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Theorem rngonegmn1l 32252
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 . . . . . . 7  |-  X  =  ran  G
2 ringneg.1 . . . . . . . 8  |-  G  =  ( 1st `  R
)
32rneqi 5067 . . . . . . 7  |-  ran  G  =  ran  ( 1st `  R
)
41, 3eqtri 2493 . . . . . 6  |-  X  =  ran  ( 1st `  R
)
5 ringneg.2 . . . . . 6  |-  H  =  ( 2nd `  R
)
6 ringneg.5 . . . . . 6  |-  U  =  (GId `  H )
74, 5, 6rngo1cl 26238 . . . . 5  |-  ( R  e.  RingOps  ->  U  e.  X
)
8 ringneg.4 . . . . . . 7  |-  N  =  ( inv `  G
)
92, 1, 8rngonegcl 32248 . . . . . 6  |-  ( ( R  e.  RingOps  /\  U  e.  X )  ->  ( N `  U )  e.  X )
107, 9mpdan 681 . . . . 5  |-  ( R  e.  RingOps  ->  ( N `  U )  e.  X
)
117, 10jca 541 . . . 4  |-  ( R  e.  RingOps  ->  ( U  e.  X  /\  ( N `
 U )  e.  X ) )
122, 5, 1rngodir 26195 . . . . . . 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 ) ) )
13123exp2 1251 . . . . . 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 ) ) ) ) ) )
1413imp42 605 . . . . 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 ) ) )
1514an32s 821 . . . 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 ) ) )
1611, 15mpidan 683 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U G ( N `  U ) ) H A )  =  ( ( U H A ) G ( ( N `  U ) H A ) ) )
17 eqid 2471 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
182, 1, 8, 17rngoaddneg1 32249 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  U  e.  X )  ->  ( U G ( N `  U ) )  =  (GId `  G )
)
197, 18mpdan 681 . . . . . 6  |-  ( R  e.  RingOps  ->  ( U G ( N `  U
) )  =  (GId
`  G ) )
2019adantr 472 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( U G ( N `  U ) )  =  (GId `  G )
)
2120oveq1d 6323 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U G ( N `  U ) ) H A )  =  ( (GId `  G ) H A ) )
2217, 1, 2, 5rngolz 26210 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
(GId `  G ) H A )  =  (GId
`  G ) )
2321, 22eqtrd 2505 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U G ( N `  U ) ) H A )  =  (GId `  G
) )
245, 4, 6rngolidm 26233 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( U H A )  =  A )
2524oveq1d 6323 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U H A ) G ( ( N `  U ) H A ) )  =  ( A G ( ( N `  U ) H A ) ) )
2616, 23, 253eqtr3rd 2514 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A G ( ( N `
 U ) H A ) )  =  (GId `  G )
)
272, 5, 1rngocl 26191 . . . . . 6  |-  ( ( R  e.  RingOps  /\  ( N `  U )  e.  X  /\  A  e.  X )  ->  (
( N `  U
) H A )  e.  X )
28273expa 1231 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  ( N `  U )  e.  X )  /\  A  e.  X )  ->  ( ( N `  U ) H A )  e.  X )
2928an32s 821 . . . 4  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  ( N `  U
)  e.  X )  ->  ( ( N `
 U ) H A )  e.  X
)
3010, 29mpidan 683 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( N `  U
) H A )  e.  X )
312rngogrpo 26199 . . . 4  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
321, 17, 8grpoinvid1 26039 . . . 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 ) ) )
3331, 32syl3an1 1325 . . 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 ) ) )
3430, 33mpd3an3 1391 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( N `  A
)  =  ( ( N `  U ) H A )  <->  ( A G ( ( N `
 U ) H A ) )  =  (GId `  G )
) )
3526, 34mpbird 240 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( N `  A )  =  ( ( N `
 U ) H A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   ran crn 4840   ` cfv 5589  (class class class)co 6308   1stc1st 6810   2ndc2nd 6811   GrpOpcgr 25995  GIdcgi 25996   invcgn 25997   RingOpscrngo 26184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-1st 6812  df-2nd 6813  df-grpo 26000  df-gid 26001  df-ginv 26002  df-ablo 26091  df-ass 26122  df-exid 26124  df-mgmOLD 26128  df-sgrOLD 26140  df-mndo 26147  df-rngo 26185
This theorem is referenced by:  rngoneglmul  32254  idlnegcl  32319
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