Definition df-iexp 8909

Description: Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 8913 and expp1 8916 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. 4-Jun-2014: The definition was extended to include negative integer exponents. The case x = 0, y < 0 gives the value (1 / 0), so we will avoid this case in our theorems. (Contributed by Jim Kingdon, 7-Jun-2020.)

Assertion

Ref	Expression
df-iexp	⊢ ↑ = (x ∈ ℂ, y ∈ ℤ ↦ if(y = 0, 1, if(0 < y, (seq1( · , (ℕ × {x}), ℂ)‘y), (1 / (seq1( · , (ℕ × {x}), ℂ)‘-y)))))

Distinct variable group:   x,y

Detailed syntax breakdown of Definition df-iexp

Step	Hyp	Ref	Expression
1		cexp 8908	. 2 class ↑
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cc 6709	. . 3 class ℂ
5		cz 8021	. . 3 class ℤ
6	3	cv 1241	. . . . 5 class y
7		cc0 6711	. . . . 5 class 0
8	6, 7	wceq 1242	. . . 4 wff y = 0
9		c1 6712	. . . 4 class 1
10		clt 6857	. . . . . 6 class <
11	7, 6, 10	wbr 3755	. . . . 5 wff 0 < y
12		cmul 6716	. . . . . . 7 class ·
13		cn 7695	. . . . . . . 8 class ℕ
14	2	cv 1241	. . . . . . . . 9 class x
15	14	csn 3367	. . . . . . . 8 class {x}
16	13, 15	cxp 4286	. . . . . . 7 class (ℕ × {x})
17	12, 4, 16, 9	cseq 8892	. . . . . 6 class seq1( · , (ℕ × {x}), ℂ)
18	6, 17	cfv 4845	. . . . 5 class (seq1( · , (ℕ × {x}), ℂ)‘y)
19	6	cneg 6980	. . . . . . 7 class -y
20	19, 17	cfv 4845	. . . . . 6 class (seq1( · , (ℕ × {x}), ℂ)‘-y)
21		cdiv 7433	. . . . . 6 class /
22	9, 20, 21	co 5455	. . . . 5 class (1 / (seq1( · , (ℕ × {x}), ℂ)‘-y))
23	11, 18, 22	cif 3325	. . . 4 class if(0 < y, (seq1( · , (ℕ × {x}), ℂ)‘y), (1 / (seq1( · , (ℕ × {x}), ℂ)‘-y)))
24	8, 9, 23	cif 3325	. . 3 class if(y = 0, 1, if(0 < y, (seq1( · , (ℕ × {x}), ℂ)‘y), (1 / (seq1( · , (ℕ × {x}), ℂ)‘-y))))
25	2, 3, 4, 5, 24	cmpt2 5457	. 2 class (x ∈ ℂ, y ∈ ℤ ↦ if(y = 0, 1, if(0 < y, (seq1( · , (ℕ × {x}), ℂ)‘y), (1 / (seq1( · , (ℕ × {x}), ℂ)‘-y)))))
26	1, 25	wceq 1242	1 wff ↑ = (x ∈ ℂ, y ∈ ℤ ↦ if(y = 0, 1, if(0 < y, (seq1( · , (ℕ × {x}), ℂ)‘y), (1 / (seq1( · , (ℕ × {x}), ℂ)‘-y)))))

This definition is referenced by:  expival  8911