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Theorem gxnval 26000
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 2453 . . . 4  |-  (GId `  G )  =  (GId
`  G )
3 gxnval.3 . . . 4  |-  N  =  ( inv `  G
)
4 gxnval.2 . . . 4  |-  P  =  ( ^g `  G
)
51, 2, 3, 4gxval 25998 . . 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 1266 . 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 9648 . . . . . 6  |-  0  e.  RR
87ltnri 9748 . . . . 5  |-  -.  0  <  0
9 breq1 4408 . . . . 5  |-  ( K  =  0  ->  ( K  <  0  <->  0  <  0 ) )
108, 9mtbiri 305 . . . 4  |-  ( K  =  0  ->  -.  K  <  0 )
11 simp3r 1038 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  K  <  0 ) )  ->  K  <  0 )
1210, 11nsyl3 123 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  K  <  0 ) )  ->  -.  K  =  0
)
1312iffalsed 3894 . 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 ) ) ) )
14 zre 10948 . . . . . 6  |-  ( K  e.  ZZ  ->  K  e.  RR )
15 ltnsym 9737 . . . . . 6  |-  ( ( K  e.  RR  /\  0  e.  RR )  ->  ( K  <  0  ->  -.  0  <  K
) )
1614, 7, 15sylancl 669 . . . . 5  |-  ( K  e.  ZZ  ->  ( K  <  0  ->  -.  0  <  K ) )
1716imp 431 . . . 4  |-  ( ( K  e.  ZZ  /\  K  <  0 )  ->  -.  0  <  K )
18173ad2ant3 1032 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  K  <  0 ) )  ->  -.  0  <  K )
1918iffalsed 3894 . 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 ) ) )
206, 13, 193eqtrd 2491 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   ifcif 3883   {csn 3970   class class class wbr 4405    X. cxp 4835   ran crn 4838   ` cfv 5585  (class class class)co 6295   RRcr 9543   0cc0 9544   1c1 9545    < clt 9680   -ucneg 9866   NNcn 10616   ZZcz 10944    seqcseq 12220   GrpOpcgr 25926  GIdcgi 25927   invcgn 25928   ^gcgx 25930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-i2m1 9612  ax-1ne0 9613  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-po 4758  df-so 4759  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-ltxr 9685  df-neg 9868  df-z 10945  df-seq 12221  df-gx 25935
This theorem is referenced by:  gxnn0neg  26003
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