Theorem rpregt0d 8629

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 8622	. 2 |- ( ph -> A e. RR )
3	1	rpgt0d 8625	. 2 |- ( ph -> 0 < A )
4	2, 3	jca 290	1 |- ( ph -> ( A e. RR /\ 0 < A ) )

Syntax hints:   -> wi 4   /\ wa 97   e. wcel 1393   class class class wbr 3764   RRcr 6888   0cc0 6889   < clt 7060   RR+crp 8583

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-rp 8584

This theorem is referenced by:  reclt1d  8636  recgt1d  8637  ltrecd  8641  lerecd  8642  ltrec1d  8643  lerec2d  8644  lediv2ad  8645  ltdiv2d  8646  lediv2d  8647  ledivdivd  8648  divge0d  8663  ltmul1d  8664  ltmul2d  8665  lemul1d  8666  lemul2d  8667  ltdiv1d  8668  lediv1d  8669  ltmuldivd  8670  ltmuldiv2d  8671  lemuldivd  8672  lemuldiv2d  8673  ltdivmuld  8674  ltdivmul2d  8675  ledivmuld  8676  ledivmul2d  8677  ltdiv23d  8683  lediv23d  8684  lt2mul2divd  8685