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Theorem cxpefd 20112
Description: Value of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
Hypotheses
Ref Expression
cxp0d.1  |-  ( ph  ->  A  e.  CC )
cxpefd.2  |-  ( ph  ->  A  =/=  0 )
cxpefd.3  |-  ( ph  ->  B  e.  CC )
Assertion
Ref Expression
cxpefd  |-  ( ph  ->  ( A  ^ c  B )  =  ( exp `  ( B  x.  ( log `  A
) ) ) )

Proof of Theorem cxpefd
StepHypRef Expression
1 cxp0d.1 . 2  |-  ( ph  ->  A  e.  CC )
2 cxpefd.2 . 2  |-  ( ph  ->  A  =/=  0 )
3 cxpefd.3 . 2  |-  ( ph  ->  B  e.  CC )
4 cxpef 20065 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  ( A  ^ c  B )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
51, 2, 3, 4syl3anc 1182 1  |-  ( ph  ->  ( A  ^ c  B )  =  ( exp `  ( B  x.  ( log `  A
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633    e. wcel 1701    =/= wne 2479   ` cfv 5292  (class class class)co 5900   CCcc 8780   0cc0 8782    x. cmul 8787   expce 12390   logclog 19965    ^ c ccxp 19966
This theorem is referenced by:  dvcxp1  20135  dvcxp2  20136  cxpcn  20138  abscxpbnd  20146  root1eq1  20148  cxpeq  20150  efiatan  20261  efiatan2  20266  efrlim  20317  cxp2limlem  20323  cxploglim  20325  amgmlem  20337  bposlem9  20584  chtppilimlem1  20675  ostth2lem4  20838  ostth2  20839  ostth3  20840  zetacvg  23973  proot1ex  26668
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-mulcl 8844  ax-i2m1 8850
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-iota 5256  df-fun 5294  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-cxp 19968
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