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Theorem rngonegmn1r 26580
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 5099 . . . . . . . . 9  |-  ran  G  =  ran  ( 1st `  R
)
41, 3eqtri 2458 . . . . . . . 8  |-  X  =  ran  ( 1st `  R
)
5 ringneg.2 . . . . . . . 8  |-  H  =  ( 2nd `  R
)
6 ringneg.5 . . . . . . . 8  |-  U  =  (GId `  H )
74, 5, 6rngo1cl 22022 . . . . . . 7  |-  ( R  e.  RingOps  ->  U  e.  X
)
8 ringneg.4 . . . . . . . 8  |-  N  =  ( inv `  G
)
92, 1, 8rngonegcl 26575 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  U  e.  X )  ->  ( N `  U )  e.  X )
107, 9mpdan 651 . . . . . 6  |-  ( R  e.  RingOps  ->  ( N `  U )  e.  X
)
1110adantr 453 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( N `  U )  e.  X )
127adantr 453 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  U  e.  X )
1311, 12jca 520 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( N `  U
)  e.  X  /\  U  e.  X )
)
142, 5, 1rngodi 21978 . . . . . 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 1172 . . . . 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 580 . . . 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 651 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( ( N `
 U ) G U ) )  =  ( ( A H ( N `  U
) ) G ( A H U ) ) )
18 eqid 2438 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
192, 1, 8, 18rngoaddneg2 26577 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  U  e.  X )  ->  (
( N `  U
) G U )  =  (GId `  G
) )
207, 19mpdan 651 . . . . . 6  |-  ( R  e.  RingOps  ->  ( ( N `
 U ) G U )  =  (GId
`  G ) )
2120adantr 453 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( N `  U
) G U )  =  (GId `  G
) )
2221oveq2d 6100 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( ( N `
 U ) G U ) )  =  ( A H (GId
`  G ) ) )
2318, 1, 2, 5rngorz 21995 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H (GId `  G
) )  =  (GId
`  G ) )
2422, 23eqtrd 2470 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( ( N `
 U ) G U ) )  =  (GId `  G )
)
255, 4, 6rngoridm 22018 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H U )  =  A )
2625oveq2d 6100 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( A H ( N `  U ) ) G ( A H U ) )  =  ( ( A H ( N `  U ) ) G A ) )
2717, 24, 263eqtr3rd 2479 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( A H ( N `  U ) ) G A )  =  (GId `  G
) )
282, 5, 1rngocl 21975 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  ( N `  U )  e.  X )  ->  ( A H ( N `  U ) )  e.  X )
2911, 28mpd3an3 1281 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( N `  U ) )  e.  X )
302rngogrpo 21983 . . . 4  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
311, 18, 8grpoinvid2 21824 . . . 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 1218 . . 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 1281 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( N `  A
)  =  ( A H ( N `  U ) )  <->  ( ( A H ( N `  U ) ) G A )  =  (GId
`  G ) ) )
3427, 33mpbird 225 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( N `  A )  =  ( A H ( N `  U
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   ran crn 4882   ` cfv 5457  (class class class)co 6084   1stc1st 6350   2ndc2nd 6351   GrpOpcgr 21779  GIdcgi 21780   invcgn 21781   RingOpscrngo 21968
This theorem is referenced by:  rngonegrmul  26582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-1st 6352  df-2nd 6353  df-riota 6552  df-grpo 21784  df-gid 21785  df-ginv 21786  df-ablo 21875  df-ass 21906  df-exid 21908  df-mgm 21912  df-sgr 21924  df-mndo 21931  df-rngo 21969
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