Theorem expval 14060

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 483	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → 𝑦 = 𝑁)
2	1	eqeq1d 2727	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (𝑦 = 0 ↔ 𝑁 = 0))
3	1	breq2d 5160	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (0 < 𝑦 ↔ 0 < 𝑁))
4		simpl 481	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → 𝑥 = 𝐴)
5	4	sneqd 4641	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → {𝑥} = {𝐴})
6	5	xpeq2d 5707	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (ℕ × {𝑥}) = (ℕ × {𝐴}))
7	6	seqeq3d 14006	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → seq1( · , (ℕ × {𝑥})) = seq1( · , (ℕ × {𝐴})))
8	7, 1	fveq12d 6901	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (seq1( · , (ℕ × {𝑥}))‘𝑦) = (seq1( · , (ℕ × {𝐴}))‘𝑁))
9	1	negeqd 11484	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → -𝑦 = -𝑁)
10	7, 9	fveq12d 6901	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (seq1( · , (ℕ × {𝑥}))‘-𝑦) = (seq1( · , (ℕ × {𝐴}))‘-𝑁))
11	10	oveq2d 7433	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))
12	3, 8, 11	ifbieq12d 4557	. . 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 4555	. 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 14059	. 2 ⊢ ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
15		1ex 11240	. . 3 ⊢ 1 ∈ V
16		fvex 6907	. . . 4 ⊢ (seq1( · , (ℕ × {𝐴}))‘𝑁) ∈ V
17		ovex 7450	. . . 4 ⊢ (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)) ∈ V
18	16, 17	ifex 4579	. . 3 ⊢ if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁))) ∈ V
19	15, 18	ifex 4579	. 2 ⊢ if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))) ∈ V
20	13, 14, 19	ovmpoa 7574	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 394   = wceq 1533   ∈ wcel 2098  ifcif 4529  {csn 4629   class class class wbr 5148   × cxp 5675  ‘cfv 6547  (class class class)co 7417  ℂcc 11136  0cc0 11138  1c1 11139   · cmul 11143   < clt 11278  -cneg 11475   / cdiv 11901  ℕcn 12242  ℤcz 12588  seqcseq 13998  ↑cexp 14058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5299  ax-nul 5306  ax-pr 5428  ax-1cn 11196

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3775  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6305  df-iota 6499  df-fun 6549  df-fv 6555  df-ov 7420  df-oprab 7421  df-mpo 7422  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-neg 11477  df-seq 13999  df-exp 14059

This theorem is referenced by:  expnnval  14061  exp0  14062  expneg  14066