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Theorem rpmulcl 8909
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
StepHypRef Expression
1 rpre 8891 . . 3  |-  ( A  e.  RR+  ->  A  e.  RR )
2 rpre 8891 . . 3  |-  ( B  e.  RR+  ->  B  e.  RR )
3 remulcl 7233 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
41, 2, 3syl2an 283 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  x.  B )  e.  RR )
5 elrp 8887 . . 3  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
6 elrp 8887 . . 3  |-  ( B  e.  RR+  <->  ( B  e.  RR  /\  0  < 
B ) )
7 mulgt0 7323 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  ( A  x.  B ) )
85, 6, 7syl2anb 285 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  0  <  ( A  x.  B
) )
9 elrp 8887 . 2  |-  ( ( A  x.  B )  e.  RR+  <->  ( ( A  x.  B )  e.  RR  /\  0  < 
( A  x.  B
) ) )
104, 8, 9sylanbrc 408 1  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  x.  B )  e.  RR+ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1434   class class class wbr 3805  (class class class)co 5564   RRcr 7112   0cc0 7113    x. cmul 7118    < clt 7285   RR+crp 8885
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7199  ax-resscn 7200  ax-1re 7202  ax-addrcl 7205  ax-mulrcl 7207  ax-rnegex 7217  ax-pre-mulgt0 7225
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-xp 4397  df-pnf 7287  df-mnf 7288  df-ltxr 7290  df-rp 8886
This theorem is referenced by:  rpmulcld  8941  rpexpcl  9662
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