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Theorem 0cxp 25162
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 ∈ ℂ ∧ 𝐴 ≠ 0) → (0↑𝑐𝐴) = 0)

Proof of Theorem 0cxp
StepHypRef Expression
1 0cn 10625 . . . 4 0 ∈ ℂ
2 cxpval 25160 . . . 4 ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (0↑𝑐𝐴) = if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0)))))
31, 2mpan 686 . . 3 (𝐴 ∈ ℂ → (0↑𝑐𝐴) = if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0)))))
4 eqid 2824 . . . 4 0 = 0
54iftruei 4476 . . 3 if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0)))) = if(𝐴 = 0, 1, 0)
63, 5syl6eq 2876 . 2 (𝐴 ∈ ℂ → (0↑𝑐𝐴) = if(𝐴 = 0, 1, 0))
7 ifnefalse 4481 . 2 (𝐴 ≠ 0 → if(𝐴 = 0, 1, 0) = 0)
86, 7sylan9eq 2880 1 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (0↑𝑐𝐴) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2106  wne 3020  ifcif 4469  cfv 6351  (class class class)co 7151  cc 10527  0cc0 10529  1c1 10530   · cmul 10534  expce 15407  logclog 25051  𝑐ccxp 25052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pr 5325  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-mulcl 10591  ax-i2m1 10597
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-iota 6311  df-fun 6353  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-cxp 25054
This theorem is referenced by:  cxpexp  25164  cxpeq0  25174  cxpge0  25179  mulcxplem  25180  cxpmul2  25185  cxple2  25193  cxpsqrt  25199  0cxpd  25206  cxpsqrtth  25225  abscxpbnd  25247
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