Theorem cxpval 26625

Description: Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)

Assertion

Ref	Expression
cxpval	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = 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.

Step	Hyp	Ref	Expression
1		simpl 482	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴)
2	1	eqeq1d 2737	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 0 ↔ 𝐴 = 0))
3		simpr 484	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
4	3	eqeq1d 2737	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 = 0 ↔ 𝐵 = 0))
5	4	ifbid 4524	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = 0, 1, 0) = if(𝐵 = 0, 1, 0))
6	1	fveq2d 6880	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (log‘𝑥) = (log‘𝐴))
7	3, 6	oveq12d 7423	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 · (log‘𝑥)) = (𝐵 · (log‘𝐴)))
8	7	fveq2d 6880	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (exp‘(𝑦 · (log‘𝑥))) = (exp‘(𝐵 · (log‘𝐴))))
9	2, 5, 8	ifbieq12d 4529	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 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 26518	. 2 ⊢ ↑𝑐 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥)))))
11		ax-1cn 11187	. . . . 5 ⊢ 1 ∈ ℂ
12		0cn 11227	. . . . 5 ⊢ 0 ∈ ℂ
13	11, 12	ifcli 4548	. . . 4 ⊢ if(𝐵 = 0, 1, 0) ∈ ℂ
14	13	elexi 3482	. . 3 ⊢ if(𝐵 = 0, 1, 0) ∈ V
15		fvex 6889	. . 3 ⊢ (exp‘(𝐵 · (log‘𝐴))) ∈ V
16	14, 15	ifex 4551	. 2 ⊢ if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴)))) ∈ V
17	9, 10, 16	ovmpoa 7562	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ifcif 4500  ‘cfv 6531  (class class class)co 7405  ℂcc 11127  0cc0 11129  1c1 11130   · cmul 11134  expce 16077  logclog 26515  ↑_𝑐ccxp 26516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-mulcl 11191  ax-i2m1 11197

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6484  df-fun 6533  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-cxp 26518

This theorem is referenced by:  cxpef  26626  0cxp  26627  cxpexp  26629  cxpcl  26635  recxpcl  26636