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Theorem rngonegmn1l 26556
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 5088 . . . . . . . 8  |-  ran  G  =  ran  ( 1st `  R
)
41, 3eqtri 2455 . . . . . . 7  |-  X  =  ran  ( 1st `  R
)
5 ringneg.2 . . . . . . 7  |-  H  =  ( 2nd `  R
)
6 ringneg.5 . . . . . . 7  |-  U  =  (GId `  H )
74, 5, 6rngo1cl 22009 . . . . . 6  |-  ( R  e.  RingOps  ->  U  e.  X
)
8 ringneg.4 . . . . . . . 8  |-  N  =  ( inv `  G
)
92, 1, 8rngonegcl 26552 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  U  e.  X )  ->  ( N `  U )  e.  X )
107, 9mpdan 650 . . . . . 6  |-  ( R  e.  RingOps  ->  ( N `  U )  e.  X
)
117, 10jca 519 . . . . 5  |-  ( R  e.  RingOps  ->  ( U  e.  X  /\  ( N `
 U )  e.  X ) )
1211adantr 452 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( U  e.  X  /\  ( N `  U )  e.  X ) )
132, 5, 1rngodir 21966 . . . . . . 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 1171 . . . . . 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 578 . . . . 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 780 . . . 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 650 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U G ( N `  U ) ) H A )  =  ( ( U H A ) G ( ( N `  U ) H A ) ) )
18 eqid 2435 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
192, 1, 8, 18rngoaddneg1 26553 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  U  e.  X )  ->  ( U G ( N `  U ) )  =  (GId `  G )
)
207, 19mpdan 650 . . . . . 6  |-  ( R  e.  RingOps  ->  ( U G ( N `  U
) )  =  (GId
`  G ) )
2120adantr 452 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( U G ( N `  U ) )  =  (GId `  G )
)
2221oveq1d 6088 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U G ( N `  U ) ) H A )  =  ( (GId `  G ) H A ) )
2318, 1, 2, 5rngolz 21981 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
(GId `  G ) H A )  =  (GId
`  G ) )
2422, 23eqtrd 2467 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U G ( N `  U ) ) H A )  =  (GId `  G
) )
255, 4, 6rngolidm 22004 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( U H A )  =  A )
2625oveq1d 6088 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U H A ) G ( ( N `  U ) H A ) )  =  ( A G ( ( N `  U ) H A ) ) )
2717, 24, 263eqtr3rd 2476 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A G ( ( N `
 U ) H A ) )  =  (GId `  G )
)
2810adantr 452 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( N `  U )  e.  X )
292, 5, 1rngocl 21962 . . . . . 6  |-  ( ( R  e.  RingOps  /\  ( N `  U )  e.  X  /\  A  e.  X )  ->  (
( N `  U
) H A )  e.  X )
30293expa 1153 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  ( N `  U )  e.  X )  /\  A  e.  X )  ->  ( ( N `  U ) H A )  e.  X )
3130an32s 780 . . . 4  |-  ( ( ( R  e.  RingOps  /\  A  e.  X )  /\  ( N `  U
)  e.  X )  ->  ( ( N `
 U ) H A )  e.  X
)
3228, 31mpdan 650 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( N `  U
) H A )  e.  X )
332rngogrpo 21970 . . . 4  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
341, 18, 8grpoinvid1 21810 . . . 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 1217 . . 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 1280 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( N `  A
)  =  ( ( N `  U ) H A )  <->  ( A G ( ( N `
 U ) H A ) )  =  (GId `  G )
) )
3727, 36mpbird 224 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( N `  A )  =  ( ( N `
 U ) H A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   ran crn 4871   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   GrpOpcgr 21766  GIdcgi 21767   invcgn 21768   RingOpscrngo 21955
This theorem is referenced by:  rngoneglmul  26558  idlnegcl  26623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-1st 6341  df-2nd 6342  df-riota 6541  df-grpo 21771  df-gid 21772  df-ginv 21773  df-ablo 21862  df-ass 21893  df-exid 21895  df-mgm 21899  df-sgr 21911  df-mndo 21918  df-rngo 21956
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