Theorem cxpval 26787

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 487	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴)
2	1	eqeq1d 2767	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 0 ↔ 𝐴 = 0))
3		simpr 489	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
4	3	eqeq1d 2767	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 = 0 ↔ 𝐵 = 0))
5	4	ifbid 4507	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = 0, 1, 0) = if(𝐵 = 0, 1, 0))
6	1	fveq2d 6875	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (log‘𝑥) = (log‘𝐴))
7	3, 6	oveq12d 7418	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 · (log‘𝑥)) = (𝐵 · (log‘𝐴)))
8	7	fveq2d 6875	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (exp‘(𝑦 · (log‘𝑥))) = (exp‘(𝐵 · (log‘𝐴))))
9	2, 5, 8	ifbieq12d 4512	. 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 26680	. 2 ⊢ ↑𝑐 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥)))))
11		ax-1cn 11146	. . . . 5 ⊢ 1 ∈ ℂ
12		0cn 11186	. . . . 5 ⊢ 0 ∈ ℂ
13	11, 12	ifcli 4531	. . . 4 ⊢ if(𝐵 = 0, 1, 0) ∈ ℂ
14	13	elexi 3479	. . 3 ⊢ if(𝐵 = 0, 1, 0) ∈ V
15		fvex 6884	. . 3 ⊢ (exp‘(𝐵 · (log‘𝐴))) ∈ V
16	14, 15	ifex 4534	. 2 ⊢ if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴)))) ∈ V
17	9, 10, 16	ovmpoa 7555	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴)))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ifcif 4483  ‘cfv 6525  (class class class)co 7400  ℂcc 11086  0cc0 11088  1c1 11089   · cmul 11093  expce 16105  logclog 26677  ↑_𝑐ccxp 26678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-mulcl 11150  ax-i2m1 11156

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-cxp 26680

This theorem is referenced by:  cxpef  26788  0cxp  26789  cxpexp  26791  cxpcl  26797  recxpcl  26798