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Theorem cxpefd 20059
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 20012 . 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 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737    x. cmul 8742   expce 12343   logclog 19912    ^ c ccxp 19913
This theorem is referenced by:  dvcxp1  20082  dvcxp2  20083  cxpcn  20085  abscxpbnd  20093  root1eq1  20095  cxpeq  20097  efiatan  20208  efiatan2  20213  efrlim  20264  cxp2limlem  20270  cxploglim  20272  amgmlem  20284  bposlem9  20531  chtppilimlem1  20622  ostth2lem4  20785  ostth2  20786  ostth3  20787  zetacvg  23689  proot1ex  27520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-mulcl 8799  ax-i2m1 8805
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-cxp 19915
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