Theorem xnegpnf 13156

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 13058	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2737	. . 3 ⊢ +∞ = +∞
3	2	iftruei 4474	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2760	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1542  ifcif 4467  +∞cpnf 11171  -∞cmnf 11172  -cneg 11373  -_𝑒cxne 13055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-if 4468  df-xneg 13058

This theorem is referenced by:  xnegcl  13160  xnegneg  13161  xltnegi  13163  xnegid  13185  xnegdi  13195  xaddass2  13197  xsubge0  13208  xlesubadd  13210  xmulneg1  13216  xmulmnf1  13223  xadddi2  13244  xrsdsreclblem  21406  xblss2ps  24380  xblss2  24381  xaddeq0  32845  supminfxr  45914  liminflbuz2  46265