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Theorem 0cxp 20029
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 8847 . . . 4  |-  0  e.  CC
2 cxpval 20027 . . . 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 2296 . . . 4  |-  0  =  0
5 iftrue 3584 . . . 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 2344 . 2  |-  ( A  e.  CC  ->  (
0  ^ c  A
)  =  if ( A  =  0 ,  1 ,  0 ) )
8 ifnefalse 3586 . 2  |-  ( A  =/=  0  ->  if ( A  =  0 ,  1 ,  0 )  =  0 )
97, 8sylan9eq 2348 1  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 0  ^ c  A )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   ifcif 3578   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    x. cmul 8758   expce 12359   logclog 19928    ^ c ccxp 19929
This theorem is referenced by:  cxpexp  20031  cxpeq0  20041  cxpge0  20046  mulcxplem  20047  cxpmul2  20052  cxple2  20060  cxpsqr  20066  0cxpd  20073  abscxpbnd  20109
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-mulcl 8815  ax-i2m1 8821
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-cxp 19931
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