MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xnegpnf Unicode version

Theorem xnegpnf 10414
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 10331 . 2  |-  - e  +oo  =  if (  +oo  =  +oo ,  -oo ,  if (  +oo  =  -oo , 
+oo ,  -u  +oo )
)
2 eqid 2253 . . 3  |-  +oo  =  +oo
3 iftrue 3476 . . 3  |-  (  +oo  =  +oo  ->  if (  +oo  =  +oo ,  -oo ,  if (  +oo  =  -oo ,  +oo ,  -u  +oo ) )  =  -oo )
42, 3ax-mp 10 . 2  |-  if ( 
+oo  =  +oo ,  -oo ,  if (  +oo  =  -oo ,  +oo ,  -u 
+oo ) )  = 
-oo
51, 4eqtri 2273 1  |-  - e  +oo  =  -oo
Colors of variables: wff set class
Syntax hints:    = wceq 1619   ifcif 3470    +oocpnf 8744    -oocmnf 8745   -ucneg 8918    - ecxne 10328
This theorem is referenced by:  xnegcl  10418  xnegneg  10419  xltnegi  10421  xnegid  10441  xnegdi  10446  xaddass2  10448  xsubge0  10459  xlesubadd  10461  xmulneg1  10467  xmulmnf1  10474  xadddi2  10495  xrsdsreclblem  16249  xblss2  17790
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-if 3471  df-xneg 10331
  Copyright terms: Public domain W3C validator