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Theorem xaddval 11185
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 2446 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  = +oo  <->  A  = +oo ) )
3 simpr 461 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
43eqeq1d 2446 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  = -oo  <->  B  = -oo ) )
54ifbid 3806 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  = -oo ,  0 , +oo )  =  if ( B  = -oo ,  0 , +oo ) )
61eqeq1d 2446 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  = -oo  <->  A  = -oo ) )
73eqeq1d 2446 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  = +oo  <->  B  = +oo ) )
87ifbid 3806 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  = +oo ,  0 , -oo )  =  if ( B  = +oo ,  0 , -oo ) )
9 oveq12 6095 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  +  y )  =  ( A  +  B ) )
104, 9ifbieq2d 3809 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  = -oo , -oo , 
( x  +  y ) )  =  if ( B  = -oo , -oo ,  ( A  +  B ) ) )
117, 10ifbieq2d 3809 . . . 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 3811 . . 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 3811 . 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 11082 . 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 9372 . . . 4  |-  0  e.  _V
16 pnfex 11085 . . . 4  |- +oo  e.  _V
1715, 16ifex 3853 . . 3  |-  if ( B  = -oo , 
0 , +oo )  e.  _V
18 mnfxr 11086 . . . . . 6  |- -oo  e.  RR*
1918elexi 2977 . . . . 5  |- -oo  e.  _V
2015, 19ifex 3853 . . . 4  |-  if ( B  = +oo , 
0 , -oo )  e.  _V
21 ovex 6111 . . . . . 6  |-  ( A  +  B )  e. 
_V
2219, 21ifex 3853 . . . . 5  |-  if ( B  = -oo , -oo ,  ( A  +  B ) )  e. 
_V
2316, 22ifex 3853 . . . 4  |-  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) )  e.  _V
2420, 23ifex 3853 . . 3  |-  if ( A  = -oo ,  if ( B  = +oo ,  0 , -oo ) ,  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo , 
( A  +  B
) ) ) )  e.  _V
2517, 24ifex 3853 . 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 6216 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 1369    e. wcel 1756   ifcif 3786  (class class class)co 6086   0cc0 9274    + caddc 9277   +oocpnf 9407   -oocmnf 9408   RR*cxr 9409   +ecxad 11079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-mulcl 9336  ax-i2m1 9342
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-iota 5376  df-fun 5415  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-pnf 9412  df-mnf 9413  df-xr 9414  df-xadd 11082
This theorem is referenced by:  xaddpnf1  11188  xaddpnf2  11189  xaddmnf1  11190  xaddmnf2  11191  pnfaddmnf  11192  mnfaddpnf  11193  rexadd  11194
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