Theorem rpmulcl 9874

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 9856	. . 3 |- ( A e. RR+ -> A e. RR )
2		rpre 9856	. . 3 |- ( B e. RR+ -> B e. RR )
3		remulcl 8127	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
4	1, 2, 3	syl2an 289	. 2 |- ( ( A e. RR+ /\ B e. RR+ ) -> ( A x. B ) e. RR )
5		elrp 9851	. . 3 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
6		elrp 9851	. . 3 |- ( B e. RR+ <-> ( B e. RR /\ 0 < B ) )
7		mulgt0 8221	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < ( A x. B ) )
8	5, 6, 7	syl2anb 291	. 2 |- ( ( A e. RR+ /\ B e. RR+ ) -> 0 < ( A x. B ) )
9		elrp 9851	. 2 |- ( ( A x. B ) e. RR+ <-> ( ( A x. B ) e. RR /\ 0 < ( A x. B ) ) )
10	4, 8, 9	sylanbrc 417	1 |- ( ( A e. RR+ /\ B e. RR+ ) -> ( A x. B ) e. RR+ )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   class class class wbr 4083  (class class class)co 6001   RRcr 7998   0cc0 7999   x. cmul 8004   < clt 8181   RR+crp 9849

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096  ax-mulrcl 8098  ax-rnegex 8108  ax-pre-mulgt0 8116

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-rp 9850

This theorem is referenced by:  rpmulcld  9909  rpexpcl  10780  expcnvap0  12013  fprodrpcl  12122  cosordlem  15523  rprelogbmul  15629  taupi  16441