Theorem rpregt0d 9895

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 9888	. 2 |- ( ph -> A e. RR )
3	1	rpgt0d 9891	. 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 4082   RRcr 7994   0cc0 7995   < clt 8177   RR+crp 9845

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 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-rp 9846

This theorem is referenced by:  reclt1d  9902  recgt1d  9903  ltrecd  9907  lerecd  9908  ltrec1d  9909  lerec2d  9910  lediv2ad  9911  ltdiv2d  9912  lediv2d  9913  ledivdivd  9914  divge0d  9929  ltmul1d  9930  ltmul2d  9931  lemul1d  9932  lemul2d  9933  ltdiv1d  9934  lediv1d  9935  ltmuldivd  9936  ltmuldiv2d  9937  lemuldivd  9938  lemuldiv2d  9939  ltdivmuld  9940  ltdivmul2d  9941  ledivmuld  9942  ledivmul2d  9943  ltdiv23d  9949  lediv23d  9950  lt2mul2divd  9957  mertenslemi1  12041  isprm6  12664