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Theorem xaddval 11297
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 simpl 457 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  x  =  A )
21eqeq1d 2453 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  = +oo  <->  A  = +oo ) )
3 simpr 461 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
43eqeq1d 2453 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  = -oo  <->  B  = -oo ) )
54ifbid 3912 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  = -oo ,  0 , +oo )  =  if ( B  = -oo ,  0 , +oo ) )
61eqeq1d 2453 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  = -oo  <->  A  = -oo ) )
73eqeq1d 2453 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  = +oo  <->  B  = +oo ) )
87ifbid 3912 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  = +oo ,  0 , -oo )  =  if ( B  = +oo ,  0 , -oo ) )
9 oveq12 6202 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  +  y )  =  ( A  +  B ) )
104, 9ifbieq2d 3915 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  = -oo , -oo , 
( x  +  y ) )  =  if ( B  = -oo , -oo ,  ( A  +  B ) ) )
117, 10ifbieq2d 3915 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) )  =  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) ) )
126, 8, 11ifbieq12d 3917 . . 3  |-  ( ( 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 ) ) ) ) )
132, 5, 12ifbieq12d 3917 . 2  |-  ( ( 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 ) ) ) ) ) )
14 df-xadd 11194 . 2  |-  +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 ) ) ) ) ) )
15 c0ex 9484 . . . 4  |-  0  e.  _V
16 pnfex 11197 . . . 4  |- +oo  e.  _V
1715, 16ifex 3959 . . 3  |-  if ( B  = -oo , 
0 , +oo )  e.  _V
18 mnfxr 11198 . . . . . 6  |- -oo  e.  RR*
1918elexi 3081 . . . . 5  |- -oo  e.  _V
2015, 19ifex 3959 . . . 4  |-  if ( B  = +oo , 
0 , -oo )  e.  _V
21 ovex 6218 . . . . . 6  |-  ( A  +  B )  e. 
_V
2219, 21ifex 3959 . . . . 5  |-  if ( B  = -oo , -oo ,  ( A  +  B ) )  e. 
_V
2316, 22ifex 3959 . . . 4  |-  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) )  e.  _V
2420, 23ifex 3959 . . 3  |-  if ( A  = -oo ,  if ( B  = +oo ,  0 , -oo ) ,  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo , 
( A  +  B
) ) ) )  e.  _V
2517, 24ifex 3959 . 2  |-  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.  _V
2613, 14, 25ovmpt2a 6324 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   ifcif 3892  (class class class)co 6193   0cc0 9386    + caddc 9389   +oocpnf 9519   -oocmnf 9520   RR*cxr 9521   +ecxad 11191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-mulcl 9448  ax-i2m1 9454
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-iota 5482  df-fun 5521  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-pnf 9524  df-mnf 9525  df-xr 9526  df-xadd 11194
This theorem is referenced by:  xaddpnf1  11300  xaddpnf2  11301  xaddmnf1  11302  xaddmnf2  11303  pnfaddmnf  11304  mnfaddpnf  11305  rexadd  11306
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