Definition df-xneg 12508

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

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2	1	cxne 12505	. 2 class -𝑒𝐴
3		cpnf 10672	. . . 4 class +∞
4	1, 3	wceq 1537	. . 3 wff 𝐴 = +∞
5		cmnf 10673	. . 3 class -∞
6	1, 5	wceq 1537	. . . 4 wff 𝐴 = -∞
7	1	cneg 10871	. . . 4 class -𝐴
8	6, 3, 7	cif 4467	. . 3 class if(𝐴 = -∞, +∞, -𝐴)
9	4, 5, 8	cif 4467	. 2 class if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
10	2, 9	wceq 1537	1 wff -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))

This definition is referenced by:  xnegeq  12601  xnegex  12602  xnegpnf  12603  xnegmnf  12604  rexneg  12605  nfxnegd  41764