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Theorem cxpval 26721
Description: Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
Assertion
Ref Expression
cxpval ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝑐𝐵) = if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴)))))

Proof of Theorem cxpval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 482 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
21eqeq1d 2737 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = 0 ↔ 𝐴 = 0))
3 simpr 484 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
43eqeq1d 2737 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 = 0 ↔ 𝐵 = 0))
54ifbid 4554 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = 0, 1, 0) = if(𝐵 = 0, 1, 0))
61fveq2d 6911 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (log‘𝑥) = (log‘𝐴))
73, 6oveq12d 7449 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 · (log‘𝑥)) = (𝐵 · (log‘𝐴)))
87fveq2d 6911 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (exp‘(𝑦 · (log‘𝑥))) = (exp‘(𝐵 · (log‘𝐴))))
92, 5, 8ifbieq12d 4559 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥)))) = if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴)))))
10 df-cxp 26614 . 2 𝑐 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥)))))
11 ax-1cn 11211 . . . . 5 1 ∈ ℂ
12 0cn 11251 . . . . 5 0 ∈ ℂ
1311, 12ifcli 4578 . . . 4 if(𝐵 = 0, 1, 0) ∈ ℂ
1413elexi 3501 . . 3 if(𝐵 = 0, 1, 0) ∈ V
15 fvex 6920 . . 3 (exp‘(𝐵 · (log‘𝐴))) ∈ V
1614, 15ifex 4581 . 2 if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴)))) ∈ V
179, 10, 16ovmpoa 7588 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝑐𝐵) = if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  ifcif 4531  cfv 6563  (class class class)co 7431  cc 11151  0cc0 11153  1c1 11154   · cmul 11158  expce 16094  logclog 26611  𝑐ccxp 26612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-mulcl 11215  ax-i2m1 11221
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-cxp 26614
This theorem is referenced by:  cxpef  26722  0cxp  26723  cxpexp  26725  cxpcl  26731  recxpcl  26732
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