Theorem rpregt0d 9797

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

Step	Hyp	Ref	Expression
1		rpred.1	. . 3 |- ( ph -> A e. RR+ )
2	1	rpred 9790	. 2 |- ( ph -> A e. RR )
3	1	rpgt0d 9793	. 2 |- ( ph -> 0 < A )
4	2, 3	jca 306	1 |- ( ph -> ( A e. RR /\ 0 < A ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2167   class class class wbr 4034   RRcr 7897   0cc0 7898   < clt 8080   RR+crp 9747

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-rp 9748

This theorem is referenced by:  reclt1d  9804  recgt1d  9805  ltrecd  9809  lerecd  9810  ltrec1d  9811  lerec2d  9812  lediv2ad  9813  ltdiv2d  9814  lediv2d  9815  ledivdivd  9816  divge0d  9831  ltmul1d  9832  ltmul2d  9833  lemul1d  9834  lemul2d  9835  ltdiv1d  9836  lediv1d  9837  ltmuldivd  9838  ltmuldiv2d  9839  lemuldivd  9840  lemuldiv2d  9841  ltdivmuld  9842  ltdivmul2d  9843  ledivmuld  9844  ledivmul2d  9845  ltdiv23d  9851  lediv23d  9852  lt2mul2divd  9859  mertenslemi1  11719  isprm6  12342