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Theorem xaddpnf1 11480
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 11376 . . 3  |- +oo  e.  RR*
2 xaddval 11477 . . 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 11381 . . . . 5  |- +oo  =/= -oo
5 ifnefalse 3899 . . . . 5  |-  ( +oo  =/= -oo  ->  if ( +oo  = -oo ,  0 , +oo )  = +oo )
64, 5mp1i 13 . . . 4  |-  ( A  =/= -oo  ->  if ( +oo  = -oo , 
0 , +oo )  = +oo )
7 ifnefalse 3899 . . . . 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 2404 . . . . . 6  |- +oo  = +oo
98iftruei 3894 . . . . 5  |-  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo , 
( A  + +oo ) ) )  = +oo
107, 9syl6eq 2461 . . . 4  |-  ( A  =/= -oo  ->  if ( A  = -oo ,  if ( +oo  = +oo ,  0 , -oo ) ,  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo ,  ( A  + +oo )
) ) )  = +oo )
116, 10ifeq12d 3907 . . 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 3924 . . 3  |-  if ( A  = +oo , +oo , +oo )  = +oo
1311, 12syl6eq 2461 . 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 2465 1  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1407    e. wcel 1844    =/= wne 2600   ifcif 3887  (class class class)co 6280   0cc0 9524    + caddc 9527   +oocpnf 9657   -oocmnf 9658   RR*cxr 9659   +ecxad 11371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-mulcl 9586  ax-i2m1 9592
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-iota 5535  df-fun 5573  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-pnf 9662  df-mnf 9663  df-xr 9664  df-xadd 11374
This theorem is referenced by:  xaddnemnf  11488  xaddcom  11492  xnegdi  11495  xaddass  11496  xleadd1a  11500  xlt2add  11507  xsubge0  11508  xlesubadd  11510  xadddilem  11541  xrsdsreclblem  18786  isxmet2d  21124  xrge0iifhom  28385  esumpr2  28527  hasheuni  28545  carsgclctunlem2  28780
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