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Theorem xaddval 9635
Description: Value of the extended real addition operation. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xaddval  |-  ( ( 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
) ) ) ) ) )

Proof of Theorem xaddval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0xr 7819 . . . . . 6  |-  0  e.  RR*
21a1i 9 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  0  e.  RR* )
3 pnfxr 7825 . . . . . 6  |- +oo  e.  RR*
43a1i 9 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> +oo  e.  RR* )
5 xrmnfdc 9633 . . . . . 6  |-  ( B  e.  RR*  -> DECID  B  = -oo )
65adantl 275 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> DECID  B  = -oo )
72, 4, 6ifcldcd 3507 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  if ( B  = -oo ,  0 , +oo )  e.  RR* )
87adantr 274 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  = +oo )  ->  if ( B  = -oo ,  0 , +oo )  e. 
RR* )
9 mnfxr 7829 . . . . . . 7  |- -oo  e.  RR*
109a1i 9 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> -oo  e.  RR* )
11 xrpnfdc 9632 . . . . . . 7  |-  ( B  e.  RR*  -> DECID  B  = +oo )
1211adantl 275 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> DECID  B  = +oo )
132, 10, 12ifcldcd 3507 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  if ( B  = +oo ,  0 , -oo )  e.  RR* )
1413ad2antrr 479 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  A  = -oo )  ->  if ( B  = +oo ,  0 , -oo )  e.  RR* )
153a1i 9 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  B  = +oo )  -> +oo  e.  RR* )
169a1i 9 . . . . . 6  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  B  = -oo )  -> -oo  e.  RR* )
17 simp-4r 531 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  -.  A  = +oo )
18 simpl 108 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  e.  RR* )
1918ad4antr 485 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  A  e.  RR* )
20 simpllr 523 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  -.  A  = -oo )
2120neqned 2315 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  A  =/= -oo )
22 xrnemnf 9571 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  <->  ( A  e.  RR  \/  A  = +oo ) )
2322biimpi 119 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A  e.  RR  \/  A  = +oo )
)
2419, 21, 23syl2anc 408 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  ( A  e.  RR  \/  A  = +oo ) )
2517, 24ecased 1327 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  A  e.  RR )
26 simplr 519 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  -.  B  = +oo )
27 simpr 109 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  e.  RR* )
2827ad4antr 485 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  B  e.  RR* )
29 simpr 109 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  -.  B  = -oo )
3029neqned 2315 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  B  =/= -oo )
31 xrnemnf 9571 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  <->  ( B  e.  RR  \/  B  = +oo ) )
3231biimpi 119 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( B  e.  RR  \/  B  = +oo )
)
3328, 30, 32syl2anc 408 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  ( B  e.  RR  \/  B  = +oo ) )
3426, 33ecased 1327 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  B  e.  RR )
3525, 34readdcld 7802 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  ( A  +  B
)  e.  RR )
3635rexrd 7822 . . . . . 6  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  ( A  +  B
)  e.  RR* )
376ad3antrrr 483 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  -> DECID  B  = -oo )
3816, 36, 37ifcldadc 3501 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  ->  if ( B  = -oo , -oo ,  ( A  +  B ) )  e.  RR* )
3912ad2antrr 479 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  -> DECID  B  = +oo )
4015, 38, 39ifcldadc 3501 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) )  e.  RR* )
41 xrmnfdc 9633 . . . . 5  |-  ( A  e.  RR*  -> DECID  A  = -oo )
4241ad2antrr 479 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  -> DECID 
A  = -oo )
4314, 40, 42ifcldadc 3501 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  ->  if ( A  = -oo ,  if ( B  = +oo ,  0 , -oo ) ,  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo , 
( A  +  B
) ) ) )  e.  RR* )
44 xrpnfdc 9632 . . . 4  |-  ( A  e.  RR*  -> DECID  A  = +oo )
4544adantr 274 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> DECID  A  = +oo )
468, 43, 45ifcldadc 3501 . 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 ) ) ) ) )  e.  RR* )
47 simpl 108 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  x  =  A )
4847eqeq1d 2148 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  = +oo  <->  A  = +oo ) )
49 simpr 109 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
5049eqeq1d 2148 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  = -oo  <->  B  = -oo ) )
5150ifbid 3493 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  = -oo ,  0 , +oo )  =  if ( B  = -oo ,  0 , +oo ) )
5247eqeq1d 2148 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  = -oo  <->  A  = -oo ) )
5349eqeq1d 2148 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  = +oo  <->  B  = +oo ) )
5453ifbid 3493 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  = +oo ,  0 , -oo )  =  if ( B  = +oo ,  0 , -oo ) )
55 oveq12 5783 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  +  y )  =  ( A  +  B ) )
5650, 55ifbieq2d 3496 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  = -oo , -oo , 
( x  +  y ) )  =  if ( B  = -oo , -oo ,  ( A  +  B ) ) )
5753, 56ifbieq2d 3496 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) )  =  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) ) )
5852, 54, 57ifbieq12d 3498 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( x  = -oo ,  if ( y  = +oo , 
0 , -oo ) ,  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) ) )  =  if ( A  = -oo ,  if ( B  = +oo ,  0 , -oo ) ,  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) ) ) )
5948, 51, 58ifbieq12d 3498 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( x  = +oo ,  if ( y  = -oo , 
0 , +oo ) ,  if ( x  = -oo ,  if ( y  = +oo , 
0 , -oo ) ,  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) ) ) )  =  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 ) ) ) ) ) )
60 df-xadd 9567 . . 3  |-  +e 
=  ( x  e. 
RR* ,  y  e.  RR*  |->  if ( x  = +oo ,  if ( y  = -oo , 
0 , +oo ) ,  if ( x  = -oo ,  if ( y  = +oo , 
0 , -oo ) ,  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) ) ) ) )
6159, 60ovmpoga 5900 . 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
) ) ) ) )  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 ) ) ) ) ) )
6246, 61mpd3an3 1316 1  |-  ( ( 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
) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 697  DECID wdc 819    = wceq 1331    e. wcel 1480    =/= wne 2308   ifcif 3474  (class class class)co 5774   RRcr 7626   0cc0 7627    + caddc 7630   +oocpnf 7804   -oocmnf 7805   RR*cxr 7806   +ecxad 9564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1re 7721  ax-addrcl 7724  ax-rnegex 7736
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7809  df-mnf 7810  df-xr 7811  df-xadd 9567
This theorem is referenced by:  xaddpnf1  9636  xaddpnf2  9637  xaddmnf1  9638  xaddmnf2  9639  pnfaddmnf  9640  mnfaddpnf  9641  rexadd  9642
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