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Theorem rpregt0 12395
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 12383 . 2 (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
21biimpi 218 1 (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wcel 2107   class class class wbr 5057  cr 10528  0cc0 10529   < clt 10667  +crp 12381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-rp 12382
This theorem is referenced by:  rpne0  12397  divlt1lt  12450  divle1le  12451  ledivge1le  12452  nnledivrp  12493  modge0  13239  modlt  13240  modid  13256  modmuladdnn0  13275  expnlbnd  13586  o1fsum  15160  isprm6  16050  gexexlem  18964  lmnn  23858  aaliou2b  24922  harmonicbnd4  25580  logfaclbnd  25790  logfacrlim  25792  chto1ub  26044  vmadivsum  26050  dchrmusumlema  26061  dchrvmasumlem2  26066  dchrisum0lem2a  26085  dchrisum0lem2  26086  dchrisum0lem3  26087  mulogsumlem  26099  mulog2sumlem2  26103  selberg2lem  26118  selberg3lem1  26125  pntrmax  26132  pntrsumo1  26133  pntibndlem3  26160  divge1b  44557  divgt1b  44558
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