Theorem rpregt0d 9706

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 9699	. 2 |- ( ph -> A e. RR )
3	1	rpgt0d 9702	. 2 |- ( ph -> 0 < A )
4	2, 3	jca 306	1 |- ( ph -> ( A e. RR /\ 0 < A ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2148   class class class wbr 4005   RRcr 7813   0cc0 7814   < clt 7995   RR+crp 9656

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-rp 9657

This theorem is referenced by:  reclt1d  9713  recgt1d  9714  ltrecd  9718  lerecd  9719  ltrec1d  9720  lerec2d  9721  lediv2ad  9722  ltdiv2d  9723  lediv2d  9724  ledivdivd  9725  divge0d  9740  ltmul1d  9741  ltmul2d  9742  lemul1d  9743  lemul2d  9744  ltdiv1d  9745  lediv1d  9746  ltmuldivd  9747  ltmuldiv2d  9748  lemuldivd  9749  lemuldiv2d  9750  ltdivmuld  9751  ltdivmul2d  9752  ledivmuld  9753  ledivmul2d  9754  ltdiv23d  9760  lediv23d  9761  lt2mul2divd  9768  mertenslemi1  11546  isprm6  12150