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Theorem xaddmnf1 10816
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 10716 . . 3  |-  -oo  e.  RR*
2 xaddval 10811 . . 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 654 . 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 3749 . . 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 pnfnemnf 10719 . . . . . . 7  |-  +oo  =/=  -oo
65necomi 2688 . . . . . 6  |-  -oo  =/=  +oo
7 ifnefalse 3749 . . . . . 6  |-  (  -oo  =/=  +oo  ->  if (  -oo  =  +oo ,  0 ,  -oo )  = 
-oo )
86, 7ax-mp 8 . . . . 5  |-  if ( 
-oo  =  +oo , 
0 ,  -oo )  =  -oo
9 ifnefalse 3749 . . . . . . 7  |-  (  -oo  =/=  +oo  ->  if (  -oo  =  +oo ,  +oo ,  if (  -oo  =  -oo ,  -oo ,  ( A  +  -oo )
) )  =  if (  -oo  =  -oo , 
-oo ,  ( A  +  -oo ) ) )
106, 9ax-mp 8 . . . . . 6  |-  if ( 
-oo  =  +oo ,  +oo ,  if (  -oo  =  -oo ,  -oo , 
( A  +  -oo ) ) )  =  if (  -oo  =  -oo ,  -oo ,  ( A  +  -oo )
)
11 eqid 2438 . . . . . . 7  |-  -oo  =  -oo
12 iftrue 3747 . . . . . . 7  |-  (  -oo  =  -oo  ->  if (  -oo  =  -oo ,  -oo ,  ( A  +  -oo ) )  =  -oo )
1311, 12ax-mp 8 . . . . . 6  |-  if ( 
-oo  =  -oo ,  -oo ,  ( A  +  -oo ) )  =  -oo
1410, 13eqtri 2458 . . . . 5  |-  if ( 
-oo  =  +oo ,  +oo ,  if (  -oo  =  -oo ,  -oo , 
( A  +  -oo ) ) )  = 
-oo
15 ifeq12 3754 . . . . 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 ) )
168, 14, 15mp2an 655 . . . 4  |-  if ( A  =  -oo ,  if (  -oo  =  +oo ,  0 ,  -oo ) ,  if (  -oo  =  +oo ,  +oo ,  if (  -oo  =  -oo ,  -oo ,  ( A  +  -oo ) ) ) )  =  if ( A  =  -oo ,  -oo , 
-oo )
17 ifid 3773 . . . 4  |-  if ( A  =  -oo ,  -oo ,  -oo )  = 
-oo
1816, 17eqtri 2458 . . 3  |-  if ( A  =  -oo ,  if (  -oo  =  +oo ,  0 ,  -oo ) ,  if (  -oo  =  +oo ,  +oo ,  if (  -oo  =  -oo ,  -oo ,  ( A  +  -oo ) ) ) )  =  -oo
194, 18syl6eq 2486 . 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 )
203, 19sylan9eq 2490 1  |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  ->  ( A + e  -oo )  =  -oo )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   ifcif 3741  (class class class)co 6083   0cc0 8992    + caddc 8995    +oocpnf 9119    -oocmnf 9120   RR*cxr 9121   + ecxad 10710
This theorem is referenced by:  xaddnepnf  10823  xaddcom  10826  xnegdi  10829  xleadd1a  10834  xsubge0  10842  xlesubadd  10844  xadddilem  10875  xblss2ps  18433  xblss2  18434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-mulcl 9054  ax-i2m1 9060
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-pnf 9124  df-mnf 9125  df-xr 9126  df-xadd 10713
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