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Theorem xaddnepnf 11535
Description: Closure of extended real addition in the subset  RR*  /  { +oo }. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xaddnepnf  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )
)  ->  ( A +e B )  =/= +oo )

Proof of Theorem xaddnepnf
StepHypRef Expression
1 xrnepnf 11427 . 2  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  <->  ( A  e.  RR  \/  A  = -oo ) )
2 xrnepnf 11427 . . . 4  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  <->  ( B  e.  RR  \/  B  = -oo ) )
3 rexadd 11532 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( A  +  B ) )
4 readdcl 9629 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
53, 4eqeltrd 2507 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  e.  RR )
65renepnfd 9698 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =/= +oo )
7 oveq2 6313 . . . . . . 7  |-  ( B  = -oo  ->  ( A +e B )  =  ( A +e -oo ) )
8 rexr 9693 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
9 renepnf 9695 . . . . . . . 8  |-  ( A  e.  RR  ->  A  =/= +oo )
10 xaddmnf1 11528 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A +e -oo )  = -oo )
118, 9, 10syl2anc 665 . . . . . . 7  |-  ( A  e.  RR  ->  ( A +e -oo )  = -oo )
127, 11sylan9eqr 2485 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A +e
B )  = -oo )
13 mnfnepnf 11425 . . . . . . 7  |- -oo  =/= +oo
1413a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  -> -oo  =/= +oo )
1512, 14eqnetrd 2713 . . . . 5  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A +e
B )  =/= +oo )
166, 15jaodan 792 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = -oo ) )  ->  ( A +e B )  =/= +oo )
172, 16sylan2b 477 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR*  /\  B  =/= +oo ) )  -> 
( A +e
B )  =/= +oo )
18 oveq1 6312 . . . . 5  |-  ( A  = -oo  ->  ( A +e B )  =  ( -oo +e B ) )
19 xaddmnf2 11529 . . . . 5  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
2018, 19sylan9eq 2483 . . . 4  |-  ( ( A  = -oo  /\  ( B  e.  RR*  /\  B  =/= +oo ) )  -> 
( A +e
B )  = -oo )
2113a1i 11 . . . 4  |-  ( ( A  = -oo  /\  ( B  e.  RR*  /\  B  =/= +oo ) )  -> -oo  =/= +oo )
2220, 21eqnetrd 2713 . . 3  |-  ( ( A  = -oo  /\  ( B  e.  RR*  /\  B  =/= +oo ) )  -> 
( A +e
B )  =/= +oo )
2317, 22jaoian 791 . 2  |-  ( ( ( A  e.  RR  \/  A  = -oo )  /\  ( B  e. 
RR*  /\  B  =/= +oo ) )  ->  ( A +e B )  =/= +oo )
241, 23sylanb 474 1  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )
)  ->  ( A +e B )  =/= +oo )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614  (class class class)co 6305   RRcr 9545    + caddc 9549   +oocpnf 9679   -oocmnf 9680   RR*cxr 9681   +ecxad 11414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-i2m1 9614
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-pnf 9684  df-mnf 9685  df-xr 9686  df-xadd 11417
This theorem is referenced by:  xlt2add  11553
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