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Theorem cxpval 22917
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 457 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  x  =  A )
21eqeq1d 2445 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  =  0  <-> 
A  =  0 ) )
3 simpr 461 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
43eqeq1d 2445 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  =  0  <-> 
B  =  0 ) )
54ifbid 3948 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  =  0 ,  1 ,  0 )  =  if ( B  =  0 ,  1 ,  0 ) )
61fveq2d 5860 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( log `  x
)  =  ( log `  A ) )
73, 6oveq12d 6299 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  x.  ( log `  x ) )  =  ( B  x.  ( log `  A ) ) )
87fveq2d 5860 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( exp `  (
y  x.  ( log `  x ) ) )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
92, 5, 8ifbieq12d 3953 . 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 22817 . 2  |-  ^c 
=  ( x  e.  CC ,  y  e.  CC  |->  if ( x  =  0 ,  if ( y  =  0 ,  1 ,  0 ) ,  ( exp `  ( y  x.  ( log `  x ) ) ) ) )
11 ax-1cn 9553 . . . . 5  |-  1  e.  CC
12 0cn 9591 . . . . 5  |-  0  e.  CC
1311, 12keepel 3994 . . . 4  |-  if ( B  =  0 ,  1 ,  0 )  e.  CC
1413elexi 3105 . . 3  |-  if ( B  =  0 ,  1 ,  0 )  e.  _V
15 fvex 5866 . . 3  |-  ( exp `  ( B  x.  ( log `  A ) ) )  e.  _V
1614, 15ifex 3995 . 2  |-  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A
) ) ) )  e.  _V
179, 10, 16ovmpt2a 6418 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   ifcif 3926   ` cfv 5578  (class class class)co 6281   CCcc 9493   0cc0 9495   1c1 9496    x. cmul 9500   expce 13675   logclog 22814    ^c ccxp 22815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-mulcl 9557  ax-i2m1 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-iota 5541  df-fun 5580  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-cxp 22817
This theorem is referenced by:  cxpef  22918  0cxp  22919  cxpexp  22921  cxpcl  22927  recxpcl  22928
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