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Theorem rexadd 10751
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 9064 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
2 rexr 9064 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
3 xaddval 10742 . . 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 464 . 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 9066 . . . . 5  |-  ( A  e.  RR  ->  A  =/=  +oo )
6 ifnefalse 3691 . . . . 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 9067 . . . . 5  |-  ( A  e.  RR  ->  A  =/=  -oo )
9 ifnefalse 3691 . . . . 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 2420 . . 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 9066 . . . . 5  |-  ( B  e.  RR  ->  B  =/=  +oo )
13 ifnefalse 3691 . . . . 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 9067 . . . . 5  |-  ( B  e.  RR  ->  B  =/=  -oo )
16 ifnefalse 3691 . . . . 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 2420 . . 3  |-  ( B  e.  RR  ->  if ( B  =  +oo , 
+oo ,  if ( B  =  -oo ,  -oo ,  ( A  +  B
) ) )  =  ( A  +  B
) )
1911, 18sylan9eq 2440 . 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 2420 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A + e B )  =  ( A  +  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551   ifcif 3683  (class class class)co 6021   RRcr 8923   0cc0 8924    + caddc 8927    +oocpnf 9051    -oocmnf 9052   RR*cxr 9053   + ecxad 10641
This theorem is referenced by:  rexsub  10752  xaddnemnf  10753  xaddnepnf  10754  xnegid  10755  xaddcom  10757  xaddid1  10758  xnegdi  10760  xaddass  10761  xpncan  10763  xleadd1a  10765  xadddilem  10806  x2times  10811  hashunx  11588  isxmet2d  18267  ismet2  18273  mettri2  18281  prdsxmetlem  18307  bl2in  18332  xmeter  18354  methaus  18441  metustexhalf  18477  metdcnlem  18739  metnrmlem3  18763  iscau3  19103  vdgrfival  21517  vdgrf  21518  vdgrfif  21519  vdgr0  21520  vdgr1d  21523  vdgr1b  21524  vdgr1a  21526  xlt2addrd  23961  xrsmulgzz  24034  xrge0iifhom  24128  esumfsupre  24258  esumpfinvallem  24261  probun  24457  cntotbnd  26197  heiborlem6  26217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-mulcl 8986  ax-i2m1 8992
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-xadd 10644
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