Theorem rpregt0d 9928

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 9921	. 2 |- ( ph -> A e. RR )
3	1	rpgt0d 9924	. 2 |- ( ph -> 0 < A )
4	2, 3	jca 306	1 |- ( ph -> ( A e. RR /\ 0 < A ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   class class class wbr 4086   RRcr 8021   0cc0 8022   < clt 8204   RR+crp 9878

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-rp 9879

This theorem is referenced by:  reclt1d  9935  recgt1d  9936  ltrecd  9940  lerecd  9941  ltrec1d  9942  lerec2d  9943  lediv2ad  9944  ltdiv2d  9945  lediv2d  9946  ledivdivd  9947  divge0d  9962  ltmul1d  9963  ltmul2d  9964  lemul1d  9965  lemul2d  9966  ltdiv1d  9967  lediv1d  9968  ltmuldivd  9969  ltmuldiv2d  9970  lemuldivd  9971  lemuldiv2d  9972  ltdivmuld  9973  ltdivmul2d  9974  ledivmuld  9975  ledivmul2d  9976  ltdiv23d  9982  lediv23d  9983  lt2mul2divd  9990  mertenslemi1  12086  isprm6  12709