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Definition df-exp 10347
 Description: Define exponentiation to nonnegative integer powers. For example, (ex-exp 13133). This definition is not meant to be used directly; instead, exp0 10351 and expp1 10354 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 (0exp0e1 10352). 4-Jun-2014: The definition was extended to include negative integer exponents. For example, (ex-exp 13133). The case gives the value , so we will avoid this case in our theorems. (Contributed by Raph Levien, 20-May-2004.) (Revised by NM, 15-Oct-2004.)
Assertion
Ref Expression
df-exp
Distinct variable group:   ,

Detailed syntax breakdown of Definition df-exp
StepHypRef Expression
1 cexp 10346 . 2
2 vx . . 3
3 vy . . 3
4 cc 7665 . . 3
5 cz 9101 . . 3
63cv 1331 . . . . 5
7 cc0 7667 . . . . 5
86, 7wceq 1332 . . . 4
9 c1 7668 . . . 4
10 clt 7847 . . . . . 6
117, 6, 10wbr 3938 . . . . 5
12 cmul 7672 . . . . . . 7
13 cn 8767 . . . . . . . 8
142cv 1331 . . . . . . . . 9
1514csn 3533 . . . . . . . 8
1613, 15cxp 4547 . . . . . . 7
1712, 16, 9cseq 10272 . . . . . 6
186, 17cfv 5133 . . . . 5
196cneg 7981 . . . . . . 7
2019, 17cfv 5133 . . . . . 6
21 cdiv 8479 . . . . . 6
229, 20, 21co 5784 . . . . 5
2311, 18, 22cif 3480 . . . 4
248, 9, 23cif 3480 . . 3
252, 3, 4, 5, 24cmpo 5786 . 2
261, 25wceq 1332 1
 Colors of variables: wff set class This definition is referenced by:  exp3val  10349
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