ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rexadd Unicode version
Theorem rexadd 10254
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 8371 . . 3 |- ( A e. RR -> A e. RR* )
2 rexr 8371 . . 3 |- ( B e. RR -> B e. RR* )
3 xaddval 10247 . . 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 289 . 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 8373 . . . . 5 |- ( A e. RR -> A =/= +oo )
6 ifnefalse 3651 . . . . 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 8374 . . . . 5 |- ( A e. RR -> A =/= -oo )
9 ifnefalse 3651 . . . . 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 2271 . . 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 8373 . . . . 5 |- ( B e. RR -> B =/= +oo )
13 ifnefalse 3651 . . . . 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 8374 . . . . 5 |- ( B e. RR -> B =/= -oo )
16 ifnefalse 3651 . . . . 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 2271 . . 3 |- ( B e. RR -> if ( B = +oo , +oo , if ( B = -oo , -oo , ( A + B ) ) ) = ( A + B ) )
1911, 18sylan9eq 2291 . 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 2271 1 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   =/= wne 2420   ifcif 3638  (class class class)co 6085   RRcr 8178   0cc0 8179   + caddc 8182   +oocpnf 8357   -oocmnf 8358   RR*cxr 8359   +ecxad 10172
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276  ax-rnegex 8288
This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-xadd 10175
This theorem is used by:  rexsub  10255  rexaddd  10256  xaddnemnf  10259  xaddnepnf  10260  xnegid  10261  xaddcom  10263  xaddid1  10264  xnn0xadd0  10269  xnegdi  10270  xaddass  10271  xltadd1  10278  isxmet2d  15449  mettri2  15463  bl2in  15504  xmeter  15537
  Copyright terms: Public domain W3C validator