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Theorem gxsub 25476
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 10895 . . . 4  |-  ( K  e.  ZZ  ->  -u K  e.  ZZ )
21anim2i 567 . . 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 25475 . . 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 1261 . 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 10865 . . . . 5  |-  ( J  e.  ZZ  ->  J  e.  CC )
8 zcn 10865 . . . . 5  |-  ( K  e.  ZZ  ->  K  e.  CC )
9 negsub 9858 . . . . 5  |-  ( ( J  e.  CC  /\  K  e.  CC )  ->  ( J  +  -u K )  =  ( J  -  K ) )
107, 8, 9syl2an 475 . . . 4  |-  ( ( J  e.  ZZ  /\  K  e.  ZZ )  ->  ( J  +  -u K )  =  ( J  -  K ) )
11103ad2ant3 1017 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  K  e.  ZZ ) )  -> 
( J  +  -u K )  =  ( J  -  K ) )
1211oveq2d 6286 . 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 25466 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P -u K )  =  ( N `  ( A P K ) ) )
15143adant3l 1222 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  K  e.  ZZ ) )  -> 
( A P -u K )  =  ( N `  ( A P K ) ) )
1615oveq2d 6286 . 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 2503 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   ran crn 4989   ` cfv 5570  (class class class)co 6270   CCcc 9479    + caddc 9484    - cmin 9796   -ucneg 9797   ZZcz 10860   GrpOpcgr 25386   invcgn 25388   ^gcgx 25390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-seq 12090  df-grpo 25391  df-gid 25392  df-ginv 25393  df-gx 25395
This theorem is referenced by: (None)
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