Theorem rpmulcld 10046

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 10011	. 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 2203  (class class class)co 6050   x. cmul 8132   RR+crp 9986

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224  ax-mulrcl 8226  ax-rnegex 8236  ax-pre-mulgt0 8244

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-xp 4755  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-rp 9987

This theorem is referenced by:  qbtwnrelemcalc  10615  cvg1nlemcxze  11667  cvg1nlemres  11670  resqrexlemnm  11703  resqrexlemcvg  11704  reccn2ap  11998  cvgratnnlembern  12209  cvgratnnlemrate  12216  cvgratnn  12217  eirraplem  12463  cosordlem  15714  rpmulcxp  15774  lgsquadlem2  15951