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Definition df-xneg 12165
Description: Define the negative of an extended real number. (Contributed by FL, 26-Dec-2011.)
Assertion
Ref Expression
df-xneg -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))

Detailed syntax breakdown of Definition df-xneg
StepHypRef Expression
1 cA . . 3 class 𝐴
21cxne 12162 . 2 class -𝑒𝐴
3 cpnf 10359 . . . 4 class +∞
41, 3wceq 1637 . . 3 wff 𝐴 = +∞
5 cmnf 10360 . . 3 class -∞
61, 5wceq 1637 . . . 4 wff 𝐴 = -∞
71cneg 10555 . . . 4 class -𝐴
86, 3, 7cif 4286 . . 3 class if(𝐴 = -∞, +∞, -𝐴)
94, 5, 8cif 4286 . 2 class if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
102, 9wceq 1637 1 wff -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
Colors of variables: wff setvar class
This definition is referenced by:  xnegeq  12259  xnegex  12260  xnegpnf  12261  xnegmnf  12262  rexneg  12263  nfxnegd  40148
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