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

Theorem rpaddcl 8838
Description: Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
Assertion
Ref Expression
rpaddcl  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  +  B )  e.  RR+ )

Proof of Theorem rpaddcl
StepHypRef Expression
1 rpre 8821 . . 3  |-  ( A  e.  RR+  ->  A  e.  RR )
2 rpre 8821 . . 3  |-  ( B  e.  RR+  ->  B  e.  RR )
3 readdcl 7161 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
41, 2, 3syl2an 283 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  +  B )  e.  RR )
5 elrp 8817 . . 3  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
6 elrp 8817 . . 3  |-  ( B  e.  RR+  <->  ( B  e.  RR  /\  0  < 
B ) )
7 addgt0 7619 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  0  <  ( A  +  B
) )
87an4s 553 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  ( A  +  B ) )
95, 6, 8syl2anb 285 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  0  <  ( A  +  B
) )
10 elrp 8817 . 2  |-  ( ( A  +  B )  e.  RR+  <->  ( ( A  +  B )  e.  RR  /\  0  < 
( A  +  B
) ) )
114, 9, 10sylanbrc 408 1  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  +  B )  e.  RR+ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1434   class class class wbr 3793  (class class class)co 5543   RRcr 7042   0cc0 7043    + caddc 7046    < clt 7215   RR+crp 8815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-i2m1 7143  ax-0id 7146  ax-rnegex 7147  ax-pre-lttrn 7152  ax-pre-ltadd 7154
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-xp 4377  df-iota 4897  df-fv 4940  df-ov 5546  df-pnf 7217  df-mnf 7218  df-ltxr 7220  df-rp 8816
This theorem is referenced by:  rpaddcld  8870
  Copyright terms: Public domain W3C validator