Theorem rpmulcl 9676

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 9658	. . 3 |- ( A e. RR+ -> A e. RR )
2		rpre 9658	. . 3 |- ( B e. RR+ -> B e. RR )
3		remulcl 7938	. . 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 9653	. . 3 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
6		elrp 9653	. . 3 |- ( B e. RR+ <-> ( B e. RR /\ 0 < B ) )
7		mulgt0 8030	. . 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 9653	. 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 2148   class class class wbr 4003  (class class class)co 5874   RRcr 7809   0cc0 7810   x. cmul 7815   < clt 7990   RR+crp 9651

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907  ax-mulrcl 7909  ax-rnegex 7919  ax-pre-mulgt0 7927

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-xp 4632  df-pnf 7992  df-mnf 7993  df-ltxr 7995  df-rp 9652

This theorem is referenced by:  rpmulcld  9711  rpexpcl  10536  expcnvap0  11505  fprodrpcl  11614  cosordlem  14163  rprelogbmul  14266  taupi  14702