Theorem rpregt0d 9772

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 9765	. 2 |- ( ph -> A e. RR )
3	1	rpgt0d 9768	. 2 |- ( ph -> 0 < A )
4	2, 3	jca 306	1 |- ( ph -> ( A e. RR /\ 0 < A ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2164   class class class wbr 4030   RRcr 7873   0cc0 7874   < clt 8056   RR+crp 9722

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-rp 9723

This theorem is referenced by:  reclt1d  9779  recgt1d  9780  ltrecd  9784  lerecd  9785  ltrec1d  9786  lerec2d  9787  lediv2ad  9788  ltdiv2d  9789  lediv2d  9790  ledivdivd  9791  divge0d  9806  ltmul1d  9807  ltmul2d  9808  lemul1d  9809  lemul2d  9810  ltdiv1d  9811  lediv1d  9812  ltmuldivd  9813  ltmuldiv2d  9814  lemuldivd  9815  lemuldiv2d  9816  ltdivmuld  9817  ltdivmul2d  9818  ledivmuld  9819  ledivmul2d  9820  ltdiv23d  9826  lediv23d  9827  lt2mul2divd  9834  mertenslemi1  11681  isprm6  12288