Theorem rpregt0d 9911

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 9904	. 2 |- ( ph -> A e. RR )
3	1	rpgt0d 9907	. 2 |- ( ph -> 0 < A )
4	2, 3	jca 306	1 |- ( ph -> ( A e. RR /\ 0 < A ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   class class class wbr 4083   RRcr 8009   0cc0 8010   < clt 8192   RR+crp 9861

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-rp 9862

This theorem is referenced by:  reclt1d  9918  recgt1d  9919  ltrecd  9923  lerecd  9924  ltrec1d  9925  lerec2d  9926  lediv2ad  9927  ltdiv2d  9928  lediv2d  9929  ledivdivd  9930  divge0d  9945  ltmul1d  9946  ltmul2d  9947  lemul1d  9948  lemul2d  9949  ltdiv1d  9950  lediv1d  9951  ltmuldivd  9952  ltmuldiv2d  9953  lemuldivd  9954  lemuldiv2d  9955  ltdivmuld  9956  ltdivmul2d  9957  ledivmuld  9958  ledivmul2d  9959  ltdiv23d  9965  lediv23d  9966  lt2mul2divd  9973  mertenslemi1  12061  isprm6  12684