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Theorem rngonegmn1l 28767
Description: Negation in a ring is the same as left multiplication by  -u 1. (Contributed by Jeff Madsen, 10-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
rngonegmn1l  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( N `  A )  =  ( ( N `
 U ) H A ) )

Proof of Theorem rngonegmn1l
StepHypRef Expression
1 ringneg.3 . . . . . . . 8  |-  X  =  ran  G
2 ringneg.1 . . . . . . . . 9  |-  G  =  ( 1st `  R
)
32rneqi 5078 . . . . . . . 8  |-  ran  G  =  ran  ( 1st `  R
)
41, 3eqtri 2463 . . . . . . 7  |-  X  =  ran  ( 1st `  R
)
5 ringneg.2 . . . . . . 7  |-  H  =  ( 2nd `  R
)
6 ringneg.5 . . . . . . 7  |-  U  =  (GId `  H )
74, 5, 6rngo1cl 23928 . . . . . 6  |-  ( R  e.  RingOps  ->  U  e.  X
)
8 ringneg.4 . . . . . . . 8  |-  N  =  ( inv `  G
)
92, 1, 8rngonegcl 28763 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  U  e.  X )  ->  ( N `  U )  e.  X )
107, 9mpdan 668 . . . . . 6  |-  ( R  e.  RingOps  ->  ( N `  U )  e.  X
)
117, 10jca 532 . . . . 5  |-  ( R  e.  RingOps  ->  ( U  e.  X  /\  ( N `
 U )  e.  X ) )
1211adantr 465 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( U  e.  X  /\  ( N `  U )  e.  X ) )
132, 5, 1rngodir 23885 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  ( U  e.  X  /\  ( N `  U )  e.  X  /\  A  e.  X ) )  -> 
( ( U G ( N `  U
) ) H A )  =  ( ( U H A ) G ( ( N `
 U ) H A ) ) )
14133exp2 1205 . . . . . 6  |-  ( R  e.  RingOps  ->  ( U  e.  X  ->  ( ( N `  U )  e.  X  ->  ( A  e.  X  ->  (
( U G ( N `  U ) ) H A )  =  ( ( U H A ) G ( ( N `  U ) H A ) ) ) ) ) )
1514imp42 594 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  ( U  e.  X  /\  ( N `  U
)  e.  X ) )  /\  A  e.  X )  ->  (
( U G ( N `  U ) ) H A )  =  ( ( U H A ) G ( ( N `  U ) H A ) ) )
1615an32s 802 . . . 4  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  ( U  e.  X  /\  ( N `  U
)  e.  X ) )  ->  ( ( U G ( N `  U ) ) H A )  =  ( ( U H A ) G ( ( N `  U ) H A ) ) )
1712, 16mpdan 668 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U G ( N `  U ) ) H A )  =  ( ( U H A ) G ( ( N `  U ) H A ) ) )
18 eqid 2443 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
192, 1, 8, 18rngoaddneg1 28764 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  U  e.  X )  ->  ( U G ( N `  U ) )  =  (GId `  G )
)
207, 19mpdan 668 . . . . . 6  |-  ( R  e.  RingOps  ->  ( U G ( N `  U
) )  =  (GId
`  G ) )
2120adantr 465 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( U G ( N `  U ) )  =  (GId `  G )
)
2221oveq1d 6118 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U G ( N `  U ) ) H A )  =  ( (GId `  G ) H A ) )
2318, 1, 2, 5rngolz 23900 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
(GId `  G ) H A )  =  (GId
`  G ) )
2422, 23eqtrd 2475 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U G ( N `  U ) ) H A )  =  (GId `  G
) )
255, 4, 6rngolidm 23923 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( U H A )  =  A )
2625oveq1d 6118 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U H A ) G ( ( N `  U ) H A ) )  =  ( A G ( ( N `  U ) H A ) ) )
2717, 24, 263eqtr3rd 2484 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A G ( ( N `
 U ) H A ) )  =  (GId `  G )
)
2810adantr 465 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( N `  U )  e.  X )
292, 5, 1rngocl 23881 . . . . . 6  |-  ( ( R  e.  RingOps  /\  ( N `  U )  e.  X  /\  A  e.  X )  ->  (
( N `  U
) H A )  e.  X )
30293expa 1187 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  ( N `  U )  e.  X )  /\  A  e.  X )  ->  ( ( N `  U ) H A )  e.  X )
3130an32s 802 . . . 4  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  ( N `  U
)  e.  X )  ->  ( ( N `
 U ) H A )  e.  X
)
3228, 31mpdan 668 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( N `  U
) H A )  e.  X )
332rngogrpo 23889 . . . 4  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
341, 18, 8grpoinvid1 23729 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  (
( N `  U
) H A )  e.  X )  -> 
( ( N `  A )  =  ( ( N `  U
) H A )  <-> 
( A G ( ( N `  U
) H A ) )  =  (GId `  G ) ) )
3533, 34syl3an1 1251 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  (
( N `  U
) H A )  e.  X )  -> 
( ( N `  A )  =  ( ( N `  U
) H A )  <-> 
( A G ( ( N `  U
) H A ) )  =  (GId `  G ) ) )
3632, 35mpd3an3 1315 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( N `  A
)  =  ( ( N `  U ) H A )  <->  ( A G ( ( N `
 U ) H A ) )  =  (GId `  G )
) )
3727, 36mpbird 232 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( N `  A )  =  ( ( N `
 U ) H A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   ran crn 4853   ` cfv 5430  (class class class)co 6103   1stc1st 6587   2ndc2nd 6588   GrpOpcgr 23685  GIdcgi 23686   invcgn 23687   RingOpscrngo 23874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-1st 6589  df-2nd 6590  df-grpo 23690  df-gid 23691  df-ginv 23692  df-ablo 23781  df-ass 23812  df-exid 23814  df-mgm 23818  df-sgr 23830  df-mndo 23837  df-rngo 23875
This theorem is referenced by:  rngoneglmul  28769  idlnegcl  28834
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