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Theorem xaddval 9775
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 7939 . . . . . 6  |-  0  e.  RR*
21a1i 9 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  0  e.  RR* )
3 pnfxr 7945 . . . . . 6  |- +oo  e.  RR*
43a1i 9 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> +oo  e.  RR* )
5 xrmnfdc 9773 . . . . . 6  |-  ( B  e.  RR*  -> DECID  B  = -oo )
65adantl 275 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> DECID  B  = -oo )
72, 4, 6ifcldcd 3553 . . . 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 7949 . . . . . . 7  |- -oo  e.  RR*
109a1i 9 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> -oo  e.  RR* )
11 xrpnfdc 9772 . . . . . . 7  |-  ( B  e.  RR*  -> DECID  B  = +oo )
1211adantl 275 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> DECID  B  = +oo )
132, 10, 12ifcldcd 3553 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  if ( B  = +oo ,  0 , -oo )  e.  RR* )
1413ad2antrr 480 . . . 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 532 . . . . . . . . 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 486 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  A  e.  RR* )
20 simpllr 524 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  -.  A  = -oo )
2120neqned 2341 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  A  =/= -oo )
22 xrnemnf 9707 . . . . . . . . . . 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 409 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  ( A  e.  RR  \/  A  = +oo ) )
2517, 24ecased 1338 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  A  e.  RR )
26 simplr 520 . . . . . . . . 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 486 . . . . . . . . . 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 2341 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  B  =/= -oo )
31 xrnemnf 9707 . . . . . . . . . . 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 409 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  ( B  e.  RR  \/  B  = +oo ) )
3426, 33ecased 1338 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  B  e.  RR )
3525, 34readdcld 7922 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  ( A  +  B
)  e.  RR )
3635rexrd 7942 . . . . . 6  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  ( A  +  B
)  e.  RR* )
376ad3antrrr 484 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  -> DECID  B  = -oo )
3816, 36, 37ifcldadc 3547 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  /\  -.  B  = +oo )  ->  if ( B  = -oo , -oo ,  ( A  +  B ) )  e.  RR* )
3912ad2antrr 480 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  -> DECID  B  = +oo )
4015, 38, 39ifcldadc 3547 . . . 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 9773 . . . . 5  |-  ( A  e.  RR*  -> DECID  A  = -oo )
4241ad2antrr 480 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  = +oo )  -> DECID 
A  = -oo )
4314, 40, 42ifcldadc 3547 . . 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 9772 . . . 4  |-  ( A  e.  RR*  -> DECID  A  = +oo )
4544adantr 274 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> DECID  A  = +oo )
468, 43, 45ifcldadc 3547 . 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 2173 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  = +oo  <->  A  = +oo ) )
49 simpr 109 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
5049eqeq1d 2173 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  = -oo  <->  B  = -oo ) )
5150ifbid 3539 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  = -oo ,  0 , +oo )  =  if ( B  = -oo ,  0 , +oo ) )
5247eqeq1d 2173 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  = -oo  <->  A  = -oo ) )
5349eqeq1d 2173 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  = +oo  <->  B  = +oo ) )
5453ifbid 3539 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  = +oo ,  0 , -oo )  =  if ( B  = +oo ,  0 , -oo ) )
55 oveq12 5848 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  +  y )  =  ( A  +  B ) )
5650, 55ifbieq2d 3542 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  = -oo , -oo , 
( x  +  y ) )  =  if ( B  = -oo , -oo ,  ( A  +  B ) ) )
5753, 56ifbieq2d 3542 . . . . 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 3544 . . . 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 3544 . . 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 9703 . . 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 5965 . 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 1327 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 698  DECID wdc 824    = wceq 1342    e. wcel 2135    =/= wne 2334   ifcif 3518  (class class class)co 5839   RRcr 7746   0cc0 7747    + caddc 7750   +oocpnf 7924   -oocmnf 7925   RR*cxr 7926   +ecxad 9700
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-pow 4150  ax-pr 4184  ax-un 4408  ax-setind 4511  ax-cnex 7838  ax-resscn 7839  ax-1re 7841  ax-addrcl 7844  ax-rnegex 7856
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2726  df-sbc 2950  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-if 3519  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-uni 3787  df-br 3980  df-opab 4041  df-id 4268  df-xp 4607  df-rel 4608  df-cnv 4609  df-co 4610  df-dm 4611  df-iota 5150  df-fun 5187  df-fv 5193  df-ov 5842  df-oprab 5843  df-mpo 5844  df-pnf 7929  df-mnf 7930  df-xr 7931  df-xadd 9703
This theorem is referenced by:  xaddpnf1  9776  xaddpnf2  9777  xaddmnf1  9778  xaddmnf2  9779  pnfaddmnf  9780  mnfaddpnf  9781  rexadd  9782
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