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Theorem rngonegmn1l 37299
Description: Negation in a ring is the same as left multiplication by -1. (Contributed by Jeff Madsen, 10-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
rngonegmn1l ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) = ((π‘β€˜π‘ˆ)𝐻𝐴))
Proof of Theorem rngonegmn1l
StepHypRef Expression
1 ringneg.3 . . . . . . 7 𝑋 = ran 𝐺
2 ringneg.1 . . . . . . . 8 𝐺 = (1st β€˜π‘…)
32rneqi 5926 . . . . . . 7 ran 𝐺 = ran (1st β€˜π‘…)
41, 3eqtri 2752 . . . . . 6 𝑋 = ran (1st β€˜π‘…)
5 ringneg.2 . . . . . 6 𝐻 = (2nd β€˜π‘…)
6 ringneg.5 . . . . . 6 π‘ˆ = (GIdβ€˜π»)
74, 5, 6rngo1cl 37297 . . . . 5 (𝑅 ∈ RingOps β†’ π‘ˆ ∈ 𝑋)
8 ringneg.4 . . . . . . 7 𝑁 = (invβ€˜πΊ)
92, 1, 8rngonegcl 37285 . . . . . 6 ((𝑅 ∈ RingOps ∧ π‘ˆ ∈ 𝑋) β†’ (π‘β€˜π‘ˆ) ∈ 𝑋)
107, 9mpdan 684 . . . . 5 (𝑅 ∈ RingOps β†’ (π‘β€˜π‘ˆ) ∈ 𝑋)
117, 10jca 511 . . . 4 (𝑅 ∈ RingOps β†’ (π‘ˆ ∈ 𝑋 ∧ (π‘β€˜π‘ˆ) ∈ 𝑋))
122, 5, 1rngodir 37263 . . . . . . 7 ((𝑅 ∈ RingOps ∧ (π‘ˆ ∈ 𝑋 ∧ (π‘β€˜π‘ˆ) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) β†’ ((π‘ˆπΊ(π‘β€˜π‘ˆ))𝐻𝐴) = ((π‘ˆπ»π΄)𝐺((π‘β€˜π‘ˆ)𝐻𝐴)))
13123exp2 1351 . . . . . 6 (𝑅 ∈ RingOps β†’ (π‘ˆ ∈ 𝑋 β†’ ((π‘β€˜π‘ˆ) ∈ 𝑋 β†’ (𝐴 ∈ 𝑋 β†’ ((π‘ˆπΊ(π‘β€˜π‘ˆ))𝐻𝐴) = ((π‘ˆπ»π΄)𝐺((π‘β€˜π‘ˆ)𝐻𝐴))))))
1413imp42 426 . . . . 5 (((𝑅 ∈ RingOps ∧ (π‘ˆ ∈ 𝑋 ∧ (π‘β€˜π‘ˆ) ∈ 𝑋)) ∧ 𝐴 ∈ 𝑋) β†’ ((π‘ˆπΊ(π‘β€˜π‘ˆ))𝐻𝐴) = ((π‘ˆπ»π΄)𝐺((π‘β€˜π‘ˆ)𝐻𝐴)))
1514an32s 649 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ (π‘ˆ ∈ 𝑋 ∧ (π‘β€˜π‘ˆ) ∈ 𝑋)) β†’ ((π‘ˆπΊ(π‘β€˜π‘ˆ))𝐻𝐴) = ((π‘ˆπ»π΄)𝐺((π‘β€˜π‘ˆ)𝐻𝐴)))
1611, 15mpidan 686 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘ˆπΊ(π‘β€˜π‘ˆ))𝐻𝐴) = ((π‘ˆπ»π΄)𝐺((π‘β€˜π‘ˆ)𝐻𝐴)))
17 eqid 2724 . . . . . . . 8 (GIdβ€˜πΊ) = (GIdβ€˜πΊ)
182, 1, 8, 17rngoaddneg1 37286 . . . . . . 7 ((𝑅 ∈ RingOps ∧ π‘ˆ ∈ 𝑋) β†’ (π‘ˆπΊ(π‘β€˜π‘ˆ)) = (GIdβ€˜πΊ))
197, 18mpdan 684 . . . . . 6 (𝑅 ∈ RingOps β†’ (π‘ˆπΊ(π‘β€˜π‘ˆ)) = (GIdβ€˜πΊ))
2019adantr 480 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π‘ˆπΊ(π‘β€˜π‘ˆ)) = (GIdβ€˜πΊ))
2120oveq1d 7416 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘ˆπΊ(π‘β€˜π‘ˆ))𝐻𝐴) = ((GIdβ€˜πΊ)𝐻𝐴))
