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Theorem xnegpnf 12078
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 11984 . 2 -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2 eqid 2651 . . 3 +∞ = +∞
32iftruei 4126 . 2 if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
41, 3eqtri 2673 1 -𝑒+∞ = -∞
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  ifcif 4119  +∞cpnf 10109  -∞cmnf 10110  -cneg 10305  -𝑒cxne 11981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-if 4120  df-xneg 11984
This theorem is referenced by:  xnegcl  12082  xnegneg  12083  xltnegi  12085  xnegid  12107  xnegdi  12116  xaddass2  12118  xsubge0  12129  xlesubadd  12131  xmulneg1  12137  xmulmnf1  12144  xadddi2  12165  xrsdsreclblem  19840  xblss2ps  22253  xblss2  22254  xaddeq0  29646  supminfxr  40007
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