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Theorem rngonegmn1r 37337
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 2755 . . . . . . . 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 37334 . . . . . . 7 (𝑅 ∈ RingOps β†’ π‘ˆ ∈ 𝑋)
8 ringneg.4 . . . . . . . 8 𝑁 = (invβ€˜πΊ)
92, 1, 8rngonegcl 37322 . . . . . . 7 ((𝑅 ∈ RingOps ∧ π‘ˆ ∈ 𝑋) β†’ (π‘β€˜π‘ˆ) ∈ 𝑋)
107, 9mpdan 686 . . . . . 6 (𝑅 ∈ RingOps β†’ (π‘β€˜π‘ˆ) ∈ 𝑋)
1110adantr 480 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π‘ˆ) ∈ 𝑋)
127adantr 480 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ π‘ˆ ∈ 𝑋)
1311, 12jca 511 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π‘ˆ) ∈ 𝑋 ∧ π‘ˆ ∈ 𝑋))
142, 5, 1rngodi 37299 . . . . . 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 1352 . . . . 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 427 . . . 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 686 . . 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 2727 . . . . . . . 8 (GIdβ€˜πΊ) = (GIdβ€˜πΊ)
192, 1, 8, 18rngoaddneg2 37324 . . . . . . 7 ((𝑅 ∈ RingOps ∧ π‘ˆ ∈ 𝑋) β†’ ((π‘β€˜π‘ˆ)πΊπ‘ˆ) = (GIdβ€˜πΊ))
207, 19mpdan 686 . . . . . 6 (𝑅 ∈ RingOps β†’ ((π‘β€˜π‘ˆ)πΊπ‘ˆ) = (GIdβ€˜πΊ))
2120adantr 480 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π‘ˆ)πΊπ‘ˆ) = (GIdβ€˜πΊ))
2221oveq2d 7430 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐻((π‘β€˜π‘ˆ)πΊπ‘ˆ)) = (𝐴𝐻(GIdβ€˜πΊ)))
2318, 1, 2, 5rngorz 37318 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐻(GIdβ€˜πΊ)) = (GIdβ€˜πΊ))
2422, 23eqtrd 2767 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐻((π‘β€˜π‘ˆ)πΊπ‘ˆ)) = (GIdβ€˜πΊ))
255, 4, 6rngoridm 37333 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π΄π»π‘ˆ) = 𝐴)
2625oveq2d 7430 . . 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 2776 . 2 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺𝐴) = (GIdβ€˜πΊ))
282, 5, 1rngocl 37296 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ (π‘β€˜π‘ˆ) ∈ 𝑋) β†’ (𝐴𝐻(π‘β€˜π‘ˆ)) ∈ 𝑋)
2911, 28mpd3an3 1459 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐻(π‘β€˜π‘ˆ)) ∈ 𝑋)
302rngogrpo 37305 . . . 4 (𝑅 ∈ RingOps β†’ 𝐺 ∈ GrpOp)
311, 18, 8grpoinvid2 30313 . . . 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 1161 . . 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 1459 . 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 257 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 395   = wceq 1534   ∈ wcel 2099  ran crn 5673  β€˜cfv 6542  (class class class)co 7414  1st c1st 7983  2nd c2nd 7984  GrpOpcgr 30273  GIdcgi 30274  invcgn 30275  RingOpscrngo 37289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  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 7370  df-ov 7417  df-1st 7985  df-2nd 7986  df-grpo 30277  df-gid 30278  df-ginv 30279  df-ablo 30329  df-ass 37238  df-exid 37240  df-mgmOLD 37244  df-sgrOLD 37256  df-mndo 37262  df-rngo 37290
This theorem is referenced by:  rngonegrmul  37339
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