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Theorem rpregt0d 9705
Description: A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
Hypothesis
Ref Expression
rpred.1 (𝜑𝐴 ∈ ℝ+)
Assertion
Ref Expression
rpregt0d (𝜑 → (𝐴 ∈ ℝ ∧ 0 < 𝐴))

Proof of Theorem rpregt0d
StepHypRef Expression
1 rpred.1 . . 3 (𝜑𝐴 ∈ ℝ+)
21rpred 9698 . 2 (𝜑𝐴 ∈ ℝ)
31rpgt0d 9701 . 2 (𝜑 → 0 < 𝐴)
42, 3jca 306 1 (𝜑 → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wcel 2148   class class class wbr 4005  cr 7812  0cc0 7813   < clt 7994  +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-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-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-rp 9656
This theorem is referenced by:  reclt1d  9712  recgt1d  9713  ltrecd  9717  lerecd  9718  ltrec1d  9719  lerec2d  9720  lediv2ad  9721  ltdiv2d  9722  lediv2d  9723  ledivdivd  9724  divge0d  9739  ltmul1d  9740  ltmul2d  9741  lemul1d  9742  lemul2d  9743  ltdiv1d  9744  lediv1d  9745  ltmuldivd  9746  ltmuldiv2d  9747  lemuldivd  9748  lemuldiv2d  9749  ltdivmuld  9750  ltdivmul2d  9751  ledivmuld  9752  ledivmul2d  9753  ltdiv23d  9759  lediv23d  9760  lt2mul2divd  9767  mertenslemi1  11545  isprm6  12149
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