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

Theorem pnfaddmnf 11441
Description: Addition of positive and negative 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
pnfaddmnf  |-  ( +oo +e -oo )  =  0

Proof of Theorem pnfaddmnf
StepHypRef Expression
1 pnfxr 11333 . . 3  |- +oo  e.  RR*
2 mnfxr 11335 . . 3  |- -oo  e.  RR*
3 xaddval 11434 . . 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 eqid 2467 . . 3  |- +oo  = +oo
65iftruei 3952 . 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 ) ) ) ) )  =  if ( -oo  = -oo , 
0 , +oo )
7 eqid 2467 . . 3  |- -oo  = -oo
87iftruei 3952 . 2  |-  if ( -oo  = -oo , 
0 , +oo )  =  0
94, 6, 83eqtri 2500 1  |-  ( +oo +e -oo )  =  0
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   ifcif 3945  (class class class)co 6295   0cc0 9504    + caddc 9507   +oocpnf 9637   -oocmnf 9638   RR*cxr 9639   +ecxad 11328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-mulcl 9566  ax-i2m1 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-pnf 9642  df-mnf 9643  df-xr 9644  df-xadd 11331
This theorem is referenced by:  xnegid  11447  xaddcom  11449  xnegdi  11452  xsubge0  11465  xlesubadd  11467  xadddilem  11498  xblss2  20773
  Copyright terms: Public domain W3C validator