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Theorem xaddnepnf 9763
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 9686 . 2  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  <->  ( A  e.  RR  \/  A  = -oo ) )
2 xrnepnf 9686 . . . 4  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  <->  ( B  e.  RR  \/  B  = -oo ) )
3 rexadd 9757 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( A  +  B ) )
4 readdcl 7859 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
53, 4eqeltrd 2234 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  e.  RR )
65renepnfd 7929 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =/= +oo )
7 oveq2 5833 . . . . . . 7  |-  ( B  = -oo  ->  ( A +e B )  =  ( A +e -oo ) )
8 rexr 7924 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
9 renepnf 7926 . . . . . . . 8  |-  ( A  e.  RR  ->  A  =/= +oo )
10 xaddmnf1 9753 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A +e -oo )  = -oo )
118, 9, 10syl2anc 409 . . . . . . 7  |-  ( A  e.  RR  ->  ( A +e -oo )  = -oo )
127, 11sylan9eqr 2212 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A +e
B )  = -oo )
13 mnfnepnf 7934 . . . . . . 7  |- -oo  =/= +oo
1413a1i 9 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  -> -oo  =/= +oo )
1512, 14eqnetrd 2351 . . . . 5  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A +e
B )  =/= +oo )
166, 15jaodan 787 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = -oo ) )  ->  ( A +e B )  =/= +oo )
172, 16sylan2b 285 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR*  /\  B  =/= +oo ) )  -> 
( A +e
B )  =/= +oo )
18 oveq1 5832 . . . . 5  |-  ( A  = -oo  ->  ( A +e B )  =  ( -oo +e B ) )
19 xaddmnf2 9754 . . . . 5  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
2018, 19sylan9eq 2210 . . . 4  |-  ( ( A  = -oo  /\  ( B  e.  RR*  /\  B  =/= +oo ) )  -> 
( A +e
B )  = -oo )
2113a1i 9 . . . 4  |-  ( ( A  = -oo  /\  ( B  e.  RR*  /\  B  =/= +oo ) )  -> -oo  =/= +oo )
2220, 21eqnetrd 2351 . . 3  |-  ( ( A  = -oo  /\  ( B  e.  RR*  /\  B  =/= +oo ) )  -> 
( A +e
B )  =/= +oo )
2317, 22jaoian 785 . 2  |-  ( ( ( A  e.  RR  \/  A  = -oo )  /\  ( B  e. 
RR*  /\  B  =/= +oo ) )  ->  ( A +e B )  =/= +oo )
241, 23sylanb 282 1  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )
)  ->  ( A +e B )  =/= +oo )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 698    = wceq 1335    e. wcel 2128    =/= wne 2327  (class class class)co 5825   RRcr 7732    + caddc 7736   +oocpnf 7910   -oocmnf 7911   RR*cxr 7912   +ecxad 9678
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-cnex 7824  ax-resscn 7825  ax-1re 7827  ax-addrcl 7830  ax-rnegex 7842
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-iota 5136  df-fun 5173  df-fv 5179  df-ov 5828  df-oprab 5829  df-mpo 5830  df-pnf 7915  df-mnf 7916  df-xr 7917  df-xadd 9681
This theorem is referenced by:  xlt2add  9785
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