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Theorem cxpef 20556
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 20555 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c  B )  =  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A ) ) ) ) )
213adant2 976 . 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 958 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  A  =/=  0 )
43neneqd 2617 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  -.  A  =  0 )
5 iffalse 3746 . . 3  |-  ( -.  A  =  0  ->  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A ) ) ) )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
64, 5syl 16 . 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 ) ) ) )
72, 6eqtrd 2468 1  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  ( A  ^ c  B )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   ifcif 3739   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    x. cmul 8995   expce 12664   logclog 20452    ^ c ccxp 20453
This theorem is referenced by:  cxpexpz  20558  logcxp  20560  1cxp  20563  ecxp  20564  rpcxpcl  20567  cxpne0  20568  cxpadd  20570  mulcxp  20576  cxpmul  20579  abscxp  20583  abscxp2  20584  cxplt  20585  cxple2  20588  cxpsqrlem  20593  cxpsqr  20594  cxpefd  20603  1cubrlem  20681  bposlem9  21076
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
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