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Definition df-iexp 9255
 Description: Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 9259 and expp1 9262 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 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 gives the value , so we will avoid this case in our theorems. (Contributed by Jim Kingdon, 7-Jun-2020.)
Assertion
Ref Expression
df-iexp
Distinct variable group:   ,

Detailed syntax breakdown of Definition df-iexp
StepHypRef Expression
1 cexp 9254 . 2
2 vx . . 3
3 vy . . 3
4 cc 6887 . . 3
5 cz 8245 . . 3
63cv 1242 . . . . 5
7 cc0 6889 . . . . 5
86, 7wceq 1243 . . . 4
9 c1 6890 . . . 4
10 clt 7060 . . . . . 6
117, 6, 10wbr 3764 . . . . 5
12 cmul 6894 . . . . . . 7
13 cn 7914 . . . . . . . 8
142cv 1242 . . . . . . . . 9
1514csn 3375 . . . . . . . 8
1613, 15cxp 4343 . . . . . . 7
1712, 4, 16, 9cseq 9211 . . . . . 6
186, 17cfv 4902 . . . . 5
196cneg 7183 . . . . . . 7
2019, 17cfv 4902 . . . . . 6
21 cdiv 7651 . . . . . 6
229, 20, 21co 5512 . . . . 5
2311, 18, 22cif 3331 . . . 4
248, 9, 23cif 3331 . . 3
252, 3, 4, 5, 24cmpt2 5514 . 2
261, 25wceq 1243 1
 Colors of variables: wff set class This definition is referenced by:  expival  9257
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