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Theorem rpmulcld 9507
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
StepHypRef Expression
1 rpred.1 . 2  |-  ( ph  ->  A  e.  RR+ )
2 rpaddcld.1 . 2  |-  ( ph  ->  B  e.  RR+ )
3 rpmulcl 9473 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  x.  B )  e.  RR+ )
41, 2, 3syl2anc 408 1  |-  ( ph  ->  ( A  x.  B
)  e.  RR+ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1480  (class class class)co 5774    x. cmul 7632   RR+crp 9448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1re 7721  ax-addrcl 7724  ax-mulrcl 7726  ax-rnegex 7736  ax-pre-mulgt0 7744
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-pnf 7809  df-mnf 7810  df-ltxr 7812  df-rp 9449
This theorem is referenced by:  qbtwnrelemcalc  10040  cvg1nlemcxze  10761  cvg1nlemres  10764  resqrexlemnm  10797  resqrexlemcvg  10798  reccn2ap  11089  cvgratnnlembern  11299  cvgratnnlemrate  11306  cvgratnn  11307  eirraplem  11489  cosordlem  12946
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