Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rpregt0 Structured version   Visualization version   GIF version

Theorem rpregt0 12049
 Description: A positive real is a positive real number. (Contributed by NM, 11-Nov-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
Assertion
Ref Expression
rpregt0 (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 < 𝐴))

Proof of Theorem rpregt0
StepHypRef Expression
1 elrp 12037 . 2 (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
21biimpi 206 1 (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∈ wcel 2145   class class class wbr 4786  ℝcr 10137  0cc0 10138   < clt 10276  ℝ+crp 12035 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-rp 12036 This theorem is referenced by:  rpne0  12051  divlt1lt  12102  divle1le  12103  ledivge1le  12104  nnledivrp  12145  modge0  12886  modlt  12887  modid  12903  modmuladdnn0  12922  expnlbnd  13201  o1fsum  14752  isprm6  15633  gexexlem  18462  lmnn  23280  aaliou2b  24316  harmonicbnd4  24958  logfaclbnd  25168  logfacrlim  25170  chto1ub  25386  vmadivsum  25392  dchrmusumlema  25403  dchrvmasumlem2  25408  dchrisum0lem2a  25427  dchrisum0lem2  25428  dchrisum0lem3  25429  mulogsumlem  25441  mulog2sumlem2  25445  selberg2lem  25460  selberg3lem1  25467  pntrmax  25474  pntrsumo1  25475  pntibndlem3  25502  divge1b  42830  divgt1b  42831
 Copyright terms: Public domain W3C validator