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Theorem rexadd 10575
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 8893 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
2 rexr 8893 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
3 xaddval 10566 . . 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 463 . 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 8895 . . . . 5  |-  ( A  e.  RR  ->  A  =/=  +oo )
6 ifnefalse 3586 . . . . 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 15 . . . 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 8896 . . . . 5  |-  ( A  e.  RR  ->  A  =/=  -oo )
9 ifnefalse 3586 . . . . 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 15 . . . 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 2328 . . 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 8895 . . . . 5  |-  ( B  e.  RR  ->  B  =/=  +oo )
13 ifnefalse 3586 . . . . 5  |-  ( B  =/=  +oo  ->  if ( B  =  +oo ,  +oo ,  if ( B  =  -oo ,  -oo ,  ( A  +  B
) ) )  =  if ( B  = 
-oo ,  -oo ,  ( A  +  B ) ) )
1412, 13syl 15 . . . 4  |-  ( B  e.  RR  ->  if ( B  =  +oo , 
+oo ,  if ( B  =  -oo ,  -oo ,  ( A  +  B
) ) )  =  if ( B  = 
-oo ,  -oo ,  ( A  +  B ) ) )
15 renemnf 8896 . . . . 5  |-  ( B  e.  RR  ->  B  =/=  -oo )
16 ifnefalse 3586 . . . . 5  |-  ( B  =/=  -oo  ->  if ( B  =  -oo ,  -oo ,  ( A  +  B ) )  =  ( A  +  B
) )
1715, 16syl 15 . . . 4  |-  ( B  e.  RR  ->  if ( B  =  -oo , 
-oo ,  ( A  +  B ) )  =  ( A  +  B
) )
1814, 17eqtrd 2328 . . 3  |-  ( B  e.  RR  ->  if ( B  =  +oo , 
+oo ,  if ( B  =  -oo ,  -oo ,  ( A  +  B
) ) )  =  ( A  +  B
) )
1911, 18sylan9eq 2348 . 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 2328 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A + e B )  =  ( A  +  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   ifcif 3578  (class class class)co 5874   RRcr 8752   0cc0 8753    + caddc 8756    +oocpnf 8880    -oocmnf 8881   RR*cxr 8882   + ecxad 10466
This theorem is referenced by:  rexsub  10576  xaddnemnf  10577  xaddnepnf  10578  xnegid  10579  xaddcom  10581  xaddid1  10582  xnegdi  10584  xaddass  10585  xpncan  10587  xleadd1a  10589  xadddilem  10630  x2times  10635  isxmet2d  17908  ismet2  17914  mettri2  17922  prdsxmetlem  17948  bl2in  17973  xmeter  17995  methaus  18082  metdcnlem  18357  metnrmlem3  18381  iscau3  18720  xlt2addrd  23268  xrsmulgzz  23322  xrge0iifhom  23334  esumpfinvallem  23457  probun  23637  cntotbnd  26623  heiborlem6  26643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-mulcl 8815  ax-i2m1 8821
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-xadd 10469
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