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Theorem rngonegmn1r 30328
Description: Negation in a ring is the same as right multiplication by  -u 1. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
ringneg.1  |-  G  =  ( 1st `  R
)
ringneg.2  |-  H  =  ( 2nd `  R
)
ringneg.3  |-  X  =  ran  G
ringneg.4  |-  N  =  ( inv `  G
)
ringneg.5  |-  U  =  (GId `  H )
Assertion
Ref Expression
rngonegmn1r  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( N `  A )  =  ( A H ( N `  U
) ) )

Proof of Theorem rngonegmn1r
StepHypRef Expression
1 ringneg.3 . . . . . . . . 9  |-  X  =  ran  G
2 ringneg.1 . . . . . . . . . 10  |-  G  =  ( 1st `  R
)
32rneqi 5219 . . . . . . . . 9  |-  ran  G  =  ran  ( 1st `  R
)
41, 3eqtri 2472 . . . . . . . 8  |-  X  =  ran  ( 1st `  R
)
5 ringneg.2 . . . . . . . 8  |-  H  =  ( 2nd `  R
)
6 ringneg.5 . . . . . . . 8  |-  U  =  (GId `  H )
74, 5, 6rngo1cl 25303 . . . . . . 7  |-  ( R  e.  RingOps  ->  U  e.  X
)
8 ringneg.4 . . . . . . . 8  |-  N  =  ( inv `  G
)
92, 1, 8rngonegcl 30323 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  U  e.  X )  ->  ( N `  U )  e.  X )
107, 9mpdan 668 . . . . . 6  |-  ( R  e.  RingOps  ->  ( N `  U )  e.  X
)
1110adantr 465 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( N `  U )  e.  X )
127adantr 465 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  U  e.  X )
1311, 12jca 532 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( N `  U
)  e.  X  /\  U  e.  X )
)
142, 5, 1rngodi 25259 . . . . . 6  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  ( N `  U )  e.  X  /\  U  e.  X ) )  -> 
( A H ( ( N `  U
) G U ) )  =  ( ( A H ( N `
 U ) ) G ( A H U ) ) )
15143exp2 1215 . . . . 5  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  ( ( N `  U )  e.  X  ->  ( U  e.  X  ->  ( A H ( ( N `
 U ) G U ) )  =  ( ( A H ( N `  U
) ) G ( A H U ) ) ) ) ) )
1615imp43 595 . . . 4  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  ( ( N `  U )  e.  X  /\  U  e.  X
) )  ->  ( A H ( ( N `
 U ) G U ) )  =  ( ( A H ( N `  U
) ) G ( A H U ) ) )
1713, 16mpdan 668 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( ( N `
 U ) G U ) )  =  ( ( A H ( N `  U
) ) G ( A H U ) ) )
18 eqid 2443 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
192, 1, 8, 18rngoaddneg2 30325 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  U  e.  X )  ->  (
( N `  U
) G U )  =  (GId `  G
) )
207, 19mpdan 668 . . . . . 6  |-  ( R  e.  RingOps  ->  ( ( N `
 U ) G U )  =  (GId
`  G ) )
2120adantr 465 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( N `  U
) G U )  =  (GId `  G
) )
2221oveq2d 6297 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( ( N `
 U ) G U ) )  =  ( A H (GId
`  G ) ) )
2318, 1, 2, 5rngorz 25276 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H (GId `  G
) )  =  (GId
`  G ) )
2422, 23eqtrd 2484 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( ( N `
 U ) G U ) )  =  (GId `  G )
)
255, 4, 6rngoridm 25299 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H U )  =  A )
2625oveq2d 6297 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( A H ( N `  U ) ) G ( A H U ) )  =  ( ( A H ( N `  U ) ) G A ) )
2717, 24, 263eqtr3rd 2493 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( A H ( N `  U ) ) G A )  =  (GId `  G
) )
282, 5, 1rngocl 25256 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  ( N `  U )  e.  X )  ->  ( A H ( N `  U ) )  e.  X )
2911, 28mpd3an3 1326 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( N `  U ) )  e.  X )
302rngogrpo 25264 . . . 4  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
311, 18, 8grpoinvid2 25105 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( A H ( N `  U ) )  e.  X )  ->  (
( N `  A
)  =  ( A H ( N `  U ) )  <->  ( ( A H ( N `  U ) ) G A )  =  (GId
`  G ) ) )
3230, 31syl3an1 1262 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  ( A H ( N `  U ) )  e.  X )  ->  (
( N `  A
)  =  ( A H ( N `  U ) )  <->  ( ( A H ( N `  U ) ) G A )  =  (GId
`  G ) ) )
3329, 32mpd3an3 1326 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( N `  A
)  =  ( A H ( N `  U ) )  <->  ( ( A H ( N `  U ) ) G A )  =  (GId
`  G ) ) )
3427, 33mpbird 232 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( N `  A )  =  ( A H ( N `  U
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   ran crn 4990   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784   GrpOpcgr 25060  GIdcgi 25061   invcgn 25062   RingOpscrngo 25249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-1st 6785  df-2nd 6786  df-grpo 25065  df-gid 25066  df-ginv 25067  df-ablo 25156  df-ass 25187  df-exid 25189  df-mgmOLD 25193  df-sgrOLD 25205  df-mndo 25212  df-rngo 25250
This theorem is referenced by:  rngonegrmul  30330
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