Theorem xnegpnf 13251

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 13154	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2737	. . 3 ⊢ +∞ = +∞
3	2	iftruei 4532	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2765	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1540  ifcif 4525  +∞cpnf 11292  -∞cmnf 11293  -cneg 11493  -_𝑒cxne 13151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-if 4526  df-xneg 13154

This theorem is referenced by:  xnegcl  13255  xnegneg  13256  xltnegi  13258  xnegid  13280  xnegdi  13290  xaddass2  13292  xsubge0  13303  xlesubadd  13305  xmulneg1  13311  xmulmnf1  13318  xadddi2  13339  xrsdsreclblem  21430  xblss2ps  24411  xblss2  24412  xaddeq0  32757  supminfxr  45475  liminflbuz2  45830