MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gxnval Structured version   Unicode version

Theorem gxnval 21840
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 2435 . . . 4  |-  (GId `  G )  =  (GId
`  G )
3 gxnval.3 . . . 4  |-  N  =  ( inv `  G
)
4 gxnval.2 . . . 4  |-  P  =  ( ^g `  G
)
51, 2, 3, 4gxval 21838 . . 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 9083 . . . . . 6  |-  0  e.  RR
87ltnri 9174 . . . . 5  |-  -.  0  <  0
9 breq1 4207 . . . . 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 3738 . . 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 10278 . . . . . 6  |-  ( K  e.  ZZ  ->  K  e.  RR )
16 ltnsym 9164 . . . . . 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 3738 . . 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 2471 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 1652    e. wcel 1725   ifcif 3731   {csn 3806   class class class wbr 4204    X. cxp 4868   ran crn 4871   ` cfv 5446  (class class class)co 6073   RRcr 8981   0cc0 8982   1c1 8983    < clt 9112   -ucneg 9284   NNcn 9992   ZZcz 10274    seq cseq 11315   GrpOpcgr 21766  GIdcgi 21767   invcgn 21768   ^gcgx 21770
This theorem is referenced by:  gxnn0neg  21843
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  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-i2m1 9050  ax-1ne0 9051  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  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-pw 3793  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-po 4495  df-so 4496  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-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-ltxr 9117  df-neg 9286  df-z 10275  df-seq 11316  df-gx 21775
  Copyright terms: Public domain W3C validator