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Theorem rpaddcl 11001
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 10987 . . 3  |-  ( A  e.  RR+  ->  A  e.  RR )
2 rpre 10987 . . 3  |-  ( B  e.  RR+  ->  B  e.  RR )
3 readdcl 9355 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
41, 2, 3syl2an 474 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  +  B )  e.  RR )
5 elrp 10983 . . 3  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
6 elrp 10983 . . 3  |-  ( B  e.  RR+  <->  ( B  e.  RR  /\  0  < 
B ) )
7 addgt0 9815 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  0  <  ( A  +  B
) )
87an4s 817 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  ( A  +  B ) )
95, 6, 8syl2anb 476 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  0  <  ( A  +  B
) )
10 elrp 10983 . 2  |-  ( ( A  +  B )  e.  RR+  <->  ( ( A  +  B )  e.  RR  /\  0  < 
( A  +  B
) ) )
114, 9, 10sylanbrc 659 1  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  +  B )  e.  RR+ )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1757   class class class wbr 4282  (class class class)co 6082   RRcr 9271   0cc0 9272    + caddc 9275    < clt 9408   RR+crp 10981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363  ax-resscn 9329  ax-1cn 9330  ax-icn 9331  ax-addcl 9332  ax-addrcl 9333  ax-mulcl 9334  ax-mulrcl 9335  ax-mulcom 9336  ax-addass 9337  ax-mulass 9338  ax-distr 9339  ax-i2m1 9340  ax-1ne0 9341  ax-1rid 9342  ax-rnegex 9343  ax-rrecex 9344  ax-cnre 9345  ax-pre-lttri 9346  ax-pre-lttrn 9347  ax-pre-ltadd 9348
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2966  df-sbc 3178  df-csb 3279  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-nul 3628  df-if 3782  df-pw 3852  df-sn 3868  df-pr 3870  df-op 3874  df-uni 4082  df-br 4283  df-opab 4341  df-mpt 4342  df-id 4625  df-po 4630  df-so 4631  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-ov 6085  df-er 7091  df-en 7301  df-dom 7302  df-sdom 7303  df-pnf 9410  df-mnf 9411  df-xr 9412  df-ltxr 9413  df-le 9414  df-rp 10982
This theorem is referenced by:  rpaddcld  11032  fsumrpcl  13200  logcnlem2  21975  logcnlem3  21976  logcnlem4  21977  loglesqr  22083  ang180lem2  22093  cxp2limlem  22256  logdifbnd  22274  emcllem4  22279  emcllem5  22280  emcllem6  22281  selberg2lem  22686  chpdifbndlem2  22690  pntpbnd1a  22721  pntpbnd1  22722  pntpbnd2  22723  pntpbnd  22724  pntibndlem1  22725  pntibndlem2  22727  pntibnd  22729  pntlemd  22730  pntlemq  22737  pntlemr  22738  pntlemj  22739  pntlemp  22746  pntleml  22747  smcnlem  23917
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