Theorem rpmulcl 9912

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 9894	. . 3 |- ( A e. RR+ -> A e. RR )
2		rpre 9894	. . 3 |- ( B e. RR+ -> B e. RR )
3		remulcl 8159	. . 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 9889	. . 3 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
6		elrp 9889	. . 3 |- ( B e. RR+ <-> ( B e. RR /\ 0 < B ) )
7		mulgt0 8253	. . 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 9889	. 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 2202   class class class wbr 4088  (class class class)co 6017   RRcr 8030   0cc0 8031   x. cmul 8036   < clt 8213   RR+crp 9887

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128  ax-mulrcl 8130  ax-rnegex 8140  ax-pre-mulgt0 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-rp 9888

This theorem is referenced by:  rpmulcld  9947  rpexpcl  10819  expcnvap0  12062  fprodrpcl  12171  cosordlem  15572  rprelogbmul  15678  taupi  16677