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Theorem gxnval 21696
Description: The result of the group power operator when the exponent is negative. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
gxnval.1  |-  X  =  ran  G
gxnval.2  |-  P  =  ( ^g `  G
)
gxnval.3  |-  N  =  ( inv `  G
)
Assertion
Ref Expression
gxnval  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  K  <  0 ) )  -> 
( A P K )  =  ( N `
 (  seq  1
( G ,  ( NN  X.  { A } ) ) `  -u K ) ) )

Proof of Theorem gxnval
StepHypRef Expression
1 gxnval.1 . . . 4  |-  X  =  ran  G
2 eqid 2387 . . . 4  |-  (GId `  G )  =  (GId
`  G )
3 gxnval.3 . . . 4  |-  N  =  ( inv `  G
)
4 gxnval.2 . . . 4  |-  P  =  ( ^g `  G
)
51, 2, 3, 4gxval 21694 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P K )  =  if ( K  =  0 ,  (GId `  G ) ,  if ( 0  <  K ,  (  seq  1
( G ,  ( NN  X.  { A } ) ) `  K ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u K ) ) ) ) )
653adant3r 1181 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  K  <  0 ) )  -> 
( A P K )  =  if ( K  =  0 ,  (GId `  G ) ,  if ( 0  < 
K ,  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  K ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u K ) ) ) ) )
7 0re 9024 . . . . . 6  |-  0  e.  RR
87ltnri 9115 . . . . 5  |-  -.  0  <  0
9 breq1 4156 . . . . 5  |-  ( K  =  0  ->  ( K  <  0  <->  0  <  0 ) )
108, 9mtbiri 295 . . . 4  |-  ( K  =  0  ->  -.  K  <  0 )
11 simp3r 986 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  K  <  0 ) )  ->  K  <  0 )
1210, 11nsyl3 113 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  K  <  0 ) )  ->  -.  K  =  0
)
13 iffalse 3689 . . 3  |-  ( -.  K  =  0  ->  if ( K  =  0 ,  (GId `  G
) ,  if ( 0  <  K , 
(  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  K
) ,  ( N `
 (  seq  1
( G ,  ( NN  X.  { A } ) ) `  -u K ) ) ) )  =  if ( 0  <  K , 
(  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  K
) ,  ( N `
 (  seq  1
( G ,  ( NN  X.  { A } ) ) `  -u K ) ) ) )
1412, 13syl 16 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  K  <  0 ) )  ->  if ( K  =  0 ,  (GId `  G
) ,  if ( 0  <  K , 
(  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  K
) ,  ( N `
 (  seq  1
( G ,  ( NN  X.  { A } ) ) `  -u K ) ) ) )  =  if ( 0  <  K , 
(  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  K
) ,  ( N `
 (  seq  1
( G ,  ( NN  X.  { A } ) ) `  -u K ) ) ) )
15 zre 10218 . . . . . 6  |-  ( K  e.  ZZ  ->  K  e.  RR )
16 ltnsym 9105 . . . . . 6  |-  ( ( K  e.  RR  /\  0  e.  RR )  ->  ( K  <  0  ->  -.  0  <  K
) )
1715, 7, 16sylancl 644 . . . . 5  |-  ( K  e.  ZZ  ->  ( K  <  0  ->  -.  0  <  K ) )
1817imp 419 . . . 4  |-  ( ( K  e.  ZZ  /\  K  <  0 )  ->  -.  0  <  K )
19183ad2ant3 980 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  K  <  0 ) )  ->  -.  0  <  K )
20 iffalse 3689 . . 3  |-  ( -.  0  <  K  ->  if ( 0  <  K ,  (  seq  1
( G ,  ( NN  X.  { A } ) ) `  K ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u K ) ) )  =  ( N `  (  seq  1 ( G ,  ( NN  X.  { A } ) ) `
 -u K ) ) )
2119, 20syl 16 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  K  <  0 ) )  ->  if ( 0  <  K ,  (  seq  1
( G ,  ( NN  X.  { A } ) ) `  K ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u K ) ) )  =  ( N `  (  seq  1 ( G ,  ( NN  X.  { A } ) ) `
 -u K ) ) )
226, 14, 213eqtrd 2423 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  K  <  0 ) )  -> 
( A P K )  =  ( N `
 (  seq  1
( G ,  ( NN  X.  { A } ) ) `  -u K ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ifcif 3682   {csn 3757   class class class wbr 4153    X. cxp 4816   ran crn 4819   ` cfv 5394  (class class class)co 6020   RRcr 8922   0cc0 8923   1c1 8924    < clt 9053   -ucneg 9224   NNcn 9932   ZZcz 10214    seq cseq 11250   GrpOpcgr 21622  GIdcgi 21623   invcgn 21624   ^gcgx 21626
This theorem is referenced by:  gxnn0neg  21699
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-i2m1 8991  ax-1ne0 8992  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-ltxr 9058  df-neg 9226  df-z 10215  df-seq 11251  df-gx 21631
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