Theorem xnegpnf 13248

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 13152	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2735	. . 3 ⊢ +∞ = +∞
3	2	iftruei 4538	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2763	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1537  ifcif 4531  +∞cpnf 11290  -∞cmnf 11291  -cneg 11491  -_𝑒cxne 13149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-if 4532  df-xneg 13152

This theorem is referenced by:  xnegcl  13252  xnegneg  13253  xltnegi  13255  xnegid  13277  xnegdi  13287  xaddass2  13289  xsubge0  13300  xlesubadd  13302  xmulneg1  13308  xmulmnf1  13315  xadddi2  13336  xrsdsreclblem  21448  xblss2ps  24427  xblss2  24428  xaddeq0  32764  supminfxr  45414  liminflbuz2  45771