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Theorem xaddnepnf 9871
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 9791 . 2  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  <->  ( A  e.  RR  \/  A  = -oo ) )
2 xrnepnf 9791 . . . 4  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  <->  ( B  e.  RR  \/  B  = -oo ) )
3 rexadd 9865 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( A  +  B ) )
4 readdcl 7950 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
53, 4eqeltrd 2264 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  e.  RR )
65renepnfd 8021 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =/= +oo )
7 oveq2 5896 . . . . . . 7  |-  ( B  = -oo  ->  ( A +e B )  =  ( A +e -oo ) )
8 rexr 8016 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
9 renepnf 8018 . . . . . . . 8  |-  ( A  e.  RR  ->  A  =/= +oo )
10 xaddmnf1 9861 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A +e -oo )  = -oo )
118, 9, 10syl2anc 411 . . . . . . 7  |-  ( A  e.  RR  ->  ( A +e -oo )  = -oo )
127, 11sylan9eqr 2242 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A +e
B )  = -oo )
13 mnfnepnf 8026 . . . . . . 7  |- -oo  =/= +oo
1413a1i 9 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  -> -oo  =/= +oo )
1512, 14eqnetrd 2381 . . . . 5  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A +e
B )  =/= +oo )
166, 15jaodan 798 . . . 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 5895 . . . . 5  |-  ( A  = -oo  ->  ( A +e B )  =  ( -oo +e B ) )
19 xaddmnf2 9862 . . . . 5  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
2018, 19sylan9eq 2240 . . . 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 2381 . . 3  |-  ( ( A  = -oo  /\  ( B  e.  RR*  /\  B  =/= +oo ) )  -> 
( A +e
B )  =/= +oo )
2317, 22jaoian 796 . 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 709    = wceq 1363    e. wcel 2158    =/= wne 2357  (class class class)co 5888   RRcr 7823    + caddc 7827   +oocpnf 8002   -oocmnf 8003   RR*cxr 8004   +ecxad 9783
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921  ax-rnegex 7933
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8007  df-mnf 8008  df-xr 8009  df-xadd 9786
This theorem is referenced by:  xlt2add  9893
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