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Theorem 0cxp 20557
Description: Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.)
Assertion
Ref Expression
0cxp  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 0  ^ c  A )  =  0 )

Proof of Theorem 0cxp
StepHypRef Expression
1 0cn 9084 . . . 4  |-  0  e.  CC
2 cxpval 20555 . . . 4  |-  ( ( 0  e.  CC  /\  A  e.  CC )  ->  ( 0  ^ c  A )  =  if ( 0  =  0 ,  if ( A  =  0 ,  1 ,  0 ) ,  ( exp `  ( A  x.  ( log `  0 ) ) ) ) )
31, 2mpan 652 . . 3  |-  ( A  e.  CC  ->  (
0  ^ c  A
)  =  if ( 0  =  0 ,  if ( A  =  0 ,  1 ,  0 ) ,  ( exp `  ( A  x.  ( log `  0
) ) ) ) )
4 eqid 2436 . . . 4  |-  0  =  0
5 iftrue 3745 . . . 4  |-  ( 0  =  0  ->  if ( 0  =  0 ,  if ( A  =  0 ,  1 ,  0 ) ,  ( exp `  ( A  x.  ( log `  0 ) ) ) )  =  if ( A  =  0 ,  1 ,  0 ) )
64, 5ax-mp 8 . . 3  |-  if ( 0  =  0 ,  if ( A  =  0 ,  1 ,  0 ) ,  ( exp `  ( A  x.  ( log `  0
) ) ) )  =  if ( A  =  0 ,  1 ,  0 )
73, 6syl6eq 2484 . 2  |-  ( A  e.  CC  ->  (
0  ^ c  A
)  =  if ( A  =  0 ,  1 ,  0 ) )
8 ifnefalse 3747 . 2  |-  ( A  =/=  0  ->  if ( A  =  0 ,  1 ,  0 )  =  0 )
97, 8sylan9eq 2488 1  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 0  ^ c  A )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   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:  cxpexp  20559  cxpeq0  20569  cxpge0  20574  mulcxplem  20575  cxpmul2  20580  cxple2  20588  cxpsqr  20594  0cxpd  20601  abscxpbnd  20637
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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