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Theorem xaddnepnf 9861
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 9781 . 2  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  <->  ( A  e.  RR  \/  A  = -oo ) )
2 xrnepnf 9781 . . . 4  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  <->  ( B  e.  RR  \/  B  = -oo ) )
3 rexadd 9855 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( A  +  B ) )
4 readdcl 7940 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
53, 4eqeltrd 2254 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  e.  RR )
65renepnfd 8011 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =/= +oo )
7 oveq2 5886 . . . . . . 7  |-  ( B  = -oo  ->  ( A +e B )  =  ( A +e -oo ) )
8 rexr 8006 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
9 renepnf 8008 . . . . . . . 8  |-  ( A  e.  RR  ->  A  =/= +oo )
10 xaddmnf1 9851 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A +e -oo )  = -oo )
118, 9, 10syl2anc 411 . . . . . . 7  |-  ( A  e.  RR  ->  ( A +e -oo )  = -oo )
127, 11sylan9eqr 2232 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A +e
B )  = -oo )
13 mnfnepnf 8016 . . . . . . 7  |- -oo  =/= +oo
1413a1i 9 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  -> -oo  =/= +oo )
1512, 14eqnetrd 2371 . . . . 5  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A +e
B )  =/= +oo )
166, 15jaodan 797 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = -oo ) )  ->  ( A +e B )  =/= +oo )
172, 16sylan2b 287 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR*  /\  B  =/= +oo ) )  -> 
( A +e
B )  =/= +oo )
18 oveq1 5885 . . . . 5  |-  ( A  = -oo  ->  ( A +e B )  =  ( -oo +e B ) )
19 xaddmnf2 9852 . . . . 5  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
2018, 19sylan9eq 2230 . . . 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 2371 . . 3  |-  ( ( A  = -oo  /\  ( B  e.  RR*  /\  B  =/= +oo ) )  -> 
( A +e
B )  =/= +oo )
2317, 22jaoian 795 . 2  |-  ( ( ( A  e.  RR  \/  A  = -oo )  /\  ( B  e. 
RR*  /\  B  =/= +oo ) )  ->  ( A +e B )  =/= +oo )
241, 23sylanb 284 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 104    \/ wo 708    = wceq 1353    e. wcel 2148    =/= wne 2347  (class class class)co 5878   RRcr 7813    + caddc 7817   +oocpnf 7992   -oocmnf 7993   RR*cxr 7994   +ecxad 9773
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1re 7908  ax-addrcl 7911  ax-rnegex 7923
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-pnf 7997  df-mnf 7998  df-xr 7999  df-xadd 9776
This theorem is referenced by:  xlt2add  9883
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