MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rpmulcl Structured version   Unicode version

Theorem rpmulcl 11000
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 10985 . . 3  |-  ( A  e.  RR+  ->  A  e.  RR )
2 rpre 10985 . . 3  |-  ( B  e.  RR+  ->  B  e.  RR )
3 remulcl 9355 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
41, 2, 3syl2an 474 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  x.  B )  e.  RR )
5 elrp 10981 . . 3  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
6 elrp 10981 . . 3  |-  ( B  e.  RR+  <->  ( B  e.  RR  /\  0  < 
B ) )
7 mulgt0 9440 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  ( A  x.  B ) )
85, 6, 7syl2anb 476 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  0  <  ( A  x.  B
) )
9 elrp 10981 . 2  |-  ( ( A  x.  B )  e.  RR+  <->  ( ( A  x.  B )  e.  RR  /\  0  < 
( A  x.  B
) ) )
104, 8, 9sylanbrc 657 1  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  x.  B )  e.  RR+ )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1755   class class class wbr 4280  (class class class)co 6080   RRcr 9269   0cc0 9270    x. cmul 9275    < clt 9406   RR+crp 10979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-i2m1 9338  ax-1ne0 9339  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-ltxr 9411  df-rp 10980
This theorem is referenced by:  rpmulcld  11031  moddi  11750  rpexpcl  11868  discr  11985  reccn2  13058  expcnv  13309  rpmsubg  17720  ovolscalem2  20839  aaliou3lem7  21700  aaliou3lem9  21701  cosordlem  21872  logfac  21934  loglesqr  22081  divsqrsumlem  22258  basellem1  22303  pclogsum  22439  bclbnd  22504  bposlem7  22514  bposlem8  22515  bposlem9  22516  chebbnd1lem2  22604  dchrisum0lem3  22653  chpdifbndlem2  22688  pntrsumbnd2  22701  pntpbnd1a  22719  pntpbnd2  22721  pntibnd  22727  pntlemd  22728  pntlema  22730  pntlemb  22731  pntlemf  22739  pntlemo  22741  minvecolem3  24100  fprodrpcl  27316  rprisefaccl  27373  ftc1anclem7  28317  ftc1anc  28319  isbnd2  28526  wallispilem4  29709  wallispi  29711  taupilem1  35188  taupilem2  35189  taupi  35190
  Copyright terms: Public domain W3C validator