Theorem xnegpnf 13138

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 13040	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2737	. . 3 ⊢ +∞ = +∞
3	2	iftruei 4488	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2760	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1542  ifcif 4481  +∞cpnf 11177  -∞cmnf 11178  -cneg 11379  -_𝑒cxne 13037

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 4482  df-xneg 13040

This theorem is referenced by:  xnegcl  13142  xnegneg  13143  xltnegi  13145  xnegid  13167  xnegdi  13177  xaddass2  13179  xsubge0  13190  xlesubadd  13192  xmulneg1  13198  xmulmnf1  13205  xadddi2  13226  xrsdsreclblem  21384  xblss2ps  24362  xblss2  24363  xaddeq0  32850  supminfxr  45851  liminflbuz2  46202