Theorem expval 13784

Description: Value of exponentiation to integer powers. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.)

Assertion

Ref	Expression
expval	⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))))

Proof of Theorem expval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 485	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → 𝑦 = 𝑁)
2	1	eqeq1d 2740	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (𝑦 = 0 ↔ 𝑁 = 0))
3	1	breq2d 5086	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (0 < 𝑦 ↔ 0 < 𝑁))
4		simpl 483	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → 𝑥 = 𝐴)
5	4	sneqd 4573	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → {𝑥} = {𝐴})
6	5	xpeq2d 5619	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (ℕ × {𝑥}) = (ℕ × {𝐴}))
7	6	seqeq3d 13729	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → seq1( · , (ℕ × {𝑥})) = seq1( · , (ℕ × {𝐴})))
8	7, 1	fveq12d 6781	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (seq1( · , (ℕ × {𝑥}))‘𝑦) = (seq1( · , (ℕ × {𝐴}))‘𝑁))
9	1	negeqd 11215	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → -𝑦 = -𝑁)
10	7, 9	fveq12d 6781	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (seq1( · , (ℕ × {𝑥}))‘-𝑦) = (seq1( · , (ℕ × {𝐴}))‘-𝑁))
11	10	oveq2d 7291	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))
12	3, 8, 11	ifbieq12d 4487	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦))) = if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁))))
13	2, 12	ifbieq2d 4485	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))))
14		df-exp 13783	. 2 ⊢ ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
15		1ex 10971	. . 3 ⊢ 1 ∈ V
16		fvex 6787	. . . 4 ⊢ (seq1( · , (ℕ × {𝐴}))‘𝑁) ∈ V
17		ovex 7308	. . . 4 ⊢ (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)) ∈ V
18	16, 17	ifex 4509	. . 3 ⊢ if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁))) ∈ V
19	15, 18	ifex 4509	. 2 ⊢ if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))) ∈ V
20	13, 14, 19	ovmpoa 7428	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ifcif 4459  {csn 4561   class class class wbr 5074   × cxp 5587  ‘cfv 6433  (class class class)co 7275  ℂcc 10869  0cc0 10871  1c1 10872   · cmul 10876   < clt 11009  -cneg 11206   / cdiv 11632  ℕcn 11973  ℤcz 12319  seqcseq 13721  ↑cexp 13782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-1cn 10929

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-neg 11208  df-seq 13722  df-exp 13783

This theorem is referenced by:  expnnval  13785  exp0  13786  expneg  13790