ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rpregt0d Unicode version

Theorem rpregt0d 9458
Description: A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
Hypothesis
Ref Expression
rpred.1  |-  ( ph  ->  A  e.  RR+ )
Assertion
Ref Expression
rpregt0d  |-  ( ph  ->  ( A  e.  RR  /\  0  <  A ) )

Proof of Theorem rpregt0d
StepHypRef Expression
1 rpred.1 . . 3  |-  ( ph  ->  A  e.  RR+ )
21rpred 9451 . 2  |-  ( ph  ->  A  e.  RR )
31rpgt0d 9454 . 2  |-  ( ph  ->  0  <  A )
42, 3jca 304 1  |-  ( ph  ->  ( A  e.  RR  /\  0  <  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1465   class class class wbr 3899   RRcr 7587   0cc0 7588    < clt 7768   RR+crp 9409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rab 2402  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-rp 9410
This theorem is referenced by:  reclt1d  9465  recgt1d  9466  ltrecd  9470  lerecd  9471  ltrec1d  9472  lerec2d  9473  lediv2ad  9474  ltdiv2d  9475  lediv2d  9476  ledivdivd  9477  divge0d  9492  ltmul1d  9493  ltmul2d  9494  lemul1d  9495  lemul2d  9496  ltdiv1d  9497  lediv1d  9498  ltmuldivd  9499  ltmuldiv2d  9500  lemuldivd  9501  lemuldiv2d  9502  ltdivmuld  9503  ltdivmul2d  9504  ledivmuld  9505  ledivmul2d  9506  ltdiv23d  9512  lediv23d  9513  lt2mul2divd  9520  mertenslemi1  11272  isprm6  11752
  Copyright terms: Public domain W3C validator