Theorem rpaddcl 10028

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 10011	. . 3 |- ( A e. RR+ -> A e. RR )
2		rpre 10011	. . 3 |- ( B e. RR+ -> B e. RR )
3		readdcl 8269	. . 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 10006	. . 3 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
6		elrp 10006	. . 3 |- ( B e. RR+ <-> ( B e. RR /\ 0 < B ) )
7		addgt0 8739	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < B ) ) -> 0 < ( A + B ) )
8	7	an4s 592	. . 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 10006	. 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 2205   class class class wbr 4114  (class class class)co 6058   RRcr 8142   0cc0 8143   + caddc 8146   < clt 8324   RR+crp 10004

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0id 8251  ax-rnegex 8252  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-xp 4760  df-iota 5317  df-fv 5365  df-ov 6061  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-rp 10005

This theorem is referenced by:  rpaddcld  10063  fsumrpcl  12115  isumrpcl  12205  efgt1p2  12406  qdiff  16959