Theorem rpregt0d 9639

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 9632	. 2 |- ( ph -> A e. RR )
3	1	rpgt0d 9635	. 2 |- ( ph -> 0 < A )
4	2, 3	jca 304	1 |- ( ph -> ( A e. RR /\ 0 < A ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2136   class class class wbr 3982   RRcr 7752   0cc0 7753   < clt 7933   RR+crp 9589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-rp 9590

This theorem is referenced by:  reclt1d  9646  recgt1d  9647  ltrecd  9651  lerecd  9652  ltrec1d  9653  lerec2d  9654  lediv2ad  9655  ltdiv2d  9656  lediv2d  9657  ledivdivd  9658  divge0d  9673  ltmul1d  9674  ltmul2d  9675  lemul1d  9676  lemul2d  9677  ltdiv1d  9678  lediv1d  9679  ltmuldivd  9680  ltmuldiv2d  9681  lemuldivd  9682  lemuldiv2d  9683  ltdivmuld  9684  ltdivmul2d  9685  ledivmuld  9686  ledivmul2d  9687  ltdiv23d  9693  lediv23d  9694  lt2mul2divd  9701  mertenslemi1  11476  isprm6  12079