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Mirrors > Home > MPE Home > Th. List > df-abs | Structured version Visualization version GIF version |
Description: Define the function for the absolute value (modulus) of a complex number. See abscli 15346 for its closure and absval 15189 or absval2i 15348 for its value. For example, (absโ-2) = 2 (ex-abs 30213). (Contributed by NM, 27-Jul-1999.) |
Ref | Expression |
---|---|
df-abs | โข abs = (๐ฅ โ โ โฆ (โโ(๐ฅ ยท (โโ๐ฅ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cabs 15185 | . 2 class abs | |
2 | vx | . . 3 setvar ๐ฅ | |
3 | cc 11107 | . . 3 class โ | |
4 | 2 | cv 1532 | . . . . 5 class ๐ฅ |
5 | ccj 15047 | . . . . . 6 class โ | |
6 | 4, 5 | cfv 6536 | . . . . 5 class (โโ๐ฅ) |
7 | cmul 11114 | . . . . 5 class ยท | |
8 | 4, 6, 7 | co 7404 | . . . 4 class (๐ฅ ยท (โโ๐ฅ)) |
9 | csqrt 15184 | . . . 4 class โ | |
10 | 8, 9 | cfv 6536 | . . 3 class (โโ(๐ฅ ยท (โโ๐ฅ))) |
11 | 2, 3, 10 | cmpt 5224 | . 2 class (๐ฅ โ โ โฆ (โโ(๐ฅ ยท (โโ๐ฅ)))) |
12 | 1, 11 | wceq 1533 | 1 wff abs = (๐ฅ โ โ โฆ (โโ(๐ฅ ยท (โโ๐ฅ)))) |
Colors of variables: wff setvar class |
This definition is referenced by: absval 15189 absf 15288 absfico 44470 |
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