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Theorem xaddpnf1 11421
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 11317 . . 3  |- +oo  e.  RR*
2 xaddval 11418 . . 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 671 . 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 pnfnemnf 11322 . . . . 5  |- +oo  =/= -oo
5 ifnefalse 3951 . . . . 5  |-  ( +oo  =/= -oo  ->  if ( +oo  = -oo ,  0 , +oo )  = +oo )
64, 5mp1i 12 . . . 4  |-  ( A  =/= -oo  ->  if ( +oo  = -oo , 
0 , +oo )  = +oo )
7 ifnefalse 3951 . . . . 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 ) ) ) )
8 eqid 2467 . . . . . 6  |- +oo  = +oo
98iftruei 3946 . . . . 5  |-  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo , 
( A  + +oo ) ) )  = +oo
107, 9syl6eq 2524 . . . 4  |-  ( A  =/= -oo  ->  if ( A  = -oo ,  if ( +oo  = +oo ,  0 , -oo ) ,  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo ,  ( A  + +oo )
) ) )  = +oo )
116, 10ifeq12d 3959 . . 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 ) )
12 ifid 3976 . . 3  |-  if ( A  = +oo , +oo , +oo )  = +oo
1311, 12syl6eq 2524 . 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 )
143, 13sylan9eq 2528 1  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   ifcif 3939  (class class class)co 6282   0cc0 9488    + caddc 9491   +oocpnf 9621   -oocmnf 9622   RR*cxr 9623   +ecxad 11312
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 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-mulcl 9550  ax-i2m1 9556
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-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-pnf 9626  df-mnf 9627  df-xr 9628  df-xadd 11315
This theorem is referenced by:  xaddnemnf  11429  xaddcom  11433  xnegdi  11436  xaddass  11437  xleadd1a  11441  xlt2add  11448  xsubge0  11449  xlesubadd  11451  xadddilem  11482  xrsdsreclblem  18232  isxmet2d  20565  xrge0iifhom  27555  esumpr2  27714  hasheuni  27731
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