Theorem xnegpnf 13176

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 13079	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2730	. . 3 ⊢ +∞ = +∞
3	2	iftruei 4498	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2753	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1540  ifcif 4491  +∞cpnf 11212  -∞cmnf 11213  -cneg 11413  -_𝑒cxne 13076

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-if 4492  df-xneg 13079

This theorem is referenced by:  xnegcl  13180  xnegneg  13181  xltnegi  13183  xnegid  13205  xnegdi  13215  xaddass2  13217  xsubge0  13228  xlesubadd  13230  xmulneg1  13236  xmulmnf1  13243  xadddi2  13264  xrsdsreclblem  21336  xblss2ps  24296  xblss2  24297  xaddeq0  32683  supminfxr  45467  liminflbuz2  45820