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Theorem rpmulcl 9680
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 9662 . . 3  |-  ( A  e.  RR+  ->  A  e.  RR )
2 rpre 9662 . . 3  |-  ( B  e.  RR+  ->  B  e.  RR )
3 remulcl 7941 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
41, 2, 3syl2an 289 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  x.  B )  e.  RR )
5 elrp 9657 . . 3  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
6 elrp 9657 . . 3  |-  ( B  e.  RR+  <->  ( B  e.  RR  /\  0  < 
B ) )
7 mulgt0 8034 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  ( A  x.  B ) )
85, 6, 7syl2anb 291 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  0  <  ( A  x.  B
) )
9 elrp 9657 . 2  |-  ( ( A  x.  B )  e.  RR+  <->  ( ( A  x.  B )  e.  RR  /\  0  < 
( A  x.  B
) ) )
104, 8, 9sylanbrc 417 1  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  x.  B )  e.  RR+ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2148   class class class wbr 4005  (class class class)co 5877   RRcr 7812   0cc0 7813    x. cmul 7818    < clt 7994   RR+crp 9655
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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910  ax-mulrcl 7912  ax-rnegex 7922  ax-pre-mulgt0 7930
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 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-xp 4634  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-rp 9656
This theorem is referenced by:  rpmulcld  9715  rpexpcl  10541  expcnvap0  11512  fprodrpcl  11621  cosordlem  14355  rprelogbmul  14458  taupi  14906
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