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Theorem rngonegmn1l 26249
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 5029 . . . . . . . 8  |-  ran  G  =  ran  ( 1st `  R
)
41, 3eqtri 2400 . . . . . . 7  |-  X  =  ran  ( 1st `  R
)
5 ringneg.2 . . . . . . 7  |-  H  =  ( 2nd `  R
)
6 ringneg.5 . . . . . . 7  |-  U  =  (GId `  H )
74, 5, 6rngo1cl 21858 . . . . . 6  |-  ( R  e.  RingOps  ->  U  e.  X
)
8 ringneg.4 . . . . . . . 8  |-  N  =  ( inv `  G
)
92, 1, 8rngonegcl 26245 . . . . . . 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 21815 . . . . . . 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 2380 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
192, 1, 8, 18rngoaddneg1 26246 . . . . . . 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 6028 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U G ( N `  U ) ) H A )  =  ( (GId `  G ) H A ) )
2318, 1, 2, 5rngolz 21830 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
(GId `  G ) H A )  =  (GId
`  G ) )
2422, 23eqtrd 2412 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U G ( N `  U ) ) H A )  =  (GId `  G
) )
255, 4, 6rngolidm 21853 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( U H A )  =  A )
2625oveq1d 6028 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U H A ) G ( ( N `  U ) H A ) )  =  ( A G ( ( N `  U ) H A ) ) )
2717, 24, 263eqtr3rd 2421 . 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 21811 . . . . . 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 21819 . . . 4  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
341, 18, 8grpoinvid1 21659 . . . 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 1649    e. wcel 1717   ran crn 4812   ` cfv 5387  (class class class)co 6013   1stc1st 6279   2ndc2nd 6280   GrpOpcgr 21615  GIdcgi 21616   invcgn 21617   RingOpscrngo 21804
This theorem is referenced by:  rngoneglmul  26251  idlnegcl  26316
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-1st 6281  df-2nd 6282  df-riota 6478  df-grpo 21620  df-gid 21621  df-ginv 21622  df-ablo 21711  df-ass 21742  df-exid 21744  df-mgm 21748  df-sgr 21760  df-mndo 21767  df-rngo 21805
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