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Definition df-iexp 9790
Description: Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 9794 and expp1 9797 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
StepHypRef Expression
1 cexp 9789 . 2 class
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 cc 7249 . . 3 class
5 cz 8644 . . 3 class
63cv 1284 . . . . 5 class 𝑦
7 cc0 7251 . . . . 5 class 0
86, 7wceq 1285 . . . 4 wff 𝑦 = 0
9 c1 7252 . . . 4 class 1
10 clt 7423 . . . . . 6 class <
117, 6, 10wbr 3811 . . . . 5 wff 0 < 𝑦
12 cmul 7256 . . . . . . 7 class ·
13 cn 8314 . . . . . . . 8 class
142cv 1284 . . . . . . . . 9 class 𝑥
1514csn 3422 . . . . . . . 8 class {𝑥}
1613, 15cxp 4397 . . . . . . 7 class (ℕ × {𝑥})
1712, 4, 16, 9cseq 9738 . . . . . 6 class seq1( · , (ℕ × {𝑥}), ℂ)
186, 17cfv 4967 . . . . 5 class (seq1( · , (ℕ × {𝑥}), ℂ)‘𝑦)
196cneg 7555 . . . . . . 7 class -𝑦
2019, 17cfv 4967 . . . . . 6 class (seq1( · , (ℕ × {𝑥}), ℂ)‘-𝑦)
21 cdiv 8035 . . . . . 6 class /
229, 20, 21co 5589 . . . . 5 class (1 / (seq1( · , (ℕ × {𝑥}), ℂ)‘-𝑦))
2311, 18, 22cif 3373 . . . 4 class if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}), ℂ)‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}), ℂ)‘-𝑦)))
248, 9, 23cif 3373 . . 3 class if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}), ℂ)‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}), ℂ)‘-𝑦))))
252, 3, 4, 5, 24cmpt2 5591 . 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}), ℂ)‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}), ℂ)‘-𝑦)))))
261, 25wceq 1285 1 wff ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}), ℂ)‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}), ℂ)‘-𝑦)))))
Colors of variables: wff set class
This definition is referenced by:  expival  9792
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