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Theorem rngonegmn1r 28681
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 5062 . . . . . . . . 9  |-  ran  G  =  ran  ( 1st `  R
)
41, 3eqtri 2461 . . . . . . . 8  |-  X  =  ran  ( 1st `  R
)
5 ringneg.2 . . . . . . . 8  |-  H  =  ( 2nd `  R
)
6 ringneg.5 . . . . . . . 8  |-  U  =  (GId `  H )
74, 5, 6rngo1cl 23851 . . . . . . 7  |-  ( R  e.  RingOps  ->  U  e.  X
)
8 ringneg.4 . . . . . . . 8  |-  N  =  ( inv `  G
)
92, 1, 8rngonegcl 28676 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  U  e.  X )  ->  ( N `  U )  e.  X )
107, 9mpdan 663 . . . . . 6  |-  ( R  e.  RingOps  ->  ( N `  U )  e.  X
)
1110adantr 462 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( N `  U )  e.  X )
127adantr 462 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  U  e.  X )
1311, 12jca 529 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( N `  U
)  e.  X  /\  U  e.  X )
)
142, 5, 1rngodi 23807 . . . . . 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 1200 . . . . 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 592 . . . 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 663 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( ( N `
 U ) G U ) )  =  ( ( A H ( N `  U
) ) G ( A H U ) ) )
18 eqid 2441 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
192, 1, 8, 18rngoaddneg2 28678 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  U  e.  X )  ->  (
( N `  U
) G U )  =  (GId `  G
) )
207, 19mpdan 663 . . . . . 6  |-  ( R  e.  RingOps  ->  ( ( N `
 U ) G U )  =  (GId
`  G ) )
2120adantr 462 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( N `  U
) G U )  =  (GId `  G
) )
2221oveq2d 6106 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( ( N `
 U ) G U ) )  =  ( A H (GId
`  G ) ) )
2318, 1, 2, 5rngorz 23824 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H (GId `  G
) )  =  (GId
`  G ) )
2422, 23eqtrd 2473 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( ( N `
 U ) G U ) )  =  (GId `  G )
)
255, 4, 6rngoridm 23847 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H U )  =  A )
2625oveq2d 6106 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( A H ( N `  U ) ) G ( A H U ) )  =  ( ( A H ( N `  U ) ) G A ) )
2717, 24, 263eqtr3rd 2482 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( A H ( N `  U ) ) G A )  =  (GId `  G
) )
282, 5, 1rngocl 23804 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  ( N `  U )  e.  X )  ->  ( A H ( N `  U ) )  e.  X )
2911, 28mpd3an3 1310 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( N `  U ) )  e.  X )
302rngogrpo 23812 . . . 4  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
311, 18, 8grpoinvid2 23653 . . . 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 1246 . . 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 1310 . 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 1364    e. wcel 1761   ran crn 4837   ` cfv 5415  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   GrpOpcgr 23608  GIdcgi 23609   invcgn 23610   RingOpscrngo 23797
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-1st 6576  df-2nd 6577  df-grpo 23613  df-gid 23614  df-ginv 23615  df-ablo 23704  df-ass 23735  df-exid 23737  df-mgm 23741  df-sgr 23753  df-mndo 23760  df-rngo 23798
This theorem is referenced by:  rngonegrmul  28683
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