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Theorem 0cxp 20013
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 8831 . . . 4  |-  0  e.  CC
2 cxpval 20011 . . . 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 651 . . 3  |-  ( A  e.  CC  ->  (
0  ^ c  A
)  =  if ( 0  =  0 ,  if ( A  =  0 ,  1 ,  0 ) ,  ( exp `  ( A  x.  ( log `  0
) ) ) ) )
4 eqid 2283 . . . 4  |-  0  =  0
5 iftrue 3571 . . . 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 2331 . 2  |-  ( A  e.  CC  ->  (
0  ^ c  A
)  =  if ( A  =  0 ,  1 ,  0 ) )
8 ifnefalse 3573 . 2  |-  ( A  =/=  0  ->  if ( A  =  0 ,  1 ,  0 )  =  0 )
97, 8sylan9eq 2335 1  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 0  ^ c  A )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   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:  cxpexp  20015  cxpeq0  20025  cxpge0  20030  mulcxplem  20031  cxpmul2  20036  cxple2  20044  cxpsqr  20050  0cxpd  20057  abscxpbnd  20093
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
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