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Mirrors > Home > ILE Home > Th. List > df-dvap | Unicode version |
Description: Define the derivative operator. This acts on functions to produce a function that is defined where the original function is differentiable, with value the derivative of the function at these points. The set here is the ambient topological space under which we are evaluating the continuity of the difference quotient. Although the definition is valid for any subset of and is well-behaved when contains no isolated points, we will restrict our attention to the cases or for the majority of the development, these corresponding respectively to real and complex differentiation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 25-Jun-2023.) |
Ref | Expression |
---|---|
df-dvap | ↾t # lim |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdv 12798 | . 2 | |
2 | vs | . . 3 | |
3 | vf | . . 3 | |
4 | cc 7623 | . . . 4 | |
5 | 4 | cpw 3510 | . . 3 |
6 | 2 | cv 1330 | . . . 4 |
7 | cpm 6543 | . . . 4 | |
8 | 4, 6, 7 | co 5774 | . . 3 |
9 | vx | . . . 4 | |
10 | 3 | cv 1330 | . . . . . 6 |
11 | 10 | cdm 4539 | . . . . 5 |
12 | cabs 10774 | . . . . . . . . 9 | |
13 | cmin 7938 | . . . . . . . . 9 | |
14 | 12, 13 | ccom 4543 | . . . . . . . 8 |
15 | cmopn 12159 | . . . . . . . 8 | |
16 | 14, 15 | cfv 5123 | . . . . . . 7 |
17 | crest 12125 | . . . . . . 7 ↾t | |
18 | 16, 6, 17 | co 5774 | . . . . . 6 ↾t |
19 | cnt 12267 | . . . . . 6 | |
20 | 18, 19 | cfv 5123 | . . . . 5 ↾t |
21 | 11, 20 | cfv 5123 | . . . 4 ↾t |
22 | 9 | cv 1330 | . . . . . 6 |
23 | 22 | csn 3527 | . . . . 5 |
24 | vz | . . . . . . 7 | |
25 | vw | . . . . . . . . . 10 | |
26 | 25 | cv 1330 | . . . . . . . . 9 |
27 | cap 8348 | . . . . . . . . 9 # | |
28 | 26, 22, 27 | wbr 3929 | . . . . . . . 8 # |
29 | 28, 25, 11 | crab 2420 | . . . . . . 7 # |
30 | 24 | cv 1330 | . . . . . . . . . 10 |
31 | 30, 10 | cfv 5123 | . . . . . . . . 9 |
32 | 22, 10 | cfv 5123 | . . . . . . . . 9 |
33 | 31, 32, 13 | co 5774 | . . . . . . . 8 |
34 | 30, 22, 13 | co 5774 | . . . . . . . 8 |
35 | cdiv 8437 | . . . . . . . 8 | |
36 | 33, 34, 35 | co 5774 | . . . . . . 7 |
37 | 24, 29, 36 | cmpt 3989 | . . . . . 6 # |
38 | climc 12797 | . . . . . 6 lim | |
39 | 37, 22, 38 | co 5774 | . . . . 5 # lim |
40 | 23, 39 | cxp 4537 | . . . 4 # lim |
41 | 9, 21, 40 | ciun 3813 | . . 3 ↾t # lim |
42 | 2, 3, 5, 8, 41 | cmpo 5776 | . 2 ↾t # lim |
43 | 1, 42 | wceq 1331 | 1 ↾t # lim |
Colors of variables: wff set class |
This definition is referenced by: reldvg 12822 dvfvalap 12824 |
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