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Theorem 0cxp 23339
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 9617 . . . 4  |-  0  e.  CC
2 cxpval 23337 . . . 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 668 . . 3  |-  ( A  e.  CC  ->  (
0  ^c  A )  =  if ( 0  =  0 ,  if ( A  =  0 ,  1 ,  0 ) ,  ( exp `  ( A  x.  ( log `  0
) ) ) ) )
4 eqid 2402 . . . 4  |-  0  =  0
54iftruei 3891 . . 3  |-  if ( 0  =  0 ,  if ( A  =  0 ,  1 ,  0 ) ,  ( exp `  ( A  x.  ( log `  0
) ) ) )  =  if ( A  =  0 ,  1 ,  0 )
63, 5syl6eq 2459 . 2  |-  ( A  e.  CC  ->  (
0  ^c  A )  =  if ( A  =  0 ,  1 ,  0 ) )
7 ifnefalse 3896 . 2  |-  ( A  =/=  0  ->  if ( A  =  0 ,  1 ,  0 )  =  0 )
86, 7sylan9eq 2463 1  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 0  ^c  A )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   ifcif 3884   ` cfv 5568  (class class class)co 6277   CCcc 9519   0cc0 9521   1c1 9522    x. cmul 9526   expce 14004   logclog 23232    ^c ccxp 23233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-mulcl 9583  ax-i2m1 9589
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-cxp 23235
This theorem is referenced by:  cxpexp  23341  cxpeq0  23351  cxpge0  23356  mulcxplem  23357  cxpmul2  23362  cxple2  23370  cxpsqrt  23376  0cxpd  23383  abscxpbnd  23421
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