MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cxpval Unicode version

Theorem cxpval 20011
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 443 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  x  =  A )
21eqeq1d 2291 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  =  0  <-> 
A  =  0 ) )
3 simpr 447 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
43eqeq1d 2291 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  =  0  <-> 
B  =  0 ) )
54ifbid 3583 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  =  0 ,  1 ,  0 )  =  if ( B  =  0 ,  1 ,  0 ) )
61fveq2d 5529 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( log `  x
)  =  ( log `  A ) )
73, 6oveq12d 5876 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  x.  ( log `  x ) )  =  ( B  x.  ( log `  A ) ) )
87fveq2d 5529 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( exp `  (
y  x.  ( log `  x ) ) )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
92, 5, 8ifbieq12d 3587 . 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 19915 . 2  |-  ^ c  =  ( x  e.  CC ,  y  e.  CC  |->  if ( x  =  0 ,  if ( y  =  0 ,  1 ,  0 ) ,  ( exp `  ( y  x.  ( log `  x ) ) ) ) )
11 ax-1cn 8795 . . . . 5  |-  1  e.  CC
12 0cn 8831 . . . . 5  |-  0  e.  CC
1311, 12keepel 3622 . . . 4  |-  if ( B  =  0 ,  1 ,  0 )  e.  CC
1413elexi 2797 . . 3  |-  if ( B  =  0 ,  1 ,  0 )  e.  _V
15 fvex 5539 . . 3  |-  ( exp `  ( B  x.  ( log `  A ) ) )  e.  _V
1614, 15ifex 3623 . 2  |-  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A
) ) ) )  e.  _V
179, 10, 16ovmpt2a 5978 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 358    = wceq 1623    e. wcel 1684   ifcif 3565   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    x. cmul 8742   expce 12343   logclog 19912    ^ c ccxp 19913
This theorem is referenced by:  cxpef  20012  0cxp  20013  cxpexp  20015  cxpcl  20021  recxpcl  20022
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-mulcl 8799  ax-i2m1 8805
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-cxp 19915
  Copyright terms: Public domain W3C validator