Theorem rpmulcl 9580

Description: Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)

Assertion

Ref	Expression
rpmulcl	|- ( ( A e. RR+ /\ B e. RR+ ) -> ( A x. B ) e. RR+ )

Proof of Theorem rpmulcl

Step	Hyp	Ref	Expression
1		rpre 9562	. . 3 |- ( A e. RR+ -> A e. RR )
2		rpre 9562	. . 3 |- ( B e. RR+ -> B e. RR )
3		remulcl 7855	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
4	1, 2, 3	syl2an 287	. 2 |- ( ( A e. RR+ /\ B e. RR+ ) -> ( A x. B ) e. RR )
5		elrp 9557	. . 3 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
6		elrp 9557	. . 3 |- ( B e. RR+ <-> ( B e. RR /\ 0 < B ) )
7		mulgt0 7947	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < ( A x. B ) )
8	5, 6, 7	syl2anb 289	. 2 |- ( ( A e. RR+ /\ B e. RR+ ) -> 0 < ( A x. B ) )
9		elrp 9557	. 2 |- ( ( A x. B ) e. RR+ <-> ( ( A x. B ) e. RR /\ 0 < ( A x. B ) ) )
10	4, 8, 9	sylanbrc 414	1 |- ( ( A e. RR+ /\ B e. RR+ ) -> ( A x. B ) e. RR+ )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2128   class class class wbr 3965  (class class class)co 5821   RRcr 7726   0cc0 7727   x. cmul 7732   < clt 7907   RR+crp 9555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-cnex 7818  ax-resscn 7819  ax-1re 7821  ax-addrcl 7824  ax-mulrcl 7826  ax-rnegex 7836  ax-pre-mulgt0 7844

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4591  df-pnf 7909  df-mnf 7910  df-ltxr 7912  df-rp 9556

This theorem is referenced by:  rpmulcld  9615  rpexpcl  10433  expcnvap0  11394  fprodrpcl  11503  cosordlem  13157  rprelogbmul  13259  taupi  13628