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Theorem cxpef 22110
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 22109 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c  B )  =  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A ) ) ) ) )
213adant2 1007 . 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 989 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  A  =/=  0 )
43neneqd 2624 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  -.  A  =  0 )
5 iffalse 3799 . . 3  |-  ( -.  A  =  0  ->  if ( A  =  0 ,  if ( B  =  0 ,  1 ,  0 ) ,  ( exp `  ( B  x.  ( log `  A ) ) ) )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
64, 5syl 16 . 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 2475 1  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  ( A  ^c  B )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   ifcif 3791   ` cfv 5418  (class class class)co 6091   CCcc 9280   0cc0 9282   1c1 9283    x. cmul 9287   expce 13347   logclog 22006    ^c ccxp 22007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-mulcl 9344  ax-i2m1 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-cxp 22009
This theorem is referenced by:  cxpexpz  22112  logcxp  22114  1cxp  22117  ecxp  22118  rpcxpcl  22121  cxpne0  22122  cxpadd  22124  mulcxp  22130  cxpmul  22133  abscxp  22137  abscxp2  22138  cxplt  22139  cxple2  22142  cxpsqrlem  22147  cxpsqr  22148  cxpefd  22157  1cubrlem  22236  bposlem9  22631
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