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Theorem rexadd 9528
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 7735 . . 3 |- ( A e. RR -> A e. RR* )
2 rexr 7735 . . 3 |- ( B e. RR -> B e. RR* )
3 xaddval 9521 . . 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 285 . 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 7737 . . . . 5 |- ( A e. RR -> A =/= +oo )
6 ifnefalse 3451 . . . . 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 14 . . . 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 7738 . . . . 5 |- ( A e. RR -> A =/= -oo )
9 ifnefalse 3451 . . . . 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 14 . . . 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 2147 . . 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 7737 . . . . 5 |- ( B e. RR -> B =/= +oo )
13 ifnefalse 3451 . . . . 5 |- ( B =/= +oo -> if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) = if ( B = -oo , -oo , ( A + B ) ) )
1412, 13syl 14 . . . 4 |- ( B e. RR -> if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) = if ( B = -oo , -oo , ( A + B ) ) )
15 renemnf 7738 . . . . 5 |- ( B e. RR -> B =/= -oo )
16 ifnefalse 3451 . . . . 5 |- ( B =/= -oo -> if ( B = -oo , -oo , ( A + B ) ) = ( A + B ) )
1715, 16syl 14 . . . 4 |- ( B e. RR -> if ( B = -oo , -oo , ( A + B ) ) = ( A + B ) )
1814, 17eqtrd 2147 . . 3 |- ( B e. RR -> if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) = ( A + B ) )
1911, 18sylan9eq 2167 . 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 2147 1 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   = wceq 1314   e. wcel 1463   =/= wne 2282   ifcif 3440  (class class class)co 5728   RRcr 7546   0cc0 7547   + caddc 7550   +oocpnf 7721   -oocmnf 7722   RR*cxr 7723   +ecxad 9450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1re 7639  ax-addrcl 7642  ax-rnegex 7654
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-iota 5046  df-fun 5083  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-pnf 7726  df-mnf 7727  df-xr 7728  df-xadd 9453
This theorem is referenced by:  rexsub  9529  rexaddd  9530  xaddnemnf  9533  xaddnepnf  9534  xnegid  9535  xaddcom  9537  xaddid1  9538  xnn0xadd0  9543  xnegdi  9544  xaddass  9545  xltadd1  9552  isxmet2d  12337  mettri2  12351  bl2in  12392  xmeter  12425
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