Theorem rpmulcl 9903

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 9885	. . 3 |- ( A e. RR+ -> A e. RR )
2		rpre 9885	. . 3 |- ( B e. RR+ -> B e. RR )
3		remulcl 8150	. . 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 9880	. . 3 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
6		elrp 9880	. . 3 |- ( B e. RR+ <-> ( B e. RR /\ 0 < B ) )
7		mulgt0 8244	. . 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 9880	. 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 4086  (class class class)co 6013   RRcr 8021   0cc0 8022   x. cmul 8027   < clt 8204   RR+crp 9878

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 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119  ax-mulrcl 8121  ax-rnegex 8131  ax-pre-mulgt0 8139

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 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-xp 4729  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-rp 9879

This theorem is referenced by:  rpmulcld  9938  rpexpcl  10810  expcnvap0  12053  fprodrpcl  12162  cosordlem  15563  rprelogbmul  15669  taupi  16613