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Theorem xaddnemnf 11458
Description: Closure of extended real addition in the subset  RR*  /  { -oo }. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xaddnemnf  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )
)  ->  ( A +e B )  =/= -oo )

Proof of Theorem xaddnemnf
StepHypRef Expression
1 xrnemnf 11353 . 2  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  <->  ( A  e.  RR  \/  A  = +oo ) )
2 xrnemnf 11353 . . . 4  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  <->  ( B  e.  RR  \/  B  = +oo ) )
3 rexadd 11456 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( A  +  B ) )
4 readdcl 9592 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
53, 4eqeltrd 2545 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  e.  RR )
65renemnfd 9662 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =/= -oo )
7 oveq2 6304 . . . . . . 7  |-  ( B  = +oo  ->  ( A +e B )  =  ( A +e +oo ) )
8 rexr 9656 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
9 renemnf 9659 . . . . . . . 8  |-  ( A  e.  RR  ->  A  =/= -oo )
10 xaddpnf1 11450 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )
118, 9, 10syl2anc 661 . . . . . . 7  |-  ( A  e.  RR  ->  ( A +e +oo )  = +oo )
127, 11sylan9eqr 2520 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A +e
B )  = +oo )
13 pnfnemnf 11351 . . . . . . 7  |- +oo  =/= -oo
1413a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  -> +oo  =/= -oo )
1512, 14eqnetrd 2750 . . . . 5  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A +e
B )  =/= -oo )
166, 15jaodan 785 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo ) )  ->  ( A +e B )  =/= -oo )
172, 16sylan2b 475 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR*  /\  B  =/= -oo ) )  -> 
( A +e
B )  =/= -oo )
18 oveq1 6303 . . . . 5  |-  ( A  = +oo  ->  ( A +e B )  =  ( +oo +e B ) )
19 xaddpnf2 11451 . . . . 5  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  = +oo )
2018, 19sylan9eq 2518 . . . 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 2750 . . 3  |-  ( ( A  = +oo  /\  ( B  e.  RR*  /\  B  =/= -oo ) )  -> 
( A +e
B )  =/= -oo )
2317, 22jaoian 784 . 2  |-  ( ( ( A  e.  RR  \/  A  = +oo )  /\  ( B  e. 
RR*  /\  B  =/= -oo ) )  ->  ( A +e B )  =/= -oo )
241, 23sylanb 472 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 368    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652  (class class class)co 6296   RRcr 9508    + caddc 9512   +oocpnf 9642   -oocmnf 9643   RR*cxr 9644   +ecxad 11341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-i2m1 9577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-xadd 11344
This theorem is referenced by:  xaddass  11466  xlt2add  11477  xadd4d  11520  xrs1mnd  18583
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