Theorem xnegpnf 12288

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 12192	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2800	. . 3 ⊢ +∞ = +∞
3	2	iftruei 4285	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2822	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1653  ifcif 4278  +∞cpnf 10361  -∞cmnf 10362  -cneg 10558  -_𝑒cxne 12189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2778

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-if 4279  df-xneg 12192

This theorem is referenced by:  xnegcl  12292  xnegneg  12293  xltnegi  12295  xnegid  12317  xnegdi  12326  xaddass2  12328  xsubge0  12339  xlesubadd  12341  xmulneg1  12347  xmulmnf1  12354  xadddi2  12375  xrsdsreclblem  20113  xblss2ps  22533  xblss2  22534  xaddeq0  30035  supminfxr  40432