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Theorem xnegpnf 11914
 Description: Minus +∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
Assertion
Ref Expression
xnegpnf -𝑒+∞ = -∞

Proof of Theorem xnegpnf
StepHypRef Expression
1 df-xneg 11822 . 2 -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2 eqid 2610 . . 3 +∞ = +∞
32iftruei 4043 . 2 if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
41, 3eqtri 2632 1 -𝑒+∞ = -∞
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  ifcif 4036  +∞cpnf 9950  -∞cmnf 9951  -cneg 10146  -𝑒cxne 11819 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-if 4037  df-xneg 11822 This theorem is referenced by:  xnegcl  11918  xnegneg  11919  xltnegi  11921  xnegid  11943  xnegdi  11950  xaddass2  11952  xsubge0  11963  xlesubadd  11965  xmulneg1  11971  xmulmnf1  11978  xadddi2  11999  xrsdsreclblem  19611  xblss2ps  22016  xblss2  22017  xaddeq0  28907
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