Theorem rpregt0d 9937

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 9930	. 2 |- ( ph -> A e. RR )
3	1	rpgt0d 9933	. 2 |- ( ph -> 0 < A )
4	2, 3	jca 306	1 |- ( ph -> ( A e. RR /\ 0 < A ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   class class class wbr 4088   RRcr 8030   0cc0 8031   < clt 8213   RR+crp 9887

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-rp 9888

This theorem is referenced by:  reclt1d  9944  recgt1d  9945  ltrecd  9949  lerecd  9950  ltrec1d  9951  lerec2d  9952  lediv2ad  9953  ltdiv2d  9954  lediv2d  9955  ledivdivd  9956  divge0d  9971  ltmul1d  9972  ltmul2d  9973  lemul1d  9974  lemul2d  9975  ltdiv1d  9976  lediv1d  9977  ltmuldivd  9978  ltmuldiv2d  9979  lemuldivd  9980  lemuldiv2d  9981  ltdivmuld  9982  ltdivmul2d  9983  ledivmuld  9984  ledivmul2d  9985  ltdiv23d  9991  lediv23d  9992  lt2mul2divd  9999  mertenslemi1  12095  isprm6  12718