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Theorem xaddpnf1 9750
Description: Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xaddpnf1  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )

Proof of Theorem xaddpnf1
StepHypRef Expression
1 pnfxr 7930 . . . 4  |- +oo  e.  RR*
2 xaddval 9749 . . . 4  |-  ( ( 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 422 . . 3  |-  ( 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 ) ) ) ) ) )
43adantr 274 . 2  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( 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 ) ) ) ) ) )
5 pnfnemnf 7932 . . . . 5  |- +oo  =/= -oo
6 ifnefalse 3516 . . . . 5  |-  ( +oo  =/= -oo  ->  if ( +oo  = -oo ,  0 , +oo )  = +oo )
75, 6mp1i 10 . . . 4  |-  ( A  =/= -oo  ->  if ( +oo  = -oo , 
0 , +oo )  = +oo )
8 ifnefalse 3516 . . . . 5  |-  ( A  =/= -oo  ->  if ( A  = -oo ,  if ( +oo  = +oo ,  0 , -oo ) ,  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo ,  ( A  + +oo )
) ) )  =  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo ,  ( A  + +oo ) ) ) )
9 eqid 2157 . . . . . 6  |- +oo  = +oo
109iftruei 3511 . . . . 5  |-  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo , 
( A  + +oo ) ) )  = +oo
118, 10eqtrdi 2206 . . . 4  |-  ( A  =/= -oo  ->  if ( A  = -oo ,  if ( +oo  = +oo ,  0 , -oo ) ,  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo ,  ( A  + +oo )
) ) )  = +oo )
127, 11ifeq12d 3524 . . 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 , +oo , +oo ) )
13 xrpnfdc 9746 . . . 4  |-  ( A  e.  RR*  -> DECID  A  = +oo )
14 ifiddc 3538 . . . 4  |-  (DECID  A  = +oo  ->  if ( A  = +oo , +oo , +oo )  = +oo )
1513, 14syl 14 . . 3  |-  ( A  e.  RR*  ->  if ( A  = +oo , +oo , +oo )  = +oo )
1612, 15sylan9eqr 2212 . 2  |-  ( ( A  e.  RR*  /\  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 )
174, 16eqtrd 2190 1  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103  DECID wdc 820    = wceq 1335    e. wcel 2128    =/= wne 2327   ifcif 3505  (class class class)co 5824   0cc0 7732    + caddc 7735   +oocpnf 7909   -oocmnf 7910   RR*cxr 7911   +ecxad 9677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1re 7826  ax-addrcl 7829  ax-rnegex 7841
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-pnf 7914  df-mnf 7915  df-xr 7916  df-xadd 9680
This theorem is referenced by:  xaddnemnf  9761  xaddcom  9765  xnn0xadd0  9771  xnegdi  9772  xaddass  9773  xleadd1a  9777  xlt2add  9784  xsubge0  9785  xposdif  9786  xlesubadd  9787  xrbdtri  11173  isxmet2d  12748
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