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

Theorem xnegmnf 10415
Description: Minus  -oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xnegmnf  |-  - e  -oo  =  +oo

Proof of Theorem xnegmnf
StepHypRef Expression
1 df-xneg 10331 . 2  |-  - e  -oo  =  if (  -oo  =  +oo ,  -oo ,  if (  -oo  =  -oo , 
+oo ,  -u  -oo )
)
2 pnfnemnf 10338 . . . 4  |-  +oo  =/=  -oo
32necomi 2494 . . 3  |-  -oo  =/=  +oo
4 ifnefalse 3478 . . 3  |-  (  -oo  =/=  +oo  ->  if (  -oo  =  +oo ,  -oo ,  if (  -oo  =  -oo ,  +oo ,  -u  -oo ) )  =  if (  -oo  =  -oo , 
+oo ,  -u  -oo )
)
53, 4ax-mp 10 . 2  |-  if ( 
-oo  =  +oo ,  -oo ,  if (  -oo  =  -oo ,  +oo ,  -u 
-oo ) )  =  if (  -oo  =  -oo ,  +oo ,  -u  -oo )
6 eqid 2253 . . 3  |-  -oo  =  -oo
7 iftrue 3476 . . 3  |-  (  -oo  =  -oo  ->  if (  -oo  =  -oo ,  +oo , 
-u  -oo )  =  +oo )
86, 7ax-mp 10 . 2  |-  if ( 
-oo  =  -oo ,  +oo ,  -u  -oo )  = 
+oo
91, 5, 83eqtri 2277 1  |-  - e  -oo  =  +oo
Colors of variables: wff set class
Syntax hints:    = wceq 1619    =/= wne 2412   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  xsubge0  10459  xmulneg1  10467  xmulpnf1n  10476  xadddi2  10495  xrsdsreclblem  16249
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-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-pow 4082  ax-un 4403  ax-cnex 8673
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-nfc 2374  df-ne 2414  df-nel 2415  df-rex 2514  df-rab 2516  df-v 2729  df-un 3083  df-in 3085  df-ss 3089  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-uni 3728  df-pnf 8749  df-mnf 8750  df-xr 8751  df-xneg 10331
  Copyright terms: Public domain W3C validator