Theorem rpaddcl 9834

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 9817	. . 3 |- ( A e. RR+ -> A e. RR )
2		rpre 9817	. . 3 |- ( B e. RR+ -> B e. RR )
3		readdcl 8086	. . 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 9812	. . 3 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
6		elrp 9812	. . 3 |- ( B e. RR+ <-> ( B e. RR /\ 0 < B ) )
7		addgt0 8556	. . . 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 9812	. 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 2178   class class class wbr 4059  (class class class)co 5967   RRcr 7959   0cc0 7960   + caddc 7963   < clt 8142   RR+crp 9810

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0id 8068  ax-rnegex 8069  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-xp 4699  df-iota 5251  df-fv 5298  df-ov 5970  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-rp 9811

This theorem is referenced by:  rpaddcld  9869  fsumrpcl  11830  isumrpcl  11920  efgt1p2  12121