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Theorem xaddmnf1 11480
Description: Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xaddmnf1  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A +e -oo )  = -oo )

Proof of Theorem xaddmnf1
StepHypRef Expression
1 mnfxr 11376 . . 3  |- -oo  e.  RR*
2 xaddval 11475 . . 3  |-  ( ( A  e.  RR*  /\ -oo  e.  RR* )  ->  ( A +e -oo )  =  if ( A  = +oo ,  if ( -oo  = -oo , 
0 , +oo ) ,  if ( A  = -oo ,  if ( -oo  = +oo , 
0 , -oo ) ,  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( A  + -oo ) ) ) ) ) )
31, 2mpan2 669 . 2  |-  ( A  e.  RR*  ->  ( A +e -oo )  =  if ( A  = +oo ,  if ( -oo  = -oo , 
0 , +oo ) ,  if ( A  = -oo ,  if ( -oo  = +oo , 
0 , -oo ) ,  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( A  + -oo ) ) ) ) ) )
4 ifnefalse 3897 . . 3  |-  ( A  =/= +oo  ->  if ( A  = +oo ,  if ( -oo  = -oo ,  0 , +oo ) ,  if ( A  = -oo ,  if ( -oo  = +oo , 
0 , -oo ) ,  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( A  + -oo ) ) ) ) )  =  if ( A  = -oo ,  if ( -oo  = +oo ,  0 , -oo ) ,  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( A  + -oo )
) ) ) )
5 mnfnepnf 11380 . . . . . 6  |- -oo  =/= +oo
6 ifnefalse 3897 . . . . . 6  |-  ( -oo  =/= +oo  ->  if ( -oo  = +oo ,  0 , -oo )  = -oo )
75, 6ax-mp 5 . . . . 5  |-  if ( -oo  = +oo , 
0 , -oo )  = -oo
8 ifnefalse 3897 . . . . . . 7  |-  ( -oo  =/= +oo  ->  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( A  + -oo )
) )  =  if ( -oo  = -oo , -oo ,  ( A  + -oo ) ) )
95, 8ax-mp 5 . . . . . 6  |-  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo , 
( A  + -oo ) ) )  =  if ( -oo  = -oo , -oo ,  ( A  + -oo )
)
10 eqid 2402 . . . . . . 7  |- -oo  = -oo
1110iftruei 3892 . . . . . 6  |-  if ( -oo  = -oo , -oo ,  ( A  + -oo ) )  = -oo
129, 11eqtri 2431 . . . . 5  |-  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo , 
( A  + -oo ) ) )  = -oo
13 ifeq12 3902 . . . . 5  |-  ( ( if ( -oo  = +oo ,  0 , -oo )  = -oo  /\  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo , 
( A  + -oo ) ) )  = -oo )  ->  if ( A  = -oo ,  if ( -oo  = +oo ,  0 , -oo ) ,  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( A  + -oo )
) ) )  =  if ( A  = -oo , -oo , -oo ) )
147, 12, 13mp2an 670 . . . 4  |-  if ( A  = -oo ,  if ( -oo  = +oo ,  0 , -oo ) ,  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( A  + -oo )
) ) )  =  if ( A  = -oo , -oo , -oo )
15 ifid 3922 . . . 4  |-  if ( A  = -oo , -oo , -oo )  = -oo
1614, 15eqtri 2431 . . 3  |-  if ( A  = -oo ,  if ( -oo  = +oo ,  0 , -oo ) ,  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( A  + -oo )
) ) )  = -oo
174, 16syl6eq 2459 . 2  |-  ( A  =/= +oo  ->  if ( A  = +oo ,  if ( -oo  = -oo ,  0 , +oo ) ,  if ( A  = -oo ,  if ( -oo  = +oo , 
0 , -oo ) ,  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( A  + -oo ) ) ) ) )  = -oo )
183, 17sylan9eq 2463 1  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A +e -oo )  = -oo )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   ifcif 3885  (class class class)co 6278   0cc0 9522    + caddc 9525   +oocpnf 9655   -oocmnf 9656   RR*cxr 9657   +ecxad 11369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-mulcl 9584  ax-i2m1 9590
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-pnf 9660  df-mnf 9661  df-xr 9662  df-xadd 11372
This theorem is referenced by:  xaddnepnf  11487  xaddcom  11490  xnegdi  11493  xleadd1a  11498  xsubge0  11506  xlesubadd  11508  xadddilem  11539  xblss2ps  21196  xblss2  21197
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