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Theorem xnegpnf 13125
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 13030 . 2 -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2 eqid 2736 . . 3 +∞ = +∞
32iftruei 4492 . 2 if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
41, 3eqtri 2764 1 -𝑒+∞ = -∞
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  ifcif 4485  +∞cpnf 11183  -∞cmnf 11184  -cneg 11383  -𝑒cxne 13027
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 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-if 4486  df-xneg 13030
This theorem is referenced by:  xnegcl  13129  xnegneg  13130  xltnegi  13132  xnegid  13154  xnegdi  13164  xaddass2  13166  xsubge0  13177  xlesubadd  13179  xmulneg1  13185  xmulmnf1  13192  xadddi2  13213  xrsdsreclblem  20839  xblss2ps  23750  xblss2  23751  xaddeq0  31553  supminfxr  43673  liminflbuz2  44026
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