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Theorem gxnn0neg 21700
Description: A negative group power is the inverse of the positive power (lemma with nonnegative exponent - use gxneg 21703 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
gxnn0neg.1  |-  X  =  ran  G
gxnn0neg.2  |-  N  =  ( inv `  G
)
gxnn0neg.3  |-  P  =  ( ^g `  G
)
Assertion
Ref Expression
gxnn0neg  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e. 
NN0 )  ->  ( A P -u K )  =  ( N `  ( A P K ) ) )

Proof of Theorem gxnn0neg
StepHypRef Expression
1 elnn0 10156 . . 3  |-  ( K  e.  NN0  <->  ( K  e.  NN  \/  K  =  0 ) )
2 nnnegz 10218 . . . . . . . 8  |-  ( K  e.  NN  ->  -u K  e.  ZZ )
3 nngt0 9962 . . . . . . . . 9  |-  ( K  e.  NN  ->  0  <  K )
4 nnre 9940 . . . . . . . . . 10  |-  ( K  e.  NN  ->  K  e.  RR )
54lt0neg2d 9530 . . . . . . . . 9  |-  ( K  e.  NN  ->  (
0  <  K  <->  -u K  <  0 ) )
63, 5mpbid 202 . . . . . . . 8  |-  ( K  e.  NN  ->  -u K  <  0 )
72, 6jca 519 . . . . . . 7  |-  ( K  e.  NN  ->  ( -u K  e.  ZZ  /\  -u K  <  0 ) )
8 gxnn0neg.1 . . . . . . . 8  |-  X  =  ran  G
9 gxnn0neg.3 . . . . . . . 8  |-  P  =  ( ^g `  G
)
10 gxnn0neg.2 . . . . . . . 8  |-  N  =  ( inv `  G
)
118, 9, 10gxnval 21697 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( -u K  e.  ZZ  /\  -u K  <  0 ) )  ->  ( A P -u K )  =  ( N `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u -u K ) ) )
127, 11syl3an3 1219 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P -u K )  =  ( N `  (  seq  1 ( G ,  ( NN  X.  { A } ) ) `
 -u -u K ) ) )
138, 9gxpval 21696 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P K )  =  (  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  K
) )
14 nncn 9941 . . . . . . . . . . 11  |-  ( K  e.  NN  ->  K  e.  CC )
1514negnegd 9335 . . . . . . . . . 10  |-  ( K  e.  NN  ->  -u -u K  =  K )
1615fveq2d 5673 . . . . . . . . 9  |-  ( K  e.  NN  ->  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u -u K )  =  (  seq  1 ( G ,  ( NN  X.  { A } ) ) `
 K ) )
17163ad2ant3 980 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u -u K )  =  (  seq  1 ( G ,  ( NN  X.  { A } ) ) `
 K ) )
1813, 17eqtr4d 2423 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P K )  =  (  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  -u -u K
) )
1918fveq2d 5673 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( N `  ( A P K ) )  =  ( N `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u -u K ) ) )
2012, 19eqtr4d 2423 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P -u K )  =  ( N `  ( A P K ) ) )
21203expia 1155 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  NN  ->  ( A P -u K
)  =  ( N `
 ( A P K ) ) ) )
22 negeq 9231 . . . . . . . . 9  |-  ( K  =  0  ->  -u K  =  -u 0 )
23 neg0 9280 . . . . . . . . 9  |-  -u 0  =  0
2422, 23syl6eq 2436 . . . . . . . 8  |-  ( K  =  0  ->  -u K  =  0 )
2524oveq2d 6037 . . . . . . 7  |-  ( K  =  0  ->  ( A P -u K )  =  ( A P 0 ) )
26 eqid 2388 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
278, 26, 9gx0 21698 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
2825, 27sylan9eqr 2442 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P -u K )  =  (GId `  G )
)
29 oveq2 6029 . . . . . . . 8  |-  ( K  =  0  ->  ( A P K )  =  ( A P 0 ) )
3029fveq2d 5673 . . . . . . 7  |-  ( K  =  0  ->  ( N `  ( A P K ) )  =  ( N `  ( A P 0 ) ) )
3127fveq2d 5673 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  ( A P 0 ) )  =  ( N `  (GId `  G ) ) )
3226, 10grpoinvid 21669 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  ( N `  (GId `  G )
)  =  (GId `  G ) )
3332adantr 452 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  (GId `  G
) )  =  (GId
`  G ) )
3431, 33eqtrd 2420 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  ( A P 0 ) )  =  (GId `  G
) )
3530, 34sylan9eqr 2442 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( N `  ( A P K ) )  =  (GId
`  G ) )
3628, 35eqtr4d 2423 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P -u K )  =  ( N `  ( A P K ) ) )
3736ex 424 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  =  0  ->  ( A P -u K
)  =  ( N `
 ( A P K ) ) ) )
3821, 37jaod 370 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( K  e.  NN  \/  K  =  0
)  ->  ( A P -u K )  =  ( N `  ( A P K ) ) ) )
391, 38syl5bi 209 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  NN0  ->  ( A P -u K )  =  ( N `  ( A P K ) ) ) )
40393impia 1150 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e. 
NN0 )  ->  ( 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   {csn 3758   class class class wbr 4154    X. cxp 4817   ran crn 4820   ` cfv 5395  (class class class)co 6021   0cc0 8924   1c1 8925    < clt 9054   -ucneg 9225   NNcn 9933   NN0cn0 10154   ZZcz 10215    seq cseq 11251   GrpOpcgr 21623  GIdcgi 21624   invcgn 21625   ^gcgx 21627
This theorem is referenced by:  gxcl  21702  gxneg  21703
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-seq 11252  df-grpo 21628  df-gid 21629  df-ginv 21630  df-gx 21632
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