Theorem rpregt0d 9778

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 9771	. 2 |- ( ph -> A e. RR )
3	1	rpgt0d 9774	. 2 |- ( ph -> 0 < A )
4	2, 3	jca 306	1 |- ( ph -> ( A e. RR /\ 0 < A ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2167   class class class wbr 4033   RRcr 7878   0cc0 7879   < clt 8061   RR+crp 9728

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-rp 9729

This theorem is referenced by:  reclt1d  9785  recgt1d  9786  ltrecd  9790  lerecd  9791  ltrec1d  9792  lerec2d  9793  lediv2ad  9794  ltdiv2d  9795  lediv2d  9796  ledivdivd  9797  divge0d  9812  ltmul1d  9813  ltmul2d  9814  lemul1d  9815  lemul2d  9816  ltdiv1d  9817  lediv1d  9818  ltmuldivd  9819  ltmuldiv2d  9820  lemuldivd  9821  lemuldiv2d  9822  ltdivmuld  9823  ltdivmul2d  9824  ledivmuld  9825  ledivmul2d  9826  ltdiv23d  9832  lediv23d  9833  lt2mul2divd  9840  mertenslemi1  11700  isprm6  12315