Theorem xnegpnf 9452

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 9400	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2100	. . 3 ⊢ +∞ = +∞
3	2	iftruei 3427	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2120	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1299  ifcif 3421  +∞cpnf 7669  -∞cmnf 7670  -cneg 7805  -_𝑒cxne 9397

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-if 3422  df-xneg 9400

This theorem is referenced by:  xnegcl  9456  xnegneg  9457  xltnegi  9459  xnegid  9483  xnegdi  9492  xaddass2  9494  xsubge0  9505  xposdif  9506  xlesubadd  9507  xblss2ps  12332  xblss2  12333