2217, 1, 2, 5rngolz 37280 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((GIdβ€˜πΊ)𝐻𝐴) = (GIdβ€˜πΊ))
2321, 22eqtrd 2764 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘ˆπΊ(π‘β€˜π‘ˆ))𝐻𝐴) = (GIdβ€˜πΊ))
245, 4, 6rngolidm 37295 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π‘ˆπ»π΄) = 𝐴)
2524oveq1d 7416 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘ˆπ»π΄)𝐺((π‘β€˜π‘ˆ)𝐻𝐴)) = (𝐴𝐺((π‘β€˜π‘ˆ)𝐻𝐴)))
2616, 23, 253eqtr3rd 2773 . 2 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐺((π‘β€˜π‘ˆ)𝐻𝐴)) = (GIdβ€˜πΊ))
272, 5, 1rngocl 37259 . . . . . 6 ((𝑅 ∈ RingOps ∧ (π‘β€˜π‘ˆ) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π‘ˆ)𝐻𝐴) ∈ 𝑋)
28273expa 1115 . . . . 5 (((𝑅 ∈ RingOps ∧ (π‘β€˜π‘ˆ) ∈ 𝑋) ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π‘ˆ)𝐻𝐴) ∈ 𝑋)
2928an32s 649 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ (π‘β€˜π‘ˆ) ∈ 𝑋) β†’ ((π‘β€˜π‘ˆ)𝐻𝐴) ∈ 𝑋)
3010, 29mpidan 686 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π‘ˆ)𝐻𝐴) ∈ 𝑋)
312rngogrpo 37268 . . . 4 (𝑅 ∈ RingOps β†’ 𝐺 ∈ GrpOp)
321, 17, 8grpoinvid1 30250 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ ((π‘β€˜π‘ˆ)𝐻𝐴) ∈ 𝑋) β†’ ((π‘β€˜π΄) = ((π‘β€˜π‘ˆ)𝐻𝐴) ↔ (𝐴𝐺((π‘β€˜π‘ˆ)𝐻𝐴)) = (GIdβ€˜πΊ)))
3331, 32syl3an1 1160 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ ((π‘β€˜π‘ˆ)𝐻𝐴) ∈ 𝑋) β†’ ((π‘β€˜π΄) = ((π‘β€˜π‘ˆ)𝐻𝐴) ↔ (𝐴𝐺((π‘β€˜π‘ˆ)𝐻𝐴)) = (GIdβ€˜πΊ)))
3430, 33mpd3an3 1458 . 2 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π΄) = ((π‘β€˜π‘ˆ)𝐻𝐴) ↔ (𝐴𝐺((π‘β€˜π‘ˆ)𝐻𝐴)) = (GIdβ€˜πΊ)))
3526, 34mpbird 257 1 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) = ((π‘β€˜π‘ˆ)𝐻𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ran crn 5667  β€˜cfv 6533  (class class class)co 7401  1st c1st 7966  2nd c2nd 7967  GrpOpcgr 30211  GIdcgi 30212  invcgn 30213  RingOpscrngo 37252
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 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-1st 7968  df-2nd 7969  df-grpo 30215  df-gid 30216  df-ginv 30217  df-ablo 30267  df-ass 37201  df-exid 37203  df-mgmOLD 37207  df-sgrOLD 37219  df-mndo 37225  df-rngo 37253
This theorem is referenced by:  rngoneglmul  37301  idlnegcl  37380
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