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Mirrors > Home > ILE Home > Th. List > df-exp | Unicode version |
Description: Define exponentiation to
nonnegative integer powers. For example,
(see ex-exp 13451).
This definition is not meant to be used directly; instead, exp0 10449 and expp1 10452 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 (see 0exp0e1 10450). 4-Jun-2014: The definition was extended to include negative integer exponents. For example, (ex-exp 13451). 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.) |
Ref | Expression |
---|---|
df-exp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cexp 10444 | . 2 | |
2 | vx | . . 3 | |
3 | vy | . . 3 | |
4 | cc 7742 | . . 3 | |
5 | cz 9182 | . . 3 | |
6 | 3 | cv 1341 | . . . . 5 |
7 | cc0 7744 | . . . . 5 | |
8 | 6, 7 | wceq 1342 | . . . 4 |
9 | c1 7745 | . . . 4 | |
10 | clt 7924 | . . . . . 6 | |
11 | 7, 6, 10 | wbr 3976 | . . . . 5 |
12 | cmul 7749 | . . . . . . 7 | |
13 | cn 8848 | . . . . . . . 8 | |
14 | 2 | cv 1341 | . . . . . . . . 9 |
15 | 14 | csn 3570 | . . . . . . . 8 |
16 | 13, 15 | cxp 4596 | . . . . . . 7 |
17 | 12, 16, 9 | cseq 10370 | . . . . . 6 |
18 | 6, 17 | cfv 5182 | . . . . 5 |
19 | 6 | cneg 8061 | . . . . . . 7 |
20 | 19, 17 | cfv 5182 | . . . . . 6 |
21 | cdiv 8559 | . . . . . 6 | |
22 | 9, 20, 21 | co 5836 | . . . . 5 |
23 | 11, 18, 22 | cif 3515 | . . . 4 |
24 | 8, 9, 23 | cif 3515 | . . 3 |
25 | 2, 3, 4, 5, 24 | cmpo 5838 | . 2 |
26 | 1, 25 | wceq 1342 | 1 |
Colors of variables: wff set class |
This definition is referenced by: exp3val 10447 |
Copyright terms: Public domain | W3C validator |