Theorem rpmulcld 9909

Description: Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)

Hypotheses

Ref	Expression
rpred.1	|- ( ph -> A e. RR+ )
rpaddcld.1	|- ( ph -> B e. RR+ )

Assertion

Ref	Expression
rpmulcld	|- ( ph -> ( A x. B ) e. RR+ )

Proof of Theorem rpmulcld

Step	Hyp	Ref	Expression
1		rpred.1	. 2 |- ( ph -> A e. RR+ )
2		rpaddcld.1	. 2 |- ( ph -> B e. RR+ )
3		rpmulcl 9874	. 2 |- ( ( A e. RR+ /\ B e. RR+ ) -> ( A x. B ) e. RR+ )
4	1, 2, 3	syl2anc 411	1 |- ( ph -> ( A x. B ) e. RR+ )

Syntax hints:   -> wi 4   e. wcel 2200  (class class class)co 6001   x. cmul 8004   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:  qbtwnrelemcalc  10475  cvg1nlemcxze  11493  cvg1nlemres  11496  resqrexlemnm  11529  resqrexlemcvg  11530  reccn2ap  11824  cvgratnnlembern  12034  cvgratnnlemrate  12041  cvgratnn  12042  eirraplem  12288  cosordlem  15523  rpmulcxp  15583  lgsquadlem2  15757