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Theorem rpaddcl 11014
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 11000 . . 3  |-  ( A  e.  RR+  ->  A  e.  RR )
2 rpre 11000 . . 3  |-  ( B  e.  RR+  ->  B  e.  RR )
3 readdcl 9368 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
41, 2, 3syl2an 477 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  +  B )  e.  RR )
5 elrp 10996 . . 3  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
6 elrp 10996 . . 3  |-  ( B  e.  RR+  <->  ( B  e.  RR  /\  0  < 
B ) )
7 addgt0 9828 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  0  <  ( A  +  B
) )
87an4s 822 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  ( A  +  B ) )
95, 6, 8syl2anb 479 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  0  <  ( A  +  B
) )
10 elrp 10996 . 2  |-  ( ( A  +  B )  e.  RR+  <->  ( ( A  +  B )  e.  RR  /\  0  < 
( A  +  B
) ) )
114, 9, 10sylanbrc 664 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 1756   class class class wbr 4295  (class class class)co 6094   RRcr 9284   0cc0 9285    + caddc 9288    < clt 9421   RR+crp 10994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-br 4296  df-opab 4354  df-mpt 4355  df-id 4639  df-po 4644  df-so 4645  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6097  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-rp 10995
This theorem is referenced by:  rpaddcld  11045  fsumrpcl  13217  logcnlem2  22091  logcnlem3  22092  logcnlem4  22093  loglesqr  22199  ang180lem2  22209  cxp2limlem  22372  logdifbnd  22390  emcllem4  22395  emcllem5  22396  emcllem6  22397  selberg2lem  22802  chpdifbndlem2  22806  pntpbnd1a  22837  pntpbnd1  22838  pntpbnd2  22839  pntpbnd  22840  pntibndlem1  22841  pntibndlem2  22843  pntibnd  22845  pntlemd  22846  pntlemq  22853  pntlemr  22854  pntlemj  22855  pntlemp  22862  pntleml  22863  smcnlem  24095
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