Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngonegmn1l Unicode version

Theorem rngonegmn1l 26580
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 4905 . . . . . . . 8  |-  ran  G  =  ran  ( 1st `  R
)
41, 3eqtri 2303 . . . . . . 7  |-  X  =  ran  ( 1st `  R
)
5 ringneg.2 . . . . . . 7  |-  H  =  ( 2nd `  R
)
6 ringneg.5 . . . . . . 7  |-  U  =  (GId `  H )
74, 5, 6rngo1cl 21096 . . . . . 6  |-  ( R  e.  RingOps  ->  U  e.  X
)
8 ringneg.4 . . . . . . . 8  |-  N  =  ( inv `  G
)
92, 1, 8rngonegcl 26576 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  U  e.  X )  ->  ( N `  U )  e.  X )
107, 9mpdan 649 . . . . . 6  |-  ( R  e.  RingOps  ->  ( N `  U )  e.  X
)
117, 10jca 518 . . . . 5  |-  ( R  e.  RingOps  ->  ( U  e.  X  /\  ( N `
 U )  e.  X ) )
1211adantr 451 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( U  e.  X  /\  ( N `  U )  e.  X ) )
132, 5, 1rngodir 21053 . . . . . . 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 1169 . . . . . 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 577 . . . . 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 779 . . . 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 649 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U G ( N `  U ) ) H A )  =  ( ( U H A ) G ( ( N `  U ) H A ) ) )
18 eqid 2283 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
192, 1, 8, 18rngoaddneg1 26577 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  U  e.  X )  ->  ( U G ( N `  U ) )  =  (GId `  G )
)
207, 19mpdan 649 . . . . . 6  |-  ( R  e.  RingOps  ->  ( U G ( N `  U
) )  =  (GId
`  G ) )
2120adantr 451 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( U G ( N `  U ) )  =  (GId `  G )
)
2221oveq1d 5873 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U G ( N `  U ) ) H A )  =  ( (GId `  G ) H A ) )
2318, 1, 2, 5rngolz 21068 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
(GId `  G ) H A )  =  (GId
`  G ) )
2422, 23eqtrd 2315 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U G ( N `  U ) ) H A )  =  (GId `  G
) )
255, 4, 6rngolidm 21091 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( U H A )  =  A )
2625oveq1d 5873 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U H A ) G ( ( N `  U ) H A ) )  =  ( A G ( ( N `  U ) H A ) ) )
2717, 24, 263eqtr3rd 2324 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A G ( ( N `
 U ) H A ) )  =  (GId `  G )
)
2810adantr 451 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( N `  U )  e.  X )
292, 5, 1rngocl 21049 . . . . . 6  |-  ( ( R  e.  RingOps  /\  ( N `  U )  e.  X  /\  A  e.  X )  ->  (
( N `  U
) H A )  e.  X )
30293expa 1151 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  ( N `  U )  e.  X )  /\  A  e.  X )  ->  ( ( N `  U ) H A )  e.  X )
3130an32s 779 . . . 4  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  ( N `  U
)  e.  X )  ->  ( ( N `
 U ) H A )  e.  X
)
3228, 31mpdan 649 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( N `  U
) H A )  e.  X )
332rngogrpo 21057 . . . 4  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
341, 18, 8grpoinvid1 20897 . . . 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 1215 . . 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 1278 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( N `  A
)  =  ( ( N `  U ) H A )  <->  ( A G ( ( N `
 U ) H A ) )  =  (GId `  G )
) )
3727, 36mpbird 223 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( N `  A )  =  ( ( N `
 U ) H A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   GrpOpcgr 20853  GIdcgi 20854   invcgn 20855   RingOpscrngo 21042
This theorem is referenced by:  rngoneglmul  26582  idlnegcl  26647
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-ablo 20949  df-ass 20980  df-exid 20982  df-mgm 20986  df-sgr 20998  df-mndo 21005  df-rngo 21043
  Copyright terms: Public domain W3C validator