Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > df-xneg | Structured version Visualization version GIF version |
Description: Define the negative of an extended real number. (Contributed by FL, 26-Dec-2011.) |
Ref | Expression |
---|---|
df-xneg | ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | 1 | cxne 12854 | . 2 class -𝑒𝐴 |
3 | cpnf 11015 | . . . 4 class +∞ | |
4 | 1, 3 | wceq 1539 | . . 3 wff 𝐴 = +∞ |
5 | cmnf 11016 | . . 3 class -∞ | |
6 | 1, 5 | wceq 1539 | . . . 4 wff 𝐴 = -∞ |
7 | 1 | cneg 11215 | . . . 4 class -𝐴 |
8 | 6, 3, 7 | cif 4460 | . . 3 class if(𝐴 = -∞, +∞, -𝐴) |
9 | 4, 5, 8 | cif 4460 | . 2 class if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) |
10 | 2, 9 | wceq 1539 | 1 wff -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) |
Colors of variables: wff setvar class |
This definition is referenced by: xnegeq 12950 xnegex 12951 xnegpnf 12952 xnegmnf 12953 rexneg 12954 nfxnegd 42988 |
Copyright terms: Public domain | W3C validator |