Theorem xnegpnf 10062

Description: Minus +∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)

Assertion

Ref	Expression
xnegpnf	⊢ -𝑒+∞ = -∞

Proof of Theorem xnegpnf

Step	Hyp	Ref	Expression
1		df-xneg 10006	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2231	. . 3 ⊢ +∞ = +∞
3	2	iftruei 3611	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2252	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1397  ifcif 3605  +∞cpnf 8210  -∞cmnf 8211  -cneg 8350  -_𝑒cxne 10003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-if 3606  df-xneg 10006

This theorem is referenced by:  xnegcl  10066  xnegneg  10067  xltnegi  10069  xnegid  10093  xnegdi  10102  xaddass2  10104  xsubge0  10115  xposdif  10116  xlesubadd  10117  xblss2ps  15127  xblss2  15128