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Theorem gxsub 21705
Description: The group power of a difference is the group quotient of the powers. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
Hypotheses
Ref Expression
gxsub.1  |-  X  =  ran  G
gxsub.2  |-  N  =  ( inv `  G
)
gxsub.3  |-  P  =  ( ^g `  G
)
Assertion
Ref Expression
gxsub  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  K  e.  ZZ ) )  -> 
( A P ( J  -  K ) )  =  ( ( A P J ) G ( N `  ( A P K ) ) ) )

Proof of Theorem gxsub
StepHypRef Expression
1 znegcl 10238 . . . 4  |-  ( K  e.  ZZ  ->  -u K  e.  ZZ )
21anim2i 553 . . 3  |-  ( ( J  e.  ZZ  /\  K  e.  ZZ )  ->  ( J  e.  ZZ  /\  -u K  e.  ZZ ) )
3 gxsub.1 . . . 4  |-  X  =  ran  G
4 gxsub.3 . . . 4  |-  P  =  ( ^g `  G
)
53, 4gxadd 21704 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  -u K  e.  ZZ ) )  -> 
( A P ( J  +  -u K
) )  =  ( ( A P J ) G ( A P -u K ) ) )
62, 5syl3an3 1219 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  K  e.  ZZ ) )  -> 
( A P ( J  +  -u K
) )  =  ( ( A P J ) G ( A P -u K ) ) )
7 zcn 10212 . . . . 5  |-  ( J  e.  ZZ  ->  J  e.  CC )
8 zcn 10212 . . . . 5  |-  ( K  e.  ZZ  ->  K  e.  CC )
9 negsub 9274 . . . . 5  |-  ( ( J  e.  CC  /\  K  e.  CC )  ->  ( J  +  -u K )  =  ( J  -  K ) )
107, 8, 9syl2an 464 . . . 4  |-  ( ( J  e.  ZZ  /\  K  e.  ZZ )  ->  ( J  +  -u K )  =  ( J  -  K ) )
11103ad2ant3 980 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  K  e.  ZZ ) )  -> 
( J  +  -u K )  =  ( J  -  K ) )
1211oveq2d 6029 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  K  e.  ZZ ) )  -> 
( A P ( J  +  -u K
) )  =  ( A P ( J  -  K ) ) )
13 gxsub.2 . . . . 5  |-  N  =  ( inv `  G
)
143, 13, 4gxneg 21695 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P -u K )  =  ( N `  ( A P K ) ) )
15143adant3l 1180 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  K  e.  ZZ ) )  -> 
( A P -u K )  =  ( N `  ( A P K ) ) )
1615oveq2d 6029 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  K  e.  ZZ ) )  -> 
( ( A P J ) G ( A P -u K
) )  =  ( ( A P J ) G ( N `
 ( A P K ) ) ) )
176, 12, 163eqtr3d 2420 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  K  e.  ZZ ) )  -> 
( A P ( J  -  K ) )  =  ( ( A P J ) G ( N `  ( A P K ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ran crn 4812   ` cfv 5387  (class class class)co 6013   CCcc 8914    + caddc 8919    - cmin 9216   -ucneg 9217   ZZcz 10207   GrpOpcgr 21615   invcgn 21617   ^gcgx 21619
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-n0 10147  df-z 10208  df-uz 10414  df-seq 11244  df-grpo 21620  df-gid 21621  df-ginv 21622  df-gx 21624
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