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Theorem 0cxp 24393
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 10017 . . . 4 0 ∈ ℂ
2 cxpval 24391 . . . 4 ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (0↑𝑐𝐴) = if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0)))))
31, 2mpan 705 . . 3 (𝐴 ∈ ℂ → (0↑𝑐𝐴) = if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0)))))
4 eqid 2620 . . . 4 0 = 0
54iftruei 4084 . . 3 if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0)))) = if(𝐴 = 0, 1, 0)
63, 5syl6eq 2670 . 2 (𝐴 ∈ ℂ → (0↑𝑐𝐴) = if(𝐴 = 0, 1, 0))
7 ifnefalse 4089 . 2 (𝐴 ≠ 0 → if(𝐴 = 0, 1, 0) = 0)
86, 7sylan9eq 2674 1 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (0↑𝑐𝐴) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  wne 2791  ifcif 4077  cfv 5876  (class class class)co 6635  cc 9919  0cc0 9921  1c1 9922   · cmul 9926  expce 14773  logclog 24282  𝑐ccxp 24283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-mulcl 9983  ax-i2m1 9989
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-cxp 24285
This theorem is referenced by:  cxpexp  24395  cxpeq0  24405  cxpge0  24410  mulcxplem  24411  cxpmul2  24416  cxple2  24424  cxpsqrt  24430  0cxpd  24437  abscxpbnd  24475
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