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Theorem xnegpnf 11525
Description: Minus +oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
Assertion
Ref Expression
xnegpnf  |-  -e +oo  = -oo

Proof of Theorem xnegpnf
StepHypRef Expression
1 df-xneg 11432 . 2  |-  -e +oo  =  if ( +oo  = +oo , -oo ,  if ( +oo  = -oo , +oo ,  -u +oo ) )
2 eqid 2471 . . 3  |- +oo  = +oo
32iftruei 3879 . 2  |-  if ( +oo  = +oo , -oo ,  if ( +oo  = -oo , +oo ,  -u +oo ) )  = -oo
41, 3eqtri 2493 1  |-  -e +oo  = -oo
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452   ifcif 3872   +oocpnf 9690   -oocmnf 9691   -ucneg 9881    -ecxne 11429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-if 3873  df-xneg 11432
This theorem is referenced by:  xnegcl  11529  xnegneg  11530  xltnegi  11532  xnegid  11553  xnegdi  11559  xaddass2  11561  xsubge0  11572  xlesubadd  11574  xmulneg1  11580  xmulmnf1  11587  xadddi2  11608  xrsdsreclblem  19091  xblss2ps  21494  xblss2  21495  xaddeq0  28405
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