Theorem rpregt0d 8861

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 8854	. 2 |- ( ph -> A e. RR )
3	1	rpgt0d 8857	. 2 |- ( ph -> 0 < A )
4	2, 3	jca 300	1 |- ( ph -> ( A e. RR /\ 0 < A ) )

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1434   class class class wbr 3793   RRcr 7042   0cc0 7043   < clt 7215   RR+crp 8815

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rab 2358  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-rp 8816

This theorem is referenced by:  reclt1d  8868  recgt1d  8869  ltrecd  8873  lerecd  8874  ltrec1d  8875  lerec2d  8876  lediv2ad  8877  ltdiv2d  8878  lediv2d  8879  ledivdivd  8880  divge0d  8895  ltmul1d  8896  ltmul2d  8897  lemul1d  8898  lemul2d  8899  ltdiv1d  8900  lediv1d  8901  ltmuldivd  8902  ltmuldiv2d  8903  lemuldivd  8904  lemuldiv2d  8905  ltdivmuld  8906  ltdivmul2d  8907  ledivmuld  8908  ledivmul2d  8909  ltdiv23d  8915  lediv23d  8916  lt2mul2divd  8917  isprm6  10670