Theorem rpaddcl 9752

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

Step	Hyp	Ref	Expression
1		rpre 9735	. . 3 |- ( A e. RR+ -> A e. RR )
2		rpre 9735	. . 3 |- ( B e. RR+ -> B e. RR )
3		readdcl 8005	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
4	1, 2, 3	syl2an 289	. 2 |- ( ( A e. RR+ /\ B e. RR+ ) -> ( A + B ) e. RR )
5		elrp 9730	. . 3 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
6		elrp 9730	. . 3 |- ( B e. RR+ <-> ( B e. RR /\ 0 < B ) )
7		addgt0 8475	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < B ) ) -> 0 < ( A + B ) )
8	7	an4s 588	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < ( A + B ) )
9	5, 6, 8	syl2anb 291	. 2 |- ( ( A e. RR+ /\ B e. RR+ ) -> 0 < ( A + B ) )
10		elrp 9730	. 2 |- ( ( A + B ) e. RR+ <-> ( ( A + B ) e. RR /\ 0 < ( A + B ) ) )
11	4, 9, 10	sylanbrc 417	1 |- ( ( A e. RR+ /\ B e. RR+ ) -> ( A + B ) e. RR+ )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2167   class class class wbr 4033  (class class class)co 5922   RRcr 7878   0cc0 7879   + caddc 7882   < clt 8061   RR+crp 9728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0id 7987  ax-rnegex 7988  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-xp 4669  df-iota 5219  df-fv 5266  df-ov 5925  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-rp 9729

This theorem is referenced by:  rpaddcld  9787  fsumrpcl  11569  isumrpcl  11659  efgt1p2  11860