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

Theorem cxpefd 22100
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 22053 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  ( A  ^c  B )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
51, 2, 3, 4syl3anc 1213 1  |-  ( ph  ->  ( A  ^c  B )  =  ( exp `  ( B  x.  ( log `  A
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 1761    =/= wne 2604   ` cfv 5415  (class class class)co 6090   CCcc 9276   0cc0 9278    x. cmul 9283   expce 13343   logclog 21949    ^c ccxp 21950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-mulcl 9340  ax-i2m1 9346
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-iota 5378  df-fun 5417  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-cxp 21952
This theorem is referenced by:  dvcxp1  22123  dvcxp2  22124  cxpcn  22126  abscxpbnd  22134  root1eq1  22136  cxpeq  22138  efiatan  22250  efiatan2  22255  efrlim  22306  cxp2limlem  22312  cxploglim  22314  amgmlem  22326  bposlem9  22574  chtppilimlem1  22665  ostth2lem4  22828  ostth2  22829  ostth3  22830  zetacvg  26915  gamcvg2lem  26959  iprodgam  27419  dvcncxp1  28386  proot1ex  29478
  Copyright terms: Public domain W3C validator