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Theorem gxnn0neg 24929
Description: A negative group power is the inverse of the positive power (lemma with nonnegative exponent - use gxneg 24932 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 10788 . . 3  |-  ( K  e.  NN0  <->  ( K  e.  NN  \/  K  =  0 ) )
2 nnnegz 10858 . . . . . . . 8  |-  ( K  e.  NN  ->  -u K  e.  ZZ )
3 nngt0 10556 . . . . . . . . 9  |-  ( K  e.  NN  ->  0  <  K )
4 nnre 10534 . . . . . . . . . 10  |-  ( K  e.  NN  ->  K  e.  RR )
54lt0neg2d 10114 . . . . . . . . 9  |-  ( K  e.  NN  ->  (
0  <  K  <->  -u K  <  0 ) )
63, 5mpbid 210 . . . . . . . 8  |-  ( K  e.  NN  ->  -u K  <  0 )
72, 6jca 532 . . . . . . 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 24926 . . . . . . 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 1258 . . . . . 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 24925 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P K )  =  (  seq 1 ( G ,  ( NN 
X.  { A }
) ) `  K
) )
14 nncn 10535 . . . . . . . . . . 11  |-  ( K  e.  NN  ->  K  e.  CC )
1514negnegd 9912 . . . . . . . . . 10  |-  ( K  e.  NN  ->  -u -u K  =  K )
1615fveq2d 5863 . . . . . . . . 9  |-  ( K  e.  NN  ->  (  seq 1 ( G , 
( NN  X.  { A } ) ) `  -u -u K )  =  (  seq 1 ( G ,  ( NN  X.  { A } ) ) `
 K ) )
17163ad2ant3 1014 . . . . . . . 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 2506 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P K )  =  (  seq 1 ( G ,  ( NN 
X.  { A }
) ) `  -u -u K
) )
1918fveq2d 5863 . . . . . 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 2506 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P -u K )  =  ( N `  ( A P K ) ) )
21203expia 1193 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  NN  ->  ( A P -u K
)  =  ( N `
 ( A P K ) ) ) )
22 negeq 9803 . . . . . . . . 9  |-  ( K  =  0  ->  -u K  =  -u 0 )
23 neg0 9856 . . . . . . . . 9  |-  -u 0  =  0
2422, 23syl6eq 2519 . . . . . . . 8  |-  ( K  =  0  ->  -u K  =  0 )
2524oveq2d 6293 . . . . . . 7  |-  ( K  =  0  ->  ( A P -u K )  =  ( A P 0 ) )
26 eqid 2462 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
278, 26, 9gx0 24927 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
2825, 27sylan9eqr 2525 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P -u K )  =  (GId `  G )
)
29 oveq2 6285 . . . . . . . 8  |-  ( K  =  0  ->  ( A P K )  =  ( A P 0 ) )
3029fveq2d 5863 . . . . . . 7  |-  ( K  =  0  ->  ( N `  ( A P K ) )  =  ( N `  ( A P 0 ) ) )
3127fveq2d 5863 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  ( A P 0 ) )  =  ( N `  (GId `  G ) ) )
3226, 10grpoinvid 24898 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  ( N `  (GId `  G )
)  =  (GId `  G ) )
3332adantr 465 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  (GId `  G
) )  =  (GId
`  G ) )
3431, 33eqtrd 2503 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  ( A P 0 ) )  =  (GId `  G
) )
3530, 34sylan9eqr 2525 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( N `  ( A P K ) )  =  (GId
`  G ) )
3628, 35eqtr4d 2506 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P -u K )  =  ( N `  ( A P K ) ) )
3736ex 434 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  =  0  ->  ( A P -u K
)  =  ( N `
 ( A P K ) ) ) )
3821, 37jaod 380 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( K  e.  NN  \/  K  =  0
)  ->  ( A P -u K )  =  ( N `  ( A P K ) ) ) )
391, 38syl5bi 217 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  NN0  ->  ( A P -u K )  =  ( N `  ( A P K ) ) ) )
40393impia 1188 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e. 
NN0 )  ->  ( A P -u K )  =  ( N `  ( A P K ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   {csn 4022   class class class wbr 4442    X. cxp 4992   ran crn 4995   ` cfv 5581  (class class class)co 6277   0cc0 9483   1c1 9484    < clt 9619   -ucneg 9797   NNcn 10527   NN0cn0 10786   ZZcz 10855    seqcseq 12065   GrpOpcgr 24852  GIdcgi 24853   invcgn 24854   ^gcgx 24856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-n0 10787  df-z 10856  df-seq 12066  df-grpo 24857  df-gid 24858  df-ginv 24859  df-gx 24861
This theorem is referenced by:  gxcl  24931  gxneg  24932
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