Theorem cxpefd 24458

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

Step	Hyp	Ref	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 24411	. 2 |- ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
5	1, 2, 3, 4	syl3anc 1326	1 |- ( ph -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   x. cmul 9941   expce 14792   logclog 24301   ^c ccxp 24302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-mulcl 9998  ax-i2m1 10004

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-cxp 24304

This theorem is referenced by:  dvcxp1  24481  dvcxp2  24482  dvcncxp1  24484  cxpcn  24486  abscxpbnd  24494  root1eq1  24496  cxpeq  24498  cxplogb  24524  efiatan  24639  efiatan2  24644  efrlim  24696  cxp2limlem  24702  cxploglim  24704  amgmlem  24716  zetacvg  24741  gamcvg2lem  24785  bposlem9  25017  chtppilimlem1  25162  ostth2lem4  25325  ostth2  25326  ostth3  25327  iprodgam  31628  proot1ex  37779  logcxp0  42329