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Theorem cxpefd 22169
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 22122 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  ( A  ^c  B )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
51, 2, 3, 4syl3anc 1218 1  |-  ( ph  ->  ( A  ^c  B )  =  ( exp `  ( B  x.  ( log `  A
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756    =/= wne 2618   ` cfv 5430  (class class class)co 6103   CCcc 9292   0cc0 9294    x. cmul 9299   expce 13359   logclog 22018    ^c ccxp 22019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-mulcl 9356  ax-i2m1 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5393  df-fun 5432  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-cxp 22021
This theorem is referenced by:  dvcxp1  22192  dvcxp2  22193  cxpcn  22195  abscxpbnd  22203  root1eq1  22205  cxpeq  22207  efiatan  22319  efiatan2  22324  efrlim  22375  cxp2limlem  22381  cxploglim  22383  amgmlem  22395  bposlem9  22643  chtppilimlem1  22734  ostth2lem4  22897  ostth2  22898  ostth3  22899  zetacvg  27013  gamcvg2lem  27057  iprodgam  27518  dvcncxp1  28489  proot1ex  29581
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