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 13042	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2731	. . 3 ⊢ +∞ = +∞
3	2	iftruei 4498	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2759	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1541  ifcif 4491  +∞cpnf 11195  -∞cmnf 11196  -cneg 11395  -_𝑒cxne 13039

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-if 4492  df-xneg 13042

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  20880  xblss2ps  23791  xblss2  23792  xaddeq0  31726  supminfxr  43819  liminflbuz2  44176