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Theorem cxpef 22871
Description: Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
Assertion
Ref Expression
cxpef  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  ( A  ^c  B )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )

Proof of Theorem cxpef
StepHypRef Expression
1 cxpval 22870 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c  B )  =  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A ) ) ) ) )
213adant2 1015 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  ( A  ^c  B )  =  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A ) ) ) ) )
3 simp2 997 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  A  =/=  0 )
43neneqd 2669 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  -.  A  =  0 )
5 iffalse 3948 . . 3  |-  ( -.  A  =  0  ->  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A ) ) ) )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
64, 5syl 16 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A
) ) ) )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
72, 6eqtrd 2508 1  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  ( A  ^c  B )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   ifcif 3939   ` cfv 5588  (class class class)co 6285   CCcc 9491   0cc0 9493   1c1 9494    x. cmul 9498   expce 13662   logclog 22767    ^c ccxp 22768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-mulcl 9555  ax-i2m1 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-cxp 22770
This theorem is referenced by:  cxpexpz  22873  logcxp  22875  1cxp  22878  ecxp  22879  rpcxpcl  22882  cxpne0  22883  cxpadd  22885  mulcxp  22891  cxpmul  22894  abscxp  22898  abscxp2  22899  cxplt  22900  cxple2  22903  cxpsqrtlem  22908  cxpsqrt  22909  cxpefd  22918  1cubrlem  22997  bposlem9  23392
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