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Theorem gxneg 21703
Description: A negative group power is the inverse of the positive power. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
Hypotheses
Ref Expression
gxneg.1  |-  X  =  ran  G
gxneg.2  |-  N  =  ( inv `  G
)
gxneg.3  |-  P  =  ( ^g `  G
)
Assertion
Ref Expression
gxneg  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P -u K )  =  ( N `  ( A P K ) ) )

Proof of Theorem gxneg
StepHypRef Expression
1 gxneg.1 . . . . . . 7  |-  X  =  ran  G
2 gxneg.2 . . . . . . 7  |-  N  =  ( inv `  G
)
3 gxneg.3 . . . . . . 7  |-  P  =  ( ^g `  G
)
41, 2, 3gxnn0neg 21700 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e. 
NN0 )  ->  ( A P -u K )  =  ( N `  ( A P K ) ) )
543adant3l 1180 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  K  e.  NN0 ) )  -> 
( A P -u K )  =  ( N `  ( A P K ) ) )
653expia 1155 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( K  e.  ZZ  /\  K  e.  NN0 )  ->  ( A P -u K )  =  ( N `  ( A P K ) ) ) )
76exp3a 426 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  ZZ  ->  ( K  e.  NN0  ->  ( A P -u K
)  =  ( N `
 ( A P K ) ) ) ) )
873impia 1150 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( K  e.  NN0  ->  ( A P -u K )  =  ( N `  ( A P K ) ) ) )
9 simp1 957 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  -u K  e.  NN0 ) )  ->  G  e.  GrpOp )
10 znegcl 10246 . . . . . . . . 9  |-  ( K  e.  ZZ  ->  -u K  e.  ZZ )
111, 3gxcl 21702 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  -u K  e.  ZZ )  ->  ( A P -u K )  e.  X )
1210, 11syl3an3 1219 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P -u K )  e.  X )
13123adant3r 1181 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  -u K  e.  NN0 ) )  -> 
( A P -u K )  e.  X
)
141, 2grpo2inv 21676 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( A P -u K )  e.  X )  -> 
( N `  ( N `  ( A P -u K ) ) )  =  ( A P -u K ) )
159, 13, 14syl2anc 643 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  -u K  e.  NN0 ) )  -> 
( N `  ( N `  ( A P -u K ) ) )  =  ( A P -u K ) )
161, 2, 3gxnn0neg 21700 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  -u K  e.  NN0 )  ->  ( A P -u -u K
)  =  ( N `
 ( A P
-u K ) ) )
17163adant3l 1180 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  -u K  e.  NN0 ) )  -> 
( A P -u -u K )  =  ( N `  ( A P -u K ) ) )
18 simp3l 985 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  -u K  e.  NN0 ) )  ->  K  e.  ZZ )
19 zcn 10220 . . . . . . . . 9  |-  ( K  e.  ZZ  ->  K  e.  CC )
20 negneg 9284 . . . . . . . . . 10  |-  ( K  e.  CC  ->  -u -u K  =  K )
2120oveq2d 6037 . . . . . . . . 9  |-  ( K  e.  CC  ->  ( A P -u -u K
)  =  ( A P K ) )
2218, 19, 213syl 19 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  -u K  e.  NN0 ) )  -> 
( A P -u -u K )  =  ( A P K ) )
2317, 22eqtr3d 2422 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  -u K  e.  NN0 ) )  -> 
( N `  ( A P -u K ) )  =  ( A P K ) )
2423fveq2d 5673 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  -u K  e.  NN0 ) )  -> 
( N `  ( N `  ( A P -u K ) ) )  =  ( N `
 ( A P K ) ) )
2515, 24eqtr3d 2422 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  -u K  e.  NN0 ) )  -> 
( A P -u K )  =  ( N `  ( A P K ) ) )
26253expia 1155 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( K  e.  ZZ  /\  -u K  e.  NN0 )  ->  ( A P
-u K )  =  ( N `  ( A P K ) ) ) )
2726exp3a 426 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  ZZ  ->  (
-u K  e.  NN0  ->  ( A P -u K )  =  ( N `  ( A P K ) ) ) ) )
28273impia 1150 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( -u K  e.  NN0  ->  ( A P -u K
)  =  ( N `
 ( A P K ) ) ) )
29 elznn0 10229 . . . 4  |-  ( K  e.  ZZ  <->  ( K  e.  RR  /\  ( K  e.  NN0  \/  -u K  e.  NN0 ) ) )
3029simprbi 451 . . 3  |-  ( K  e.  ZZ  ->  ( K  e.  NN0  \/  -u K  e.  NN0 ) )
31303ad2ant3 980 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( K  e.  NN0  \/  -u K  e.  NN0 ) )
328, 28, 31mpjaod 371 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P -u K )  =  ( N `  ( A P K ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ran crn 4820   ` cfv 5395  (class class class)co 6021   CCcc 8922   RRcr 8923   -ucneg 9225   NN0cn0 10154   ZZcz 10215   GrpOpcgr 21623   invcgn 21625   ^gcgx 21627
This theorem is referenced by:  gxneg2  21704  gxm1  21705  gxcom  21706  gxinv  21707  gxsuc  21709  gxid  21710  gxadd  21712  gxsub  21713  gxmul  21715  gxdi  21733
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-n0 10155  df-z 10216  df-uz 10422  df-seq 11252  df-grpo 21628  df-gid 21629  df-ginv 21630  df-gx 21632
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