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

Theorem mnfaddpnf 11421
Description: Addition of negative and positive infinity. This is often taken to be a "null" value or out of the domain, but we define it (somewhat arbitrarily) to be zero so that the resulting function is total, which simplifies proofs. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
mnfaddpnf  |-  ( -oo +e +oo )  =  0

Proof of Theorem mnfaddpnf
StepHypRef Expression
1 mnfxr 11314 . . 3  |- -oo  e.  RR*
2 pnfxr 11312 . . 3  |- +oo  e.  RR*
3 xaddval 11413 . . 3  |-  ( ( -oo  e.  RR*  /\ +oo  e.  RR* )  ->  ( -oo +e +oo )  =  if ( -oo  = +oo ,  if ( +oo  = -oo ,  0 , +oo ) ,  if ( -oo  = -oo ,  if ( +oo  = +oo ,  0 , -oo ) ,  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo ,  ( -oo  + +oo )
) ) ) ) )
41, 2, 3mp2an 672 . 2  |-  ( -oo +e +oo )  =  if ( -oo  = +oo ,  if ( +oo  = -oo ,  0 , +oo ) ,  if ( -oo  = -oo ,  if ( +oo  = +oo ,  0 , -oo ) ,  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo ,  ( -oo  + +oo )
) ) ) )
5 mnfnepnf 11318 . . . 4  |- -oo  =/= +oo
6 ifnefalse 3946 . . . 4  |-  ( -oo  =/= +oo  ->  if ( -oo  = +oo ,  if ( +oo  = -oo , 
0 , +oo ) ,  if ( -oo  = -oo ,  if ( +oo  = +oo ,  0 , -oo ) ,  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo , 
( -oo  + +oo ) ) ) ) )  =  if ( -oo  = -oo ,  if ( +oo  = +oo ,  0 , -oo ) ,  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo ,  ( -oo  + +oo )
) ) ) )
75, 6ax-mp 5 . . 3  |-  if ( -oo  = +oo ,  if ( +oo  = -oo ,  0 , +oo ) ,  if ( -oo  = -oo ,  if ( +oo  = +oo , 
0 , -oo ) ,  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo ,  ( -oo  + +oo ) ) ) ) )  =  if ( -oo  = -oo ,  if ( +oo  = +oo ,  0 , -oo ) ,  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo ,  ( -oo  + +oo )
) ) )
8 eqid 2462 . . . . 5  |- -oo  = -oo
98iftruei 3941 . . . 4  |-  if ( -oo  = -oo ,  if ( +oo  = +oo ,  0 , -oo ) ,  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo ,  ( -oo  + +oo )
) ) )  =  if ( +oo  = +oo ,  0 , -oo )
10 eqid 2462 . . . . 5  |- +oo  = +oo
1110iftruei 3941 . . . 4  |-  if ( +oo  = +oo , 
0 , -oo )  =  0
129, 11eqtri 2491 . . 3  |-  if ( -oo  = -oo ,  if ( +oo  = +oo ,  0 , -oo ) ,  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo ,  ( -oo  + +oo )
) ) )  =  0
137, 12eqtri 2491 . 2  |-  if ( -oo  = +oo ,  if ( +oo  = -oo ,  0 , +oo ) ,  if ( -oo  = -oo ,  if ( +oo  = +oo , 
0 , -oo ) ,  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo ,  ( -oo  + +oo ) ) ) ) )  =  0
144, 13eqtri 2491 1  |-  ( -oo +e +oo )  =  0
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    e. wcel 1762    =/= wne 2657   ifcif 3934  (class class class)co 6277   0cc0 9483    + caddc 9486   +oocpnf 9616   -oocmnf 9617   RR*cxr 9618   +ecxad 11307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-mulcl 9545  ax-i2m1 9551
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-pnf 9621  df-mnf 9622  df-xr 9623  df-xadd 11310
This theorem is referenced by:  xnegid  11426  xaddcom  11428  xnegdi  11431  xsubge0  11444  xadddilem  11477
  Copyright terms: Public domain W3C validator