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Theorem 0cxp 22911
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 9600 . . . 4  |-  0  e.  CC
2 cxpval 22909 . . . 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 670 . . 3  |-  ( A  e.  CC  ->  (
0  ^c  A )  =  if ( 0  =  0 ,  if ( A  =  0 ,  1 ,  0 ) ,  ( exp `  ( A  x.  ( log `  0
) ) ) ) )
4 eqid 2467 . . . 4  |-  0  =  0
54iftruei 3952 . . 3  |-  if ( 0  =  0 ,  if ( A  =  0 ,  1 ,  0 ) ,  ( exp `  ( A  x.  ( log `  0
) ) ) )  =  if ( A  =  0 ,  1 ,  0 )
63, 5syl6eq 2524 . 2  |-  ( A  e.  CC  ->  (
0  ^c  A )  =  if ( A  =  0 ,  1 ,  0 ) )
7 ifnefalse 3957 . 2  |-  ( A  =/=  0  ->  if ( A  =  0 ,  1 ,  0 )  =  0 )
86, 7sylan9eq 2528 1  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 0  ^c  A )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   ifcif 3945   ` cfv 5594  (class class class)co 6295   CCcc 9502   0cc0 9504   1c1 9505    x. cmul 9509   expce 13675   logclog 22806    ^c ccxp 22807
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 4574  ax-nul 4582  ax-pr 4692  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-mulcl 9566  ax-i2m1 9572
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-cxp 22809
This theorem is referenced by:  cxpexp  22913  cxpeq0  22923  cxpge0  22928  mulcxplem  22929  cxpmul2  22934  cxple2  22942  cxpsqrt  22948  0cxpd  22955  abscxpbnd  22991
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