Definition df-xneg 13011

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 13008	. 2 class -𝑒𝐴
3		cpnf 11143	. . . 4 class +∞
4	1, 3	wceq 1541	. . 3 wff 𝐴 = +∞
5		cmnf 11144	. . 3 class -∞
6	1, 5	wceq 1541	. . . 4 wff 𝐴 = -∞
7	1	cneg 11345	. . . 4 class -𝐴
8	6, 3, 7	cif 4472	. . 3 class if(𝐴 = -∞, +∞, -𝐴)
9	4, 5, 8	cif 4472	. 2 class if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
10	2, 9	wceq 1541	1 wff -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))

This definition is referenced by:  xnegeq  13106  xnegex  13107  xnegpnf  13108  xnegmnf  13109  rexneg  13110  nfxnegd  45549