Definition df-exp 13431

Description: Define exponentiation to nonnegative integer powers. For example, (5↑2) = 25 (ex-exp 28229). Terminology: In general, "exponentiation" is the operation of raising a "base" 𝑥 to the power of the "exponent" 𝑦, resulting in the "power" (𝑥↑𝑦), 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" (𝑥↑𝑦).

This definition is not meant to be used directly; instead, exp0 13434 and expp1 13437 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 13435).

4-Jun-2014: The definition was extended to include negative integer exponents. For example, (-3↑-2) = (1 / 9) (ex-exp 28229). 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 25141 (complex exponentiation). Both definitions are equivalent for integer exponents, see cxpexpz 25250. (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

Step	Hyp	Ref	Expression
1		cexp 13430	. 2 class ↑
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 10535	. . 3 class ℂ
5		cz 11982	. . 3 class ℤ
6	3	cv 1536	. . . . 5 class 𝑦
7		cc0 10537	. . . . 5 class 0
8	6, 7	wceq 1537	. . . 4 wff 𝑦 = 0
9		c1 10538	. . . 4 class 1
10		clt 10675	. . . . . 6 class <
11	7, 6, 10	wbr 5066	. . . . 5 wff 0 < 𝑦
12		cmul 10542	. . . . . . 7 class ·
13		cn 11638	. . . . . . . 8 class ℕ
14	2	cv 1536	. . . . . . . . 9 class 𝑥
15	14	csn 4567	. . . . . . . 8 class {𝑥}
16	13, 15	cxp 5553	. . . . . . 7 class (ℕ × {𝑥})
17	12, 16, 9	cseq 13370	. . . . . 6 class seq1( · , (ℕ × {𝑥}))
18	6, 17	cfv 6355	. . . . 5 class (seq1( · , (ℕ × {𝑥}))‘𝑦)
19	6	cneg 10871	. . . . . . 7 class -𝑦
20	19, 17	cfv 6355	. . . . . 6 class (seq1( · , (ℕ × {𝑥}))‘-𝑦)
21		cdiv 11297	. . . . . 6 class /
22	9, 20, 21	co 7156	. . . . 5 class (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦))
23	11, 18, 22	cif 4467	. . . 4 class if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))
24	8, 9, 23	cif 4467	. . 3 class if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦))))
25	2, 3, 4, 5, 24	cmpo 7158	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
26	1, 25	wceq 1537	1 wff ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))

This definition is referenced by:  expval  13432