Theorem rpregt0d 9660

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 9653	. 2 |- ( ph -> A e. RR )
3	1	rpgt0d 9656	. 2 |- ( ph -> 0 < A )
4	2, 3	jca 304	1 |- ( ph -> ( A e. RR /\ 0 < A ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   class class class wbr 3989   RRcr 7773   0cc0 7774   < clt 7954   RR+crp 9610

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-rp 9611

This theorem is referenced by:  reclt1d  9667  recgt1d  9668  ltrecd  9672  lerecd  9673  ltrec1d  9674  lerec2d  9675  lediv2ad  9676  ltdiv2d  9677  lediv2d  9678  ledivdivd  9679  divge0d  9694  ltmul1d  9695  ltmul2d  9696  lemul1d  9697  lemul2d  9698  ltdiv1d  9699  lediv1d  9700  ltmuldivd  9701  ltmuldiv2d  9702  lemuldivd  9703  lemuldiv2d  9704  ltdivmuld  9705  ltdivmul2d  9706  ledivmuld  9707  ledivmul2d  9708  ltdiv23d  9714  lediv23d  9715  lt2mul2divd  9722  mertenslemi1  11498  isprm6  12101