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Theorem cxpval 20555
Description: Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
Assertion
Ref Expression
cxpval  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c  B )  =  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A ) ) ) ) )

Proof of Theorem cxpval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 444 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  x  =  A )
21eqeq1d 2444 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  =  0  <-> 
A  =  0 ) )
3 simpr 448 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
43eqeq1d 2444 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  =  0  <-> 
B  =  0 ) )
54ifbid 3757 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  =  0 ,  1 ,  0 )  =  if ( B  =  0 ,  1 ,  0 ) )
61fveq2d 5732 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( log `  x
)  =  ( log `  A ) )
73, 6oveq12d 6099 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  x.  ( log `  x ) )  =  ( B  x.  ( log `  A ) ) )
87fveq2d 5732 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( exp `  (
y  x.  ( log `  x ) ) )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
92, 5, 8ifbieq12d 3761 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( x  =  0 ,  if ( y  =  0 ,  1 ,  0 ) ,  ( exp `  (
y  x.  ( log `  x ) ) ) )  =  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A
) ) ) ) )
10 df-cxp 20455 . 2  |-  ^ c  =  ( x  e.  CC ,  y  e.  CC  |->  if ( x  =  0 ,  if ( y  =  0 ,  1 ,  0 ) ,  ( exp `  ( y  x.  ( log `  x ) ) ) ) )
11 ax-1cn 9048 . . . . 5  |-  1  e.  CC
12 0cn 9084 . . . . 5  |-  0  e.  CC
1311, 12keepel 3796 . . . 4  |-  if ( B  =  0 ,  1 ,  0 )  e.  CC
1413elexi 2965 . . 3  |-  if ( B  =  0 ,  1 ,  0 )  e.  _V
15 fvex 5742 . . 3  |-  ( exp `  ( B  x.  ( log `  A ) ) )  e.  _V
1614, 15ifex 3797 . 2  |-  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A
) ) ) )  e.  _V
179, 10, 16ovmpt2a 6204 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c  B )  =  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   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:  cxpef  20556  0cxp  20557  cxpexp  20559  cxpcl  20565  recxpcl  20566
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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