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Definition df-iexp 8889
Description: Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 8893 and expp1 8896 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 Kingodon, 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 8888 . 2 class
2 vx . . 3 setvar x
3 vy . . 3 setvar y
4 cc 6689 . . 3 class
5 cz 8001 . . 3 class
63cv 1241 . . . . 5 class y
7 cc0 6691 . . . . 5 class 0
86, 7wceq 1242 . . . 4 wff y = 0
9 c1 6692 . . . 4 class 1
10 clt 6837 . . . . . 6 class <
117, 6, 10wbr 3755 . . . . 5 wff 0 < y
12 cmul 6696 . . . . . . 7 class ·
13 cn 7675 . . . . . . . 8 class
142cv 1241 . . . . . . . . 9 class x
1514csn 3367 . . . . . . . 8 class {x}
1613, 15cxp 4286 . . . . . . 7 class (ℕ × {x})
1712, 4, 16, 9cseq 8872 . . . . . 6 class seq1( · , (ℕ × {x}), ℂ)
186, 17cfv 4845 . . . . 5 class (seq1( · , (ℕ × {x}), ℂ)‘y)
196cneg 6960 . . . . . . 7 class -y
2019, 17cfv 4845 . . . . . 6 class (seq1( · , (ℕ × {x}), ℂ)‘-y)
21 cdiv 7413 . . . . . 6 class /
229, 20, 21co 5455 . . . . 5 class (1 / (seq1( · , (ℕ × {x}), ℂ)‘-y))
2311, 18, 22cif 3325 . . . 4 class if(0 < y, (seq1( · , (ℕ × {x}), ℂ)‘y), (1 / (seq1( · , (ℕ × {x}), ℂ)‘-y)))
248, 9, 23cif 3325 . . 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 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)))))
261, 25wceq 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)))))
Colors of variables: wff set class
This definition is referenced by:  expival  8891
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