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Theorem xnegpnf 10552
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 10468 . 2  |-  - e  +oo  =  if (  +oo  =  +oo ,  -oo ,  if (  +oo  =  -oo , 
+oo ,  -u  +oo )
)
2 eqid 2296 . . 3  |-  +oo  =  +oo
3 iftrue 3584 . . 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 2316 1  |-  - e  +oo  =  -oo
Colors of variables: wff set class
Syntax hints:    = wceq 1632   ifcif 3578    +oocpnf 8880    -oocmnf 8881   -ucneg 9054    - ecxne 10465
This theorem is referenced by:  xnegcl  10556  xnegneg  10557  xltnegi  10559  xnegid  10579  xnegdi  10584  xaddass2  10586  xsubge0  10597  xlesubadd  10599  xmulneg1  10605  xmulmnf1  10612  xadddi2  10633  xrsdsreclblem  16433  xblss2  17974  xaddeq0  23319
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-if 3579  df-xneg 10468
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