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Theorem cxpval 24701
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 474 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
21eqeq1d 2767 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = 0 ↔ 𝐴 = 0))
3 simpr 477 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
43eqeq1d 2767 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 = 0 ↔ 𝐵 = 0))
54ifbid 4265 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = 0, 1, 0) = if(𝐵 = 0, 1, 0))
61fveq2d 6379 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (log‘𝑥) = (log‘𝐴))
73, 6oveq12d 6860 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 · (log‘𝑥)) = (𝐵 · (log‘𝐴)))
87fveq2d 6379 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (exp‘(𝑦 · (log‘𝑥))) = (exp‘(𝐵 · (log‘𝐴))))
92, 5, 8ifbieq12d 4270 . 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 24595 . 2 𝑐 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥)))))
11 ax-1cn 10247 . . . . 5 1 ∈ ℂ
12 0cn 10285 . . . . 5 0 ∈ ℂ
1311, 12ifcli 4289 . . . 4 if(𝐵 = 0, 1, 0) ∈ ℂ
1413elexi 3366 . . 3 if(𝐵 = 0, 1, 0) ∈ V
15 fvex 6388 . . 3 (exp‘(𝐵 · (log‘𝐴))) ∈ V
1614, 15ifex 4291 . 2 if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴)))) ∈ V
179, 10, 16ovmpt2a 6989 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 384   = wceq 1652  wcel 2155  ifcif 4243  cfv 6068  (class class class)co 6842  cc 10187  0cc0 10189  1c1 10190   · cmul 10194  expce 15074  logclog 24592  𝑐ccxp 24593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-mulcl 10251  ax-i2m1 10257
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-iota 6031  df-fun 6070  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-cxp 24595
This theorem is referenced by:  cxpef  24702  0cxp  24703  cxpexp  24705  cxpcl  24711  recxpcl  24712
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