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Theorem rexadd 9628
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 7804 . . 3 |- ( A e. RR -> A e. RR* )
2 rexr 7804 . . 3 |- ( B e. RR -> B e. RR* )
3 xaddval 9621 . . 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 287 . 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 7806 . . . . 5 |- ( A e. RR -> A =/= +oo )
6 ifnefalse 3480 . . . . 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 7807 . . . . 5 |- ( A e. RR -> A =/= -oo )
9 ifnefalse 3480 . . . . 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 2170 . . 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 7806 . . . . 5 |- ( B e. RR -> B =/= +oo )
13 ifnefalse 3480 . . . . 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 7807 . . . . 5 |- ( B e. RR -> B =/= -oo )
16 ifnefalse 3480 . . . . 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 2170 . . 3 |- ( B e. RR -> if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) = ( A + B ) )
1911, 18sylan9eq 2190 . 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 2170 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 1331   e. wcel 1480   =/= wne 2306   ifcif 3469  (class class class)co 5767   RRcr 7612   0cc0 7613   + caddc 7616   +oocpnf 7790   -oocmnf 7791   RR*cxr 7792   +ecxad 9550
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 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1re 7707  ax-addrcl 7710  ax-rnegex 7722
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 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-xadd 9553
This theorem is referenced by:  rexsub  9629  rexaddd  9630  xaddnemnf  9633  xaddnepnf  9634  xnegid  9635  xaddcom  9637  xaddid1  9638  xnn0xadd0  9643  xnegdi  9644  xaddass  9645  xltadd1  9652  isxmet2d  12506  mettri2  12520  bl2in  12561  xmeter  12594
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