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Theorem rngonegmn1r 26087
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 5008 . . . . . . . . 9  |-  ran  G  =  ran  ( 1st `  R
)
41, 3eqtri 2386 . . . . . . . 8  |-  X  =  ran  ( 1st `  R
)
5 ringneg.2 . . . . . . . 8  |-  H  =  ( 2nd `  R
)
6 ringneg.5 . . . . . . . 8  |-  U  =  (GId `  H )
74, 5, 6rngo1cl 21407 . . . . . . 7  |-  ( R  e.  RingOps  ->  U  e.  X
)
8 ringneg.4 . . . . . . . 8  |-  N  =  ( inv `  G
)
92, 1, 8rngonegcl 26082 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  U  e.  X )  ->  ( N `  U )  e.  X )
107, 9mpdan 649 . . . . . 6  |-  ( R  e.  RingOps  ->  ( N `  U )  e.  X
)
1110adantr 451 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( N `  U )  e.  X )
127adantr 451 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  U  e.  X )
1311, 12jca 518 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( N `  U
)  e.  X  /\  U  e.  X )
)
142, 5, 1rngodi 21363 . . . . . 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 1170 . . . . 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 578 . . . 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 649 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( ( N `
 U ) G U ) )  =  ( ( A H ( N `  U
) ) G ( A H U ) ) )
18 eqid 2366 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
192, 1, 8, 18rngoaddneg2 26084 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  U  e.  X )  ->  (
( N `  U
) G U )  =  (GId `  G
) )
207, 19mpdan 649 . . . . . 6  |-  ( R  e.  RingOps  ->  ( ( N `
 U ) G U )  =  (GId
`  G ) )
2120adantr 451 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( N `  U
) G U )  =  (GId `  G
) )
2221oveq2d 5997 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( ( N `
 U ) G U ) )  =  ( A H (GId
`  G ) ) )
2318, 1, 2, 5rngorz 21380 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H (GId `  G
) )  =  (GId
`  G ) )
2422, 23eqtrd 2398 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( ( N `
 U ) G U ) )  =  (GId `  G )
)
255, 4, 6rngoridm 21403 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H U )  =  A )
2625oveq2d 5997 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( A H ( N `  U ) ) G ( A H U ) )  =  ( ( A H ( N `  U ) ) G A ) )
2717, 24, 263eqtr3rd 2407 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( A H ( N `  U ) ) G A )  =  (GId `  G
) )
282, 5, 1rngocl 21360 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  ( N `  U )  e.  X )  ->  ( A H ( N `  U ) )  e.  X )
2911, 28mpd3an3 1279 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( N `  U ) )  e.  X )
302rngogrpo 21368 . . . 4  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
311, 18, 8grpoinvid2 21209 . . . 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 1216 . . 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 1279 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( N `  A
)  =  ( A H ( N `  U ) )  <->  ( ( A H ( N `  U ) ) G A )  =  (GId
`  G ) ) )
3427, 33mpbird 223 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( N `  A )  =  ( A H ( N `  U
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715   ran crn 4793   ` cfv 5358  (class class class)co 5981   1stc1st 6247   2ndc2nd 6248   GrpOpcgr 21164  GIdcgi 21165   invcgn 21166   RingOpscrngo 21353
This theorem is referenced by:  rngonegrmul  26089
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-1st 6249  df-2nd 6250  df-riota 6446  df-grpo 21169  df-gid 21170  df-ginv 21171  df-ablo 21260  df-ass 21291  df-exid 21293  df-mgm 21297  df-sgr 21309  df-mndo 21316  df-rngo 21354
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