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Theorem cxpval 22770
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 2469 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  =  0  <-> 
A  =  0 ) )
3 simpr 461 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
43eqeq1d 2469 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  =  0  <-> 
B  =  0 ) )
54ifbid 3961 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  =  0 ,  1 ,  0 )  =  if ( B  =  0 ,  1 ,  0 ) )
61fveq2d 5868 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( log `  x
)  =  ( log `  A ) )
73, 6oveq12d 6300 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  x.  ( log `  x ) )  =  ( B  x.  ( log `  A ) ) )
87fveq2d 5868 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( exp `  (
y  x.  ( log `  x ) ) )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
92, 5, 8ifbieq12d 3966 . 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 22670 . 2  |-  ^c 
=  ( x  e.  CC ,  y  e.  CC  |->  if ( x  =  0 ,  if ( y  =  0 ,  1 ,  0 ) ,  ( exp `  ( y  x.  ( log `  x ) ) ) ) )
11 ax-1cn 9546 . . . . 5  |-  1  e.  CC
12 0cn 9584 . . . . 5  |-  0  e.  CC
1311, 12keepel 4007 . . . 4  |-  if ( B  =  0 ,  1 ,  0 )  e.  CC
1413elexi 3123 . . 3  |-  if ( B  =  0 ,  1 ,  0 )  e.  _V
15 fvex 5874 . . 3  |-  ( exp `  ( B  x.  ( log `  A ) ) )  e.  _V
1614, 15ifex 4008 . 2  |-  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A
) ) ) )  e.  _V
179, 10, 16ovmpt2a 6415 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 1379    e. wcel 1767   ifcif 3939   ` cfv 5586  (class class class)co 6282   CCcc 9486   0cc0 9488   1c1 9489    x. cmul 9493   expce 13652   logclog 22667    ^c ccxp 22668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-mulcl 9550  ax-i2m1 9556
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-cxp 22670
This theorem is referenced by:  cxpef  22771  0cxp  22772  cxpexp  22774  cxpcl  22780  recxpcl  22781
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