Theorem rpregt0d 9630

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 9623	. 2 |- ( ph -> A e. RR )
3	1	rpgt0d 9626	. 2 |- ( ph -> 0 < A )
4	2, 3	jca 304	1 |- ( ph -> ( A e. RR /\ 0 < A ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2135   class class class wbr 3976   RRcr 7743   0cc0 7744   < clt 7924   RR+crp 9580

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rab 2451  df-v 2723  df-un 3115  df-in 3117  df-ss 3124  df-sn 3576  df-pr 3577  df-op 3579  df-br 3977  df-rp 9581

This theorem is referenced by:  reclt1d  9637  recgt1d  9638  ltrecd  9642  lerecd  9643  ltrec1d  9644  lerec2d  9645  lediv2ad  9646  ltdiv2d  9647  lediv2d  9648  ledivdivd  9649  divge0d  9664  ltmul1d  9665  ltmul2d  9666  lemul1d  9667  lemul2d  9668  ltdiv1d  9669  lediv1d  9670  ltmuldivd  9671  ltmuldiv2d  9672  lemuldivd  9673  lemuldiv2d  9674  ltdivmuld  9675  ltdivmul2d  9676  ledivmuld  9677  ledivmul2d  9678  ltdiv23d  9684  lediv23d  9685  lt2mul2divd  9692  mertenslemi1  11462  isprm6  12056