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Theorem cxpval 26629
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 2738 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = 0 ↔ 𝐴 = 0))
3 simpr 484 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
43eqeq1d 2738 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 = 0 ↔ 𝐵 = 0))
54ifbid 4503 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = 0, 1, 0) = if(𝐵 = 0, 1, 0))
61fveq2d 6838 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (log‘𝑥) = (log‘𝐴))
73, 6oveq12d 7376 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 · (log‘𝑥)) = (𝐵 · (log‘𝐴)))
87fveq2d 6838 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (exp‘(𝑦 · (log‘𝑥))) = (exp‘(𝐵 · (log‘𝐴))))
92, 5, 8ifbieq12d 4508 . 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 26522 . 2 𝑐 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥)))))
11 ax-1cn 11084 . . . . 5 1 ∈ ℂ
12 0cn 11124 . . . . 5 0 ∈ ℂ
1311, 12ifcli 4527 . . . 4 if(𝐵 = 0, 1, 0) ∈ ℂ
1413elexi 3463 . . 3 if(𝐵 = 0, 1, 0) ∈ V
15 fvex 6847 . . 3 (exp‘(𝐵 · (log‘𝐴))) ∈ V
1614, 15ifex 4530 . 2 if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴)))) ∈ V
179, 10, 16ovmpoa 7513 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 1541  wcel 2113  ifcif 4479  cfv 6492  (class class class)co 7358  cc 11024  0cc0 11026  1c1 11027   · cmul 11031  expce 15984  logclog 26519  𝑐ccxp 26520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-mulcl 11088  ax-i2m1 11094
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-cxp 26522
This theorem is referenced by:  cxpef  26630  0cxp  26631  cxpexp  26633  cxpcl  26639  recxpcl  26640
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