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Theorem 0cxp 23667
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 9666 . . . 4  |-  0  e.  CC
2 cxpval 23665 . . . 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 681 . . 3  |-  ( A  e.  CC  ->  (
0  ^c  A )  =  if ( 0  =  0 ,  if ( A  =  0 ,  1 ,  0 ) ,  ( exp `  ( A  x.  ( log `  0
) ) ) ) )
4 eqid 2462 . . . 4  |-  0  =  0
54iftruei 3900 . . 3  |-  if ( 0  =  0 ,  if ( A  =  0 ,  1 ,  0 ) ,  ( exp `  ( A  x.  ( log `  0
) ) ) )  =  if ( A  =  0 ,  1 ,  0 )
63, 5syl6eq 2512 . 2  |-  ( A  e.  CC  ->  (
0  ^c  A )  =  if ( A  =  0 ,  1 ,  0 ) )
7 ifnefalse 3905 . 2  |-  ( A  =/=  0  ->  if ( A  =  0 ,  1 ,  0 )  =  0 )
86, 7sylan9eq 2516 1  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 0  ^c  A )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1455    e. wcel 1898    =/= wne 2633   ifcif 3893   ` cfv 5605  (class class class)co 6320   CCcc 9568   0cc0 9570   1c1 9571    x. cmul 9575   expce 14169   logclog 23560    ^c ccxp 23561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pr 4656  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-mulcl 9632  ax-i2m1 9638
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-iota 5569  df-fun 5607  df-fv 5613  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-cxp 23563
This theorem is referenced by:  cxpexp  23669  cxpeq0  23679  cxpge0  23684  mulcxplem  23685  cxpmul2  23690  cxple2  23698  cxpsqrt  23704  0cxpd  23711  abscxpbnd  23749
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