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Theorem cxpef 20028
Description: Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
Assertion
Ref Expression
cxpef  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  ( A  ^ c  B )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )

Proof of Theorem cxpef
StepHypRef Expression
1 cxpval 20027 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c  B )  =  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A ) ) ) ) )
213adant2 974 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  ( A  ^ c  B )  =  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A ) ) ) ) )
3 simp2 956 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  A  =/=  0 )
43neneqd 2475 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  -.  A  =  0 )
5 iffalse 3585 . . 3  |-  ( -.  A  =  0  ->  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A ) ) ) )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
64, 5syl 15 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A
) ) ) )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
72, 6eqtrd 2328 1  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  ( A  ^ c  B )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 934    = 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:  cxpexpz  20030  logcxp  20032  1cxp  20035  ecxp  20036  rpcxpcl  20039  cxpne0  20040  cxpadd  20042  mulcxp  20048  cxpmul  20051  abscxp  20055  abscxp2  20056  cxplt  20057  cxple2  20060  cxpsqrlem  20065  cxpsqr  20066  cxpefd  20075  1cubrlem  20153  bposlem9  20547
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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