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Theorem gxnval 20943
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 2296 . . . 4  |-  (GId `  G )  =  (GId
`  G )
3 gxnval.3 . . . 4  |-  N  =  ( inv `  G
)
4 gxnval.2 . . . 4  |-  P  =  ( ^g `  G
)
51, 2, 3, 4gxval 20941 . . 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 1179 . 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 8854 . . . . . 6  |-  0  e.  RR
87ltnri 8945 . . . . 5  |-  -.  0  <  0
9 breq1 4042 . . . . 5  |-  ( K  =  0  ->  ( K  <  0  <->  0  <  0 ) )
108, 9mtbiri 294 . . . 4  |-  ( K  =  0  ->  -.  K  <  0 )
11 simp3r 984 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  K  <  0 ) )  ->  K  <  0 )
1210, 11nsyl3 111 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  K  <  0 ) )  ->  -.  K  =  0
)
13 iffalse 3585 . . 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 15 . 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 10044 . . . . . 6  |-  ( K  e.  ZZ  ->  K  e.  RR )
16 ltnsym 8935 . . . . . 6  |-  ( ( K  e.  RR  /\  0  e.  RR )  ->  ( K  <  0  ->  -.  0  <  K
) )
1715, 7, 16sylancl 643 . . . . 5  |-  ( K  e.  ZZ  ->  ( K  <  0  ->  -.  0  <  K ) )
1817imp 418 . . . 4  |-  ( ( K  e.  ZZ  /\  K  <  0 )  ->  -.  0  <  K )
19183ad2ant3 978 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  K  <  0 ) )  ->  -.  0  <  K )
20 iffalse 3585 . . 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 15 . 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 2332 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ifcif 3578   {csn 3653   class class class wbr 4039    X. cxp 4703   ran crn 4706   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    < clt 8883   -ucneg 9054   NNcn 9762   ZZcz 10040    seq cseq 11062   GrpOpcgr 20869  GIdcgi 20870   invcgn 20871   ^gcgx 20873
This theorem is referenced by:  gxnn0neg  20946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-neg 9056  df-z 10041  df-seq 11063  df-gx 20878
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