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Theorem cxpef 22915
Description: Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
Assertion
Ref Expression
cxpef  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  ( A  ^c  B )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )

Proof of Theorem cxpef
StepHypRef Expression
1 cxpval 22914 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c  B )  =  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A ) ) ) ) )
213adant2 1014 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  ( A  ^c  B )  =  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A ) ) ) ) )
3 simp2 996 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  A  =/=  0 )
43neneqd 2643 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  -.  A  =  0 )
54iffalsed 3934 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A
) ) ) )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
62, 5eqtrd 2482 1  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  ( A  ^c  B )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 972    = wceq 1381    e. wcel 1802    =/= wne 2636   ifcif 3923   ` cfv 5575  (class class class)co 6278   CCcc 9490   0cc0 9492   1c1 9493    x. cmul 9497   expce 13672   logclog 22811    ^c ccxp 22812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pr 4673  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-mulcl 9554  ax-i2m1 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-br 4435  df-opab 4493  df-id 4782  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-iota 5538  df-fun 5577  df-fv 5583  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-cxp 22814
This theorem is referenced by:  cxpexpz  22917  logcxp  22919  1cxp  22922  ecxp  22923  rpcxpcl  22926  cxpne0  22927  cxpadd  22929  mulcxp  22935  cxpmul  22938  abscxp  22942  abscxp2  22943  cxplt  22944  cxple2  22947  cxpsqrtlem  22952  cxpsqrt  22953  cxpefd  22962  1cubrlem  23041  bposlem9  23436
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