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Theorem rpregt0d 9702
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 9695 . 2 (𝜑𝐴 ∈ ℝ)
31rpgt0d 9698 . 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 4003  cr 7809  0cc0 7810   < clt 7991  +crp 9652
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 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-rp 9653
This theorem is referenced by:  reclt1d  9709  recgt1d  9710  ltrecd  9714  lerecd  9715  ltrec1d  9716  lerec2d  9717  lediv2ad  9718  ltdiv2d  9719  lediv2d  9720  ledivdivd  9721  divge0d  9736  ltmul1d  9737  ltmul2d  9738  lemul1d  9739  lemul2d  9740  ltdiv1d  9741  lediv1d  9742  ltmuldivd  9743  ltmuldiv2d  9744  lemuldivd  9745  lemuldiv2d  9746  ltdivmuld  9747  ltdivmul2d  9748  ledivmuld  9749  ledivmul2d  9750  ltdiv23d  9756  lediv23d  9757  lt2mul2divd  9764  mertenslemi1  11542  isprm6  12146
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