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Definition df-exp 12167
Description: Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 12170 and expp1 12173 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, 0 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 12166 . 2
2 vx . . 3
3 vy . . 3
4 cc 9511 . . 3
5 cz 10889 . . 3
63cv 1394 . . . . 5
7 cc0 9513 . . . . 5
86, 7wceq 1395 . . . 4
9 c1 9514 . . . 4
10 clt 9649 . . . . . 6
117, 6, 10wbr 4452 . . . . 5
12 cmul 9518 . . . . . . 7
13 cn 10561 . . . . . . . 8
142cv 1394 . . . . . . . . 9
1514csn 4029 . . . . . . . 8
1613, 15cxp 5002 . . . . . . 7
1712, 16, 9cseq 12107 . . . . . 6
186, 17cfv 5593 . . . . 5
196cneg 9829 . . . . . . 7
2019, 17cfv 5593 . . . . . 6
21 cdiv 10231 . . . . . 6
229, 20, 21co 6296 . . . . 5
2311, 18, 22cif 3941 . . . 4
248, 9, 23cif 3941 . . 3
252, 3, 4, 5, 24cmpt2 6298 . 2
261, 25wceq 1395 1
Colors of variables: wff setvar class
This definition is referenced by:  expval  12168
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