Theorem cxpval 26610

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 2735	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 0 ↔ 𝐴 = 0))
3		simpr 484	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
4	3	eqeq1d 2735	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 = 0 ↔ 𝐵 = 0))
5	4	ifbid 4500	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = 0, 1, 0) = if(𝐵 = 0, 1, 0))
6	1	fveq2d 6835	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (log‘𝑥) = (log‘𝐴))
7	3, 6	oveq12d 7373	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 · (log‘𝑥)) = (𝐵 · (log‘𝐴)))
8	7	fveq2d 6835	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (exp‘(𝑦 · (log‘𝑥))) = (exp‘(𝐵 · (log‘𝐴))))
9	2, 5, 8	ifbieq12d 4505	. 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 26503	. 2 ⊢ ↑𝑐 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥)))))
11		ax-1cn 11074	. . . . 5 ⊢ 1 ∈ ℂ
12		0cn 11114	. . . . 5 ⊢ 0 ∈ ℂ
13	11, 12	ifcli 4524	. . . 4 ⊢ if(𝐵 = 0, 1, 0) ∈ ℂ
14	13	elexi 3461	. . 3 ⊢ if(𝐵 = 0, 1, 0) ∈ V
15		fvex 6844	. . 3 ⊢ (exp‘(𝐵 · (log‘𝐴))) ∈ V
16	14, 15	ifex 4527	. 2 ⊢ if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴)))) ∈ V
17	9, 10, 16	ovmpoa 7510	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 1541   ∈ wcel 2113  ifcif 4476  ‘cfv 6489  (class class class)co 7355  ℂcc 11014  0cc0 11016  1c1 11017   · cmul 11021  expce 15978  logclog 26500  ↑_𝑐ccxp 26501

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 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-1cn 11074  ax-icn 11075  ax-addcl 11076  ax-mulcl 11078  ax-i2m1 11084

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-cxp 26503

This theorem is referenced by:  cxpef  26611  0cxp  26612  cxpexp  26614  cxpcl  26620  recxpcl  26621