Theorem expval 14101

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 484	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → 𝑦 = 𝑁)
2	1	eqeq1d 2737	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (𝑦 = 0 ↔ 𝑁 = 0))
3	1	breq2d 5160	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (0 < 𝑦 ↔ 0 < 𝑁))
4		simpl 482	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → 𝑥 = 𝐴)
5	4	sneqd 4643	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → {𝑥} = {𝐴})
6	5	xpeq2d 5719	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (ℕ × {𝑥}) = (ℕ × {𝐴}))
7	6	seqeq3d 14047	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → seq1( · , (ℕ × {𝑥})) = seq1( · , (ℕ × {𝐴})))
8	7, 1	fveq12d 6914	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (seq1( · , (ℕ × {𝑥}))‘𝑦) = (seq1( · , (ℕ × {𝐴}))‘𝑁))
9	1	negeqd 11500	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → -𝑦 = -𝑁)
10	7, 9	fveq12d 6914	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (seq1( · , (ℕ × {𝑥}))‘-𝑦) = (seq1( · , (ℕ × {𝐴}))‘-𝑁))
11	10	oveq2d 7447	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝑁) → (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))
12	3, 8, 11	ifbieq12d 4559	. . 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 4557	. 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 14100	. 2 ⊢ ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
15		1ex 11255	. . 3 ⊢ 1 ∈ V
16		fvex 6920	. . . 4 ⊢ (seq1( · , (ℕ × {𝐴}))‘𝑁) ∈ V
17		ovex 7464	. . . 4 ⊢ (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)) ∈ V
18	16, 17	ifex 4581	. . 3 ⊢ if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁))) ∈ V
19	15, 18	ifex 4581	. 2 ⊢ if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))) ∈ V
20	13, 14, 19	ovmpoa 7588	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 395   = wceq 1537   ∈ wcel 2106  ifcif 4531  {csn 4631   class class class wbr 5148   × cxp 5687  ‘cfv 6563  (class class class)co 7431  ℂcc 11151  0cc0 11153  1c1 11154   · cmul 11158   < clt 11293  -cneg 11491   / cdiv 11918  ℕcn 12264  ℤcz 12611  seqcseq 14039  ↑cexp 14099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-1cn 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-neg 11493  df-seq 14040  df-exp 14100

This theorem is referenced by:  expnnval  14102  exp0  14103  expneg  14107