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Theorem gxnn0neg 20930
Description: A negative group power is the inverse of the positive power (lemma with nonnegative exponent - use gxneg 20933 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 9967 . . 3  |-  ( K  e.  NN0  <->  ( K  e.  NN  \/  K  =  0 ) )
2 nnnegz 10027 . . . . . . . 8  |-  ( K  e.  NN  ->  -u K  e.  ZZ )
3 nngt0 9775 . . . . . . . . 9  |-  ( K  e.  NN  ->  0  <  K )
4 nnre 9753 . . . . . . . . . 10  |-  ( K  e.  NN  ->  K  e.  RR )
54lt0neg2d 9343 . . . . . . . . 9  |-  ( K  e.  NN  ->  (
0  <  K  <->  -u K  <  0 ) )
63, 5mpbid 201 . . . . . . . 8  |-  ( K  e.  NN  ->  -u K  <  0 )
72, 6jca 518 . . . . . . 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 20927 . . . . . . 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 1217 . . . . . 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 20926 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P K )  =  (  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  K
) )
14 nncn 9754 . . . . . . . . . . 11  |-  ( K  e.  NN  ->  K  e.  CC )
1514negnegd 9148 . . . . . . . . . 10  |-  ( K  e.  NN  ->  -u -u K  =  K )
1615fveq2d 5529 . . . . . . . . 9  |-  ( K  e.  NN  ->  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u -u K )  =  (  seq  1 ( G ,  ( NN  X.  { A } ) ) `
 K ) )
17163ad2ant3 978 . . . . . . . 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 2318 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P K )  =  (  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  -u -u K
) )
1918fveq2d 5529 . . . . . 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 2318 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P -u K )  =  ( N `  ( A P K ) ) )
21203expia 1153 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  NN  ->  ( A P -u K
)  =  ( N `
 ( A P K ) ) ) )
22 negeq 9044 . . . . . . . . 9  |-  ( K  =  0  ->  -u K  =  -u 0 )
23 neg0 9093 . . . . . . . . 9  |-  -u 0  =  0
2422, 23syl6eq 2331 . . . . . . . 8  |-  ( K  =  0  ->  -u K  =  0 )
2524oveq2d 5874 . . . . . . 7  |-  ( K  =  0  ->  ( A P -u K )  =  ( A P 0 ) )
26 eqid 2283 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
278, 26, 9gx0 20928 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
2825, 27sylan9eqr 2337 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P -u K )  =  (GId `  G )
)
29 oveq2 5866 . . . . . . . 8  |-  ( K  =  0  ->  ( A P K )  =  ( A P 0 ) )
3029fveq2d 5529 . . . . . . 7  |-  ( K  =  0  ->  ( N `  ( A P K ) )  =  ( N `  ( A P 0 ) ) )
3127fveq2d 5529 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  ( A P 0 ) )  =  ( N `  (GId `  G ) ) )
3226, 10grpoinvid 20899 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  ( N `  (GId `  G )
)  =  (GId `  G ) )
3332adantr 451 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  (GId `  G
) )  =  (GId
`  G ) )
3431, 33eqtrd 2315 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  ( A P 0 ) )  =  (GId `  G
) )
3530, 34sylan9eqr 2337 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( N `  ( A P K ) )  =  (GId
`  G ) )
3628, 35eqtr4d 2318 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P -u K )  =  ( N `  ( A P K ) ) )
3736ex 423 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  =  0  ->  ( A P -u K
)  =  ( N `
 ( A P K ) ) ) )
3821, 37jaod 369 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( K  e.  NN  \/  K  =  0
)  ->  ( A P -u K )  =  ( N `  ( A P K ) ) ) )
391, 38syl5bi 208 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  NN0  ->  ( A P -u K )  =  ( N `  ( A P K ) ) ) )
40393impia 1148 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 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {csn 3640   class class class wbr 4023    X. cxp 4687   ran crn 4690   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738    < clt 8867   -ucneg 9038   NNcn 9746   NN0cn0 9965   ZZcz 10024    seq cseq 11046   GrpOpcgr 20853  GIdcgi 20854   invcgn 20855   ^gcgx 20857
This theorem is referenced by:  gxcl  20932  gxneg  20933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-seq 11047  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gx 20862
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