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Theorem xnegpnf 10784
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 10699 . 2  |-  - e  +oo  =  if (  +oo  =  +oo ,  -oo ,  if (  +oo  =  -oo , 
+oo ,  -u  +oo )
)
2 eqid 2435 . . 3  |-  +oo  =  +oo
3 iftrue 3737 . . 3  |-  (  +oo  =  +oo  ->  if (  +oo  =  +oo ,  -oo ,  if (  +oo  =  -oo ,  +oo ,  -u  +oo ) )  =  -oo )
42, 3ax-mp 8 . 2  |-  if ( 
+oo  =  +oo ,  -oo ,  if (  +oo  =  -oo ,  +oo ,  -u 
+oo ) )  = 
-oo
51, 4eqtri 2455 1  |-  - e  +oo  =  -oo
Colors of variables: wff set class
Syntax hints:    = wceq 1652   ifcif 3731    +oocpnf 9106    -oocmnf 9107   -ucneg 9281    - ecxne 10696
This theorem is referenced by:  xnegcl  10788  xnegneg  10789  xltnegi  10791  xnegid  10811  xnegdi  10816  xaddass2  10818  xsubge0  10829  xlesubadd  10831  xmulneg1  10837  xmulmnf1  10844  xadddi2  10865  xrsdsreclblem  16732  xblss2ps  18419  xblss2  18420  xaddeq0  24107
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-if 3732  df-xneg 10699
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