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Theorem rngonegmn1r 37471
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 5933 . . . . . . . . 9 ran 𝐺 = ran (1st β€˜π‘…)
41, 3eqtri 2753 . . . . . . . 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 37468 . . . . . . 7 (𝑅 ∈ RingOps β†’ π‘ˆ ∈ 𝑋)
8 ringneg.4 . . . . . . . 8 𝑁 = (invβ€˜πΊ)
92, 1, 8rngonegcl 37456 . . . . . . 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 479 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π‘ˆ) ∈ 𝑋)
127adantr 479 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ π‘ˆ ∈ 𝑋)
1311, 12jca 510 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π‘ˆ) ∈ 𝑋 ∧ π‘ˆ ∈ 𝑋))
142, 5, 1rngodi 37433 . . . . . 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 1351 . . . . 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 426 . . . 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 2725 . . . . . . . 8 (GIdβ€˜πΊ) = (GIdβ€˜πΊ)
192, 1, 8, 18rngoaddneg2 37458 . . . . . . 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 479 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π‘ˆ)πΊπ‘ˆ) = (GIdβ€˜πΊ))
2221oveq2d 7431 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐻((π‘β€˜π‘ˆ)πΊπ‘ˆ)) = (𝐴𝐻(GIdβ€˜πΊ)))
2318, 1, 2, 5rngorz 37452 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐻(GIdβ€˜πΊ)) = (GIdβ€˜πΊ))
2422, 23eqtrd 2765 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐻((π‘β€˜π‘ˆ)πΊπ‘ˆ)) = (GIdβ€˜πΊ))
255, 4, 6rngoridm 37467 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π΄π»π‘ˆ) = 𝐴)
2625oveq2d 7431 . . 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 2774 . 2 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺𝐴) = (GIdβ€˜πΊ))
282, 5, 1rngocl 37430 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ (π‘β€˜π‘ˆ) ∈ 𝑋) β†’ (𝐴𝐻(π‘β€˜π‘ˆ)) ∈ 𝑋)
2911, 28mpd3an3 1458 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐻(π‘β€˜π‘ˆ)) ∈ 𝑋)
302rngogrpo 37439 . . . 4 (𝑅 ∈ RingOps β†’ 𝐺 ∈ GrpOp)
311, 18, 8grpoinvid2 30381 . . . 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 1160 . . 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 1458 . 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 394   = wceq 1533   ∈ wcel 2098  ran crn 5673  β€˜cfv 6542  (class class class)co 7415  1st c1st 7987  2nd c2nd 7988  GrpOpcgr 30341  GIdcgi 30342  invcgn 30343  RingOpscrngo 37423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-1st 7989  df-2nd 7990  df-grpo 30345  df-gid 30346  df-ginv 30347  df-ablo 30397  df-ass 37372  df-exid 37374  df-mgmOLD 37378  df-sgrOLD 37390  df-mndo 37396  df-rngo 37424
This theorem is referenced by:  rngonegrmul  37473
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