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Theorem cxpval 25174
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 483 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
21eqeq1d 2820 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = 0 ↔ 𝐴 = 0))
3 simpr 485 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
43eqeq1d 2820 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 = 0 ↔ 𝐵 = 0))
54ifbid 4485 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = 0, 1, 0) = if(𝐵 = 0, 1, 0))
61fveq2d 6667 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (log‘𝑥) = (log‘𝐴))
73, 6oveq12d 7163 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 · (log‘𝑥)) = (𝐵 · (log‘𝐴)))
87fveq2d 6667 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (exp‘(𝑦 · (log‘𝑥))) = (exp‘(𝐵 · (log‘𝐴))))
92, 5, 8ifbieq12d 4490 . 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 25068 . 2 𝑐 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥)))))
11 ax-1cn 10583 . . . . 5 1 ∈ ℂ
12 0cn 10621 . . . . 5 0 ∈ ℂ
1311, 12ifcli 4509 . . . 4 if(𝐵 = 0, 1, 0) ∈ ℂ
1413elexi 3511 . . 3 if(𝐵 = 0, 1, 0) ∈ V
15 fvex 6676 . . 3 (exp‘(𝐵 · (log‘𝐴))) ∈ V
1614, 15ifex 4511 . 2 if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴)))) ∈ V
179, 10, 16ovmpoa 7294 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 396   = wceq 1528  wcel 2105  ifcif 4463  cfv 6348  (class class class)co 7145  cc 10523  0cc0 10525  1c1 10526   · cmul 10530  expce 15403  logclog 25065  𝑐ccxp 25066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-mulcl 10587  ax-i2m1 10593
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-cxp 25068
This theorem is referenced by:  cxpef  25175  0cxp  25176  cxpexp  25178  cxpcl  25184  recxpcl  25185
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