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Definition df-exp 13087
Description: Define exponentiation to nonnegative integer powers. For example, (5↑2) = 25 (ex-exp 27644).

This definition is not meant to be used directly; instead, exp0 13090 and expp1 13093 provide the standard recursive definition. The up-arrow notation is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976) and is convenient for us since we don't have superscripts.

10-Jun-2005: The definition was extended to include zero exponents, so that 0↑0 = 1 per the convention of Definition 10-4.1 of [Gleason] p. 134 (0exp0e1 13091).

4-Jun-2014: The definition was extended to include negative integer exponents. For example, (-3↑-2) = (1 / 9) (ex-exp 27644). The case 𝑥 = 0, 𝑦 < 0 gives the value (1 / 0), so we will avoid this case in our theorems. (Contributed by Raph Levien, 20-May-2004.) (Revised by NM, 15-Oct-2004.)

Assertion
Ref Expression
df-exp ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-exp
StepHypRef Expression
1 cexp 13086 . 2 class
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 cc 10222 . . 3 class
5 cz 11646 . . 3 class
63cv 1636 . . . . 5 class 𝑦
7 cc0 10224 . . . . 5 class 0
86, 7wceq 1637 . . . 4 wff 𝑦 = 0
9 c1 10225 . . . 4 class 1
10 clt 10362 . . . . . 6 class <
117, 6, 10wbr 4851 . . . . 5 wff 0 < 𝑦
12 cmul 10229 . . . . . . 7 class ·
13 cn 11308 . . . . . . . 8 class
142cv 1636 . . . . . . . . 9 class 𝑥
1514csn 4377 . . . . . . . 8 class {𝑥}
1613, 15cxp 5316 . . . . . . 7 class (ℕ × {𝑥})
1712, 16, 9cseq 13027 . . . . . 6 class seq1( · , (ℕ × {𝑥}))
186, 17cfv 6104 . . . . 5 class (seq1( · , (ℕ × {𝑥}))‘𝑦)
196cneg 10555 . . . . . . 7 class -𝑦
2019, 17cfv 6104 . . . . . 6 class (seq1( · , (ℕ × {𝑥}))‘-𝑦)
21 cdiv 10972 . . . . . 6 class /
229, 20, 21co 6877 . . . . 5 class (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦))
2311, 18, 22cif 4286 . . . 4 class if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))
248, 9, 23cif 4286 . . 3 class if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦))))
252, 3, 4, 5, 24cmpt2 6879 . 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
261, 25wceq 1637 1 wff ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
Colors of variables: wff setvar class
This definition is referenced by:  expval  13088
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