MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cxpefd Structured version   Unicode version

Theorem cxpefd 20603
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 20556 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  ( A  ^ c  B )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
51, 2, 3, 4syl3anc 1184 1  |-  ( ph  ->  ( A  ^ c  B )  =  ( exp `  ( B  x.  ( log `  A
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    =/= wne 2599   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990    x. cmul 8995   expce 12664   logclog 20452    ^ c ccxp 20453
This theorem is referenced by:  dvcxp1  20626  dvcxp2  20627  cxpcn  20629  abscxpbnd  20637  root1eq1  20639  cxpeq  20641  efiatan  20752  efiatan2  20757  efrlim  20808  cxp2limlem  20814  cxploglim  20816  amgmlem  20828  bposlem9  21076  chtppilimlem1  21167  ostth2lem4  21330  ostth2  21331  ostth3  21332  zetacvg  24799  gamcvg2lem  24843  iprodgam  25319  proot1ex  27497
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-mulcl 9052  ax-i2m1 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-cxp 20455
  Copyright terms: Public domain W3C validator