ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xaddnemnf Unicode version

Theorem xaddnemnf 9801
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 9721 . 2  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  <->  ( A  e.  RR  \/  A  = +oo ) )
2 xrnemnf 9721 . . . 4  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  <->  ( B  e.  RR  \/  B  = +oo ) )
3 rexadd 9796 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( A  +  B ) )
4 readdcl 7887 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
53, 4eqeltrd 2247 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  e.  RR )
65renemnfd 7958 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =/= -oo )
7 oveq2 5858 . . . . . . 7  |-  ( B  = +oo  ->  ( A +e B )  =  ( A +e +oo ) )
8 rexr 7952 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
9 renemnf 7955 . . . . . . . 8  |-  ( A  e.  RR  ->  A  =/= -oo )
10 xaddpnf1 9790 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )
118, 9, 10syl2anc 409 . . . . . . 7  |-  ( A  e.  RR  ->  ( A +e +oo )  = +oo )
127, 11sylan9eqr 2225 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A +e
B )  = +oo )
13 pnfnemnf 7961 . . . . . . 7  |- +oo  =/= -oo
1413a1i 9 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  -> +oo  =/= -oo )
1512, 14eqnetrd 2364 . . . . 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 285 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR*  /\  B  =/= -oo ) )  -> 
( A +e
B )  =/= -oo )
18 oveq1 5857 . . . . 5  |-  ( A  = +oo  ->  ( A +e B )  =  ( +oo +e B ) )
19 xaddpnf2 9791 . . . . 5  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  = +oo )
2018, 19sylan9eq 2223 . . . 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 2364 . . 3  |-  ( ( A  = +oo  /\  ( B  e.  RR*  /\  B  =/= -oo ) )  -> 
( A +e
B )  =/= -oo )
2317, 22jaoian 790 . 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 703    = wceq 1348    e. wcel 2141    =/= wne 2340  (class class class)co 5850   RRcr 7760    + caddc 7764   +oocpnf 7938   -oocmnf 7939   RR*cxr 7940   +ecxad 9714
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7852  ax-resscn 7853  ax-1re 7855  ax-addrcl 7858  ax-rnegex 7870
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-pnf 7943  df-mnf 7944  df-xr 7945  df-xadd 9717
This theorem is referenced by:  xaddass  9813  xlt2add  9824  xadd4d  9829  xleaddadd  9831
  Copyright terms: Public domain W3C validator