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Theorem cxpval 22107
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 2449 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  =  0  <-> 
A  =  0 ) )
3 simpr 461 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
43eqeq1d 2449 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  =  0  <-> 
B  =  0 ) )
54ifbid 3809 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  =  0 ,  1 ,  0 )  =  if ( B  =  0 ,  1 ,  0 ) )
61fveq2d 5693 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( log `  x
)  =  ( log `  A ) )
73, 6oveq12d 6107 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  x.  ( log `  x ) )  =  ( B  x.  ( log `  A ) ) )
87fveq2d 5693 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( exp `  (
y  x.  ( log `  x ) ) )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
92, 5, 8ifbieq12d 3814 . 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 22007 . 2  |-  ^c 
=  ( x  e.  CC ,  y  e.  CC  |->  if ( x  =  0 ,  if ( y  =  0 ,  1 ,  0 ) ,  ( exp `  ( y  x.  ( log `  x ) ) ) ) )
11 ax-1cn 9338 . . . . 5  |-  1  e.  CC
12 0cn 9376 . . . . 5  |-  0  e.  CC
1311, 12keepel 3855 . . . 4  |-  if ( B  =  0 ,  1 ,  0 )  e.  CC
1413elexi 2980 . . 3  |-  if ( B  =  0 ,  1 ,  0 )  e.  _V
15 fvex 5699 . . 3  |-  ( exp `  ( B  x.  ( log `  A ) ) )  e.  _V
1614, 15ifex 3856 . 2  |-  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A
) ) ) )  e.  _V
179, 10, 16ovmpt2a 6219 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 1369    e. wcel 1756   ifcif 3789   ` cfv 5416  (class class class)co 6089   CCcc 9278   0cc0 9280   1c1 9281    x. cmul 9285   expce 13345   logclog 22004    ^c ccxp 22005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-mulcl 9342  ax-i2m1 9348
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-iota 5379  df-fun 5418  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-cxp 22007
This theorem is referenced by:  cxpef  22108  0cxp  22109  cxpexp  22111  cxpcl  22117  recxpcl  22118
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