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Theorem gxnn0neg 25665
Description: A negative group power is the inverse of the positive power (lemma with nonnegative exponent - use gxneg 25668 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 10837 . . 3  |-  ( K  e.  NN0  <->  ( K  e.  NN  \/  K  =  0 ) )
2 nnnegz 10907 . . . . . . . 8  |-  ( K  e.  NN  ->  -u K  e.  ZZ )
3 nngt0 10604 . . . . . . . . 9  |-  ( K  e.  NN  ->  0  <  K )
4 nnre 10582 . . . . . . . . . 10  |-  ( K  e.  NN  ->  K  e.  RR )
54lt0neg2d 10162 . . . . . . . . 9  |-  ( K  e.  NN  ->  (
0  <  K  <->  -u K  <  0 ) )
63, 5mpbid 210 . . . . . . . 8  |-  ( K  e.  NN  ->  -u K  <  0 )
72, 6jca 530 . . . . . . 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 25662 . . . . . . 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 1265 . . . . . 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 25661 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P K )  =  (  seq 1 ( G ,  ( NN 
X.  { A }
) ) `  K
) )
14 nncn 10583 . . . . . . . . . . 11  |-  ( K  e.  NN  ->  K  e.  CC )
1514negnegd 9957 . . . . . . . . . 10  |-  ( K  e.  NN  ->  -u -u K  =  K )
1615fveq2d 5852 . . . . . . . . 9  |-  ( K  e.  NN  ->  (  seq 1 ( G , 
( NN  X.  { A } ) ) `  -u -u K )  =  (  seq 1 ( G ,  ( NN  X.  { A } ) ) `
 K ) )
17163ad2ant3 1020 . . . . . . . 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 2446 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P K )  =  (  seq 1 ( G ,  ( NN 
X.  { A }
) ) `  -u -u K
) )
1918fveq2d 5852 . . . . . 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 2446 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P -u K )  =  ( N `  ( A P K ) ) )
21203expia 1199 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  NN  ->  ( A P -u K
)  =  ( N `
 ( A P K ) ) ) )
22 negeq 9847 . . . . . . . . 9  |-  ( K  =  0  ->  -u K  =  -u 0 )
23 neg0 9900 . . . . . . . . 9  |-  -u 0  =  0
2422, 23syl6eq 2459 . . . . . . . 8  |-  ( K  =  0  ->  -u K  =  0 )
2524oveq2d 6293 . . . . . . 7  |-  ( K  =  0  ->  ( A P -u K )  =  ( A P 0 ) )
26 eqid 2402 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
278, 26, 9gx0 25663 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
2825, 27sylan9eqr 2465 . . . . . 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 5852 . . . . . . 7  |-  ( K  =  0  ->  ( N `  ( A P K ) )  =  ( N `  ( A P 0 ) ) )
3127fveq2d 5852 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  ( A P 0 ) )  =  ( N `  (GId `  G ) ) )
3226, 10grpoinvid 25634 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  ( N `  (GId `  G )
)  =  (GId `  G ) )
3332adantr 463 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  (GId `  G
) )  =  (GId
`  G ) )
3431, 33eqtrd 2443 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  ( A P 0 ) )  =  (GId `  G
) )
3530, 34sylan9eqr 2465 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( N `  ( A P K ) )  =  (GId
`  G ) )
3628, 35eqtr4d 2446 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P -u K )  =  ( N `  ( A P K ) ) )
3736ex 432 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  =  0  ->  ( A P -u K
)  =  ( N `
 ( A P K ) ) ) )
3821, 37jaod 378 . . 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 1194 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 366    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   {csn 3971   class class class wbr 4394    X. cxp 4820   ran crn 4823   ` cfv 5568  (class class class)co 6277   0cc0 9521   1c1 9522    < clt 9657   -ucneg 9841   NNcn 10575   NN0cn0 10835   ZZcz 10904    seqcseq 12149   GrpOpcgr 25588  GIdcgi 25589   invcgn 25590   ^gcgx 25592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-n0 10836  df-z 10905  df-seq 12150  df-grpo 25593  df-gid 25594  df-ginv 25595  df-gx 25597
This theorem is referenced by:  gxcl  25667  gxneg  25668
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