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Theorem rngonegmn1r 29943
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 5220 . . . . . . . . 9  |-  ran  G  =  ran  ( 1st `  R
)
41, 3eqtri 2489 . . . . . . . 8  |-  X  =  ran  ( 1st `  R
)
5 ringneg.2 . . . . . . . 8  |-  H  =  ( 2nd `  R
)
6 ringneg.5 . . . . . . . 8  |-  U  =  (GId `  H )
74, 5, 6rngo1cl 25093 . . . . . . 7  |-  ( R  e.  RingOps  ->  U  e.  X
)
8 ringneg.4 . . . . . . . 8  |-  N  =  ( inv `  G
)
92, 1, 8rngonegcl 29938 . . . . . . 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 25049 . . . . . 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 1209 . . . . 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 2460 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
192, 1, 8, 18rngoaddneg2 29940 . . . . . . 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 6291 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( ( N `
 U ) G U ) )  =  ( A H (GId
`  G ) ) )
2318, 1, 2, 5rngorz 25066 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H (GId `  G
) )  =  (GId
`  G ) )
2422, 23eqtrd 2501 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( ( N `
 U ) G U ) )  =  (GId `  G )
)
255, 4, 6rngoridm 25089 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H U )  =  A )
2625oveq2d 6291 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( A H ( N `  U ) ) G ( A H U ) )  =  ( ( A H ( N `  U ) ) G A ) )
2717, 24, 263eqtr3rd 2510 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( A H ( N `  U ) ) G A )  =  (GId `  G
) )
282, 5, 1rngocl 25046 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  ( N `  U )  e.  X )  ->  ( A H ( N `  U ) )  e.  X )
2911, 28mpd3an3 1320 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( N `  U ) )  e.  X )
302rngogrpo 25054 . . . 4  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
311, 18, 8grpoinvid2 24895 . . . 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 1256 . . 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 1320 . 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 1374    e. wcel 1762   ran crn 4993   ` cfv 5579  (class class class)co 6275   1stc1st 6772   2ndc2nd 6773   GrpOpcgr 24850  GIdcgi 24851   invcgn 24852   RingOpscrngo 25039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-1st 6774  df-2nd 6775  df-grpo 24855  df-gid 24856  df-ginv 24857  df-ablo 24946  df-ass 24977  df-exid 24979  df-mgm 24983  df-sgr 24995  df-mndo 25002  df-rngo 25040
This theorem is referenced by:  rngonegrmul  29945
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