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Theorem rexadd 11198
Description: The extended real addition operation when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
rexadd  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( A  +  B ) )

Proof of Theorem rexadd
StepHypRef Expression
1 rexr 9425 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
2 rexr 9425 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
3 xaddval 11189 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  =  if ( A  = +oo ,  if ( B  = -oo ,  0 , +oo ) ,  if ( A  = -oo ,  if ( B  = +oo ,  0 , -oo ) ,  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo , 
( A  +  B
) ) ) ) ) )
41, 2, 3syl2an 474 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  if ( A  = +oo ,  if ( B  = -oo ,  0 , +oo ) ,  if ( A  = -oo ,  if ( B  = +oo ,  0 , -oo ) ,  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) ) ) ) )
5 renepnf 9427 . . . . 5  |-  ( A  e.  RR  ->  A  =/= +oo )
6 ifnefalse 3798 . . . . 5  |-  ( A  =/= +oo  ->  if ( A  = +oo ,  if ( B  = -oo ,  0 , +oo ) ,  if ( A  = -oo ,  if ( B  = +oo ,  0 , -oo ) ,  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo , 
( A  +  B
) ) ) ) )  =  if ( A  = -oo ,  if ( B  = +oo ,  0 , -oo ) ,  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo , 
( A  +  B
) ) ) ) )
75, 6syl 16 . . . 4  |-  ( A  e.  RR  ->  if ( A  = +oo ,  if ( B  = -oo ,  0 , +oo ) ,  if ( A  = -oo ,  if ( B  = +oo ,  0 , -oo ) ,  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) ) ) )  =  if ( A  = -oo ,  if ( B  = +oo ,  0 , -oo ) ,  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) ) ) )
8 renemnf 9428 . . . . 5  |-  ( A  e.  RR  ->  A  =/= -oo )
9 ifnefalse 3798 . . . . 5  |-  ( A  =/= -oo  ->  if ( A  = -oo ,  if ( B  = +oo ,  0 , -oo ) ,  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo , 
( A  +  B
) ) ) )  =  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo , 
( A  +  B
) ) ) )
108, 9syl 16 . . . 4  |-  ( A  e.  RR  ->  if ( A  = -oo ,  if ( B  = +oo ,  0 , -oo ) ,  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) ) )  =  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) ) )
117, 10eqtrd 2473 . . 3  |-  ( A  e.  RR  ->  if ( A  = +oo ,  if ( B  = -oo ,  0 , +oo ) ,  if ( A  = -oo ,  if ( B  = +oo ,  0 , -oo ) ,  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) ) ) )  =  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) ) )
12 renepnf 9427 . . . . 5  |-  ( B  e.  RR  ->  B  =/= +oo )
13 ifnefalse 3798 . . . . 5  |-  ( B  =/= +oo  ->  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) )  =  if ( B  = -oo , -oo ,  ( A  +  B ) ) )
1412, 13syl 16 . . . 4  |-  ( B  e.  RR  ->  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) )  =  if ( B  = -oo , -oo ,  ( A  +  B ) ) )
15 renemnf 9428 . . . . 5  |-  ( B  e.  RR  ->  B  =/= -oo )
16 ifnefalse 3798 . . . . 5  |-  ( B  =/= -oo  ->  if ( B  = -oo , -oo ,  ( A  +  B ) )  =  ( A  +  B
) )
1715, 16syl 16 . . . 4  |-  ( B  e.  RR  ->  if ( B  = -oo , -oo ,  ( A  +  B ) )  =  ( A  +  B ) )
1814, 17eqtrd 2473 . . 3  |-  ( B  e.  RR  ->  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) )  =  ( A  +  B ) )
1911, 18sylan9eq 2493 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  if ( A  = +oo ,  if ( B  = -oo , 
0 , +oo ) ,  if ( A  = -oo ,  if ( B  = +oo , 
0 , -oo ) ,  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) ) ) )  =  ( A  +  B
) )
204, 19eqtrd 2473 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( A  +  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604   ifcif 3788  (class class class)co 6090   RRcr 9277   0cc0 9278    + caddc 9281   +oocpnf 9411   -oocmnf 9412   RR*cxr 9413   +ecxad 11083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-mulcl 9340  ax-i2m1 9346
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-xadd 11086
This theorem is referenced by:  rexsub  11199  xaddnemnf  11200  xaddnepnf  11201  xnegid  11202  xaddcom  11204  xaddid1  11205  xnegdi  11207  xaddass  11208  xpncan  11210  xleadd1a  11212  xadddilem  11253  x2times  11258  hashunx  12145  isxmet2d  19861  ismet2  19867  mettri2  19875  prdsxmetlem  19902  bl2in  19934  xblss2ps  19935  xmeter  19967  methaus  20054  metustexhalfOLD  20097  metustexhalf  20098  metdcnlem  20372  metnrmlem3  20396  iscau3  20748  vdgrfival  23502  vdgrf  23503  vdgrfif  23504  vdgr0  23505  vdgr1d  23508  vdgr1b  23509  vdgr1a  23511  xlt2addrd  25986  xrsmulgzz  26072  xrge0slmod  26248  xrge0iifhom  26303  esumfsupre  26456  esumpfinvallem  26459  probun  26732  heicant  28351  cntotbnd  28620  heiborlem6  28640
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