈ 𝑋)
127adantr 481 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ π‘ˆ ∈ 𝑋)
1311, 12jca 512 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π‘ˆ) ∈ 𝑋 ∧ π‘ˆ ∈ 𝑋))
142, 5, 1rngodi 36767 . . . . . 6 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ (π‘β€˜π‘ˆ) ∈ 𝑋 ∧ π‘ˆ ∈ 𝑋)) β†’ (𝐴𝐻((π‘β€˜π‘ˆ)πΊπ‘ˆ)) = ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺(π΄π»π‘ˆ)))
15143exp2 1354 . . . . 5 (𝑅 ∈ RingOps β†’ (𝐴 ∈ 𝑋 β†’ ((π‘β€˜π‘ˆ) ∈ 𝑋 β†’ (π‘ˆ ∈ 𝑋 β†’ (𝐴𝐻((π‘β€˜π‘ˆ)πΊπ‘ˆ)) = ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺(π΄π»π‘ˆ))))))
1615imp43 428 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ ((π‘β€˜π‘ˆ) ∈ 𝑋 ∧ π‘ˆ ∈ 𝑋)) β†’ (𝐴𝐻((π‘β€˜π‘ˆ)πΊπ‘ˆ)) = ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺(π΄π»π‘ˆ)))
1713, 16mpdan 685 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐻((π‘β€˜π‘ˆ)πΊπ‘ˆ)) = ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺(π΄π»π‘ˆ)))
18 eqid 2732 . . . . . . . 8 (GIdβ€˜πΊ) = (GIdβ€˜πΊ)
192, 1, 8, 18rngoaddneg2 36792 . . . . . . 7 ((𝑅 ∈ RingOps ∧ π‘ˆ ∈ 𝑋) β†’ ((π‘β€˜π‘ˆ)πΊπ‘ˆ) = (GIdβ€˜πΊ))
207, 19mpdan 685 . . . . . 6 (𝑅 ∈ RingOps β†’ ((π‘β€˜π‘ˆ)πΊπ‘ˆ) = (GIdβ€˜πΊ))
2120adantr 481 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π‘ˆ)πΊπ‘ˆ) = (GIdβ€˜πΊ))
2221oveq2d 7424 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐻((π‘β€˜π‘ˆ)πΊπ‘ˆ)) = (𝐴𝐻(GIdβ€˜πΊ)))
2318, 1, 2, 5rngorz 36786 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐻(GIdβ€˜πΊ)) = (GIdβ€˜πΊ))
2422, 23eqtrd 2772 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐻((π‘β€˜π‘ˆ)πΊπ‘ˆ)) = (GIdβ€˜πΊ))
255, 4, 6rngoridm 36801 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π΄π»π‘ˆ) = 𝐴)
2625oveq2d 7424 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺(π΄π»π‘ˆ)) = ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺𝐴))
2717, 24, 263eqtr3rd 2781 . 2 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺𝐴) = (GIdβ€˜πΊ))
282, 5, 1rngocl 36764 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ (π‘β€˜π‘ˆ) ∈ 𝑋) β†’ (𝐴𝐻(π‘β€˜π‘ˆ)) ∈ 𝑋)
2911, 28mpd3an3 1462 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐻(π‘β€˜π‘ˆ)) ∈ 𝑋)
302rngogrpo 36773 . . . 4 (𝑅 ∈ RingOps β†’ 𝐺 ∈ GrpOp)
311, 18, 8grpoinvid2 29777 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐻(π‘β€˜π‘ˆ)) ∈ 𝑋) β†’ ((π‘β€˜π΄) = (𝐴𝐻(π‘β€˜π‘ˆ)) ↔ ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺𝐴) = (GIdβ€˜πΊ)))
3230, 31syl3an1 1163 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐻(π‘β€˜π‘ˆ)) ∈ 𝑋) β†’ ((π‘β€˜π΄) = (𝐴𝐻(π‘β€˜π‘ˆ)) ↔ ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺𝐴) = (GIdβ€˜πΊ)))
3329, 32mpd3an3 1462 . 2 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π΄) = (𝐴𝐻(π‘β€˜π‘ˆ)) ↔ ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺𝐴) = (GIdβ€˜πΊ)))
3427, 33mpbird 256 1 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) = (𝐴𝐻(π‘β€˜π‘ˆ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ran crn 5677  β€˜cfv 6543  (class class class)co 7408  1st c1st 7972  2nd c2nd 7973  GrpOpcgr 29737  GIdcgi 29738  invcgn 29739  RingOpscrngo 36757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-1st 7974  df-2nd 7975  df-grpo 29741  df-gid 29742  df-ginv 29743  df-ablo 29793  df-ass 36706  df-exid 36708  df-mgmOLD 36712  df-sgrOLD 36724  df-mndo 36730  df-rngo 36758
This theorem is referenced by:  rngonegrmul  36807
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