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Definition df-exp 13430
Description: Define exponentiation to nonnegative integer powers. For example, (5↑2) = 25 (ex-exp 28238). Terminology: In general, "exponentiation" is the operation of raising a "base" 𝑥 to the power of the "exponent" 𝑦, resulting in the "power" (𝑥𝑦), also called "x to the power of y". In this case, "integer exponentiation" is the operation of raising a complex "base" 𝑥 to the power of an integer 𝑦, resulting in the "integer power" (𝑥𝑦).

This definition is not meant to be used directly; instead, exp0 13433 and expp1 13436 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 13434).

4-Jun-2014: The definition was extended to include negative integer exponents. For example, (-3↑-2) = (1 / 9) (ex-exp 28238). The case 𝑥 = 0, 𝑦 < 0 gives the value (1 / 0), so we will avoid this case in our theorems.

For a definition of exponentiation including complex exponents see df-cxp 25152 (complex exponentiation). Both definitions are equivalent for integer exponents, see cxpexpz 25261. (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 13429 . 2 class
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 cc 10528 . . 3 class
5 cz 11973 . . 3 class
63cv 1537 . . . . 5 class 𝑦
7 cc0 10530 . . . . 5 class 0
86, 7wceq 1538 . . . 4 wff 𝑦 = 0
9 c1 10531 . . . 4 class 1
10 clt 10668 . . . . . 6 class <
117, 6, 10wbr 5033 . . . . 5 wff 0 < 𝑦
12 cmul 10535 . . . . . . 7 class ·
13 cn 11629 . . . . . . . 8 class
142cv 1537 . . . . . . . . 9 class 𝑥
1514csn 4528 . . . . . . . 8 class {𝑥}
1613, 15cxp 5521 . . . . . . 7 class (ℕ × {𝑥})
1712, 16, 9cseq 13368 . . . . . 6 class seq1( · , (ℕ × {𝑥}))
186, 17cfv 6328 . . . . 5 class (seq1( · , (ℕ × {𝑥}))‘𝑦)
196cneg 10864 . . . . . . 7 class -𝑦
2019, 17cfv 6328 . . . . . 6 class (seq1( · , (ℕ × {𝑥}))‘-𝑦)
21 cdiv 11290 . . . . . 6 class /
229, 20, 21co 7139 . . . . 5 class (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦))
2311, 18, 22cif 4428 . . . 4 class if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))
248, 9, 23cif 4428 . . 3 class if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦))))
252, 3, 4, 5, 24cmpo 7141 . 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
261, 25wceq 1538 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  13431
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