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Theorem gxneg 21846
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 21843 . . . . . 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 10305 . . . . . . . . 9  |-  ( K  e.  ZZ  ->  -u K  e.  ZZ )
111, 3gxcl 21845 . . . . . . . . 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 21819 . . . . . . 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 21843 . . . . . . . . 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 10279 . . . . . . . . 9  |-  ( K  e.  ZZ  ->  K  e.  CC )
20 negneg 9343 . . . . . . . . . 10  |-  ( K  e.  CC  ->  -u -u K  =  K )
2120oveq2d 6089 . . . . . . . . 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 2469 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  -u K  e.  NN0 ) )  -> 
( N `  ( A P -u K ) )  =  ( A P K ) )
2423fveq2d 5724 . . . . . 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 2469 . . . . 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 10288 . . . 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 1652    e. wcel 1725   ran crn 4871   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   -ucneg 9284   NN0cn0 10213   ZZcz 10274   GrpOpcgr 21766   invcgn 21768   ^gcgx 21770
This theorem is referenced by:  gxneg2  21847  gxm1  21848  gxcom  21849  gxinv  21850  gxsuc  21852  gxid  21853  gxadd  21855  gxsub  21856  gxmul  21858  gxdi  21876
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-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  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-riota 6541  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-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-seq 11316  df-grpo 21771  df-gid 21772  df-ginv 21773  df-gx 21775
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