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Theorem rexadd 10233
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 8361 . . 3 |- ( A e. RR -> A e. RR* )
2 rexr 8361 . . 3 |- ( B e. RR -> B e. RR* )
3 xaddval 10226 . . 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 8363 . . . . 5 |- ( A e. RR -> A =/= +oo )
6 ifnefalse 3648 . . . . 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 8364 . . . . 5 |- ( A e. RR -> A =/= -oo )
9 ifnefalse 3648 . . . . 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 8363 . . . . 5 |- ( B e. RR -> B =/= +oo )
13 ifnefalse 3648 . . . . 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 8364 . . . . 5 |- ( B e. RR -> B =/= -oo )
16 ifnefalse 3648 . . . . 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
Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   =/= wne 2420   ifcif 3635  (class class class)co 6075   RRcr 8168   0cc0 8169   + caddc 8172   +oocpnf 8347   -oocmnf 8348   RR*cxr 8349   +ecxad 10151
This theorem was proved from 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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266  ax-rnegex 8278
This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-xadd 10154
This theorem is referenced by:  rexsub  10234  rexaddd  10235  xaddnemnf  10238  xaddnepnf  10239  xnegid  10240  xaddcom  10242  xaddid1  10243  xnn0xadd0  10248  xnegdi  10249  xaddass  10250  xltadd1  10257  isxmet2d  15372  mettri2  15386  bl2in  15427  xmeter  15460
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