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Theorem rngonegmn1l 36809
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 5937 . . . . . . 7 ran 𝐺 = ran (1st β€˜π‘…)
41, 3eqtri 2761 . . . . . 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 36807 . . . . 5 (𝑅 ∈ RingOps β†’ π‘ˆ ∈ 𝑋)
8 ringneg.4 . . . . . . 7 𝑁 = (invβ€˜πΊ)
92, 1, 8rngonegcl 36795 . . . . . 6 ((𝑅 ∈ RingOps ∧ π‘ˆ ∈ 𝑋) β†’ (π‘β€˜π‘ˆ) ∈ 𝑋)
107, 9mpdan 686 . . . . 5 (𝑅 ∈ RingOps β†’ (π‘β€˜π‘ˆ) ∈ 𝑋)
117, 10jca 513 . . . 4 (𝑅 ∈ RingOps β†’ (π‘ˆ ∈ 𝑋 ∧ (π‘β€˜π‘ˆ) ∈ 𝑋))
122, 5, 1rngodir 36773 . . . . . . 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 1355 . . . . . 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 428 . . . . 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 651 . . . 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 688 . . 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 2733 . . . . . . . 8 (GIdβ€˜πΊ) = (GIdβ€˜πΊ)
182, 1, 8, 17rngoaddneg1 36796 . . . . . . 7 ((𝑅 ∈ RingOps ∧ π‘ˆ ∈ 𝑋) β†’ (π‘ˆπΊ(π‘β€˜π‘ˆ)) = (GIdβ€˜πΊ))
197, 18mpdan 686 . . . . . 6 (𝑅 ∈ RingOps β†’ (π‘ˆπΊ(π‘β€˜π‘ˆ)) = (GIdβ€˜πΊ))
2019adantr 482 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π‘ˆπΊ(π‘β€˜π‘ˆ)) = (GIdβ€˜πΊ))
2120oveq1d 7424 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘ˆπΊ(π‘β€˜π‘ˆ))𝐻𝐴) = ((GIdβ€˜πΊ)𝐻𝐴))
2217, 1, 2, 5rngolz 36790 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((GIdβ€˜πΊ)𝐻𝐴) = (GIdβ€˜πΊ))
2321, 22eqtrd 2773 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘ˆπΊ(π‘β€˜π‘ˆ))𝐻𝐴) = (GIdβ€˜πΊ))
245, 4, 6rngolidm 36805 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π‘ˆπ»π΄) = 𝐴)
2524oveq1d 7424 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘ˆπ»π΄)𝐺((π‘β€˜π‘ˆ)𝐻𝐴)) = (𝐴𝐺((π‘β€˜π‘ˆ)𝐻𝐴)))
2616, 23, 253eqtr3rd 2782 . 2 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐺((π‘β€˜π‘ˆ)𝐻𝐴)) = (GIdβ€˜πΊ))
272, 5, 1rngocl 36769 . . . . . 6 ((𝑅 ∈ RingOps ∧ (π‘β€˜π‘ˆ) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π‘ˆ)𝐻𝐴) ∈ 𝑋)
28273expa 1119 . . . . 5 (((𝑅 ∈ RingOps ∧ (π‘β€˜π‘ˆ) ∈ 𝑋) ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π‘ˆ)𝐻𝐴) ∈ 𝑋)
2928an32s 651 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ (π‘β€˜π‘ˆ) ∈ 𝑋) β†’ ((π‘β€˜π‘ˆ)𝐻𝐴) ∈ 𝑋)
3010, 29mpidan 688 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π‘ˆ)𝐻𝐴) ∈ 𝑋)
312rngogrpo 36778 . . . 4 (𝑅 ∈ RingOps β†’ 𝐺 ∈ GrpOp)
321, 17, 8grpoinvid1 29781 . . . 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 1164 . . 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 1463 . 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 397   = wceq 1542   ∈ wcel 2107  ran crn 5678  β€˜cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  GrpOpcgr 29742  GIdcgi 29743  invcgn 29744  RingOpscrngo 36762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-1st 7975  df-2nd 7976  df-grpo 29746  df-gid 29747  df-ginv 29748  df-ablo 29798  df-ass 36711  df-exid 36713  df-mgmOLD 36717  df-sgrOLD 36729  df-mndo 36735  df-rngo 36763
This theorem is referenced by:  rngoneglmul  36811  idlnegcl  36890
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