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Theorem rpregt0d 11021
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
StepHypRef Expression
1 rpred.1 . . 3  |-  ( ph  ->  A  e.  RR+ )
21rpred 11015 . 2  |-  ( ph  ->  A  e.  RR )
31rpgt0d 11018 . 2  |-  ( ph  ->  0  <  A )
42, 3jca 529 1  |-  ( ph  ->  ( A  e.  RR  /\  0  <  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1755   class class class wbr 4280   RRcr 9269   0cc0 9270    < clt 9406   RR+crp 10979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-br 4281  df-rp 10980
This theorem is referenced by:  reclt1d  11028  recgt1d  11029  ltrecd  11033  lerecd  11034  ltrec1d  11035  lerec2d  11036  lediv2ad  11037  ltdiv2d  11038  lediv2d  11039  ledivdivd  11040  divge0d  11051  ltmul1d  11052  ltmul2d  11053  lemul1d  11054  lemul2d  11055  ltdiv1d  11056  lediv1d  11057  ltmuldivd  11058  ltmuldiv2d  11059  lemuldivd  11060  lemuldiv2d  11061  ltdivmuld  11062  ltdivmul2d  11063  ledivmuld  11064  ledivmul2d  11065  ltdiv23d  11071  lediv23d  11072  lt2mul2divd  11073  mertenslem1  13327  isprm6  13778  nmoi  20149  icopnfhmeo  20357  nmoleub2lem3  20512  lmnn  20616  ovolscalem1  20838  aaliou2b  21692  birthdaylem3  22232  fsumharmonic  22290  bcmono  22501  chtppilim  22609  dchrisum0lem1a  22620  dchrvmasumiflem1  22635  dchrisum0lem1b  22649  dchrisum0lem1  22650  mulog2sumlem2  22669  selberg3lem1  22691  pntrsumo1  22699  pntibndlem1  22723  pntibndlem3  22726  pntlemr  22736  pntlemj  22737  ostth3  22772  ttgcontlem1  22954  minvecolem3  24100  lnconi  25260  stoweidlem14  29655  stoweidlem34  29675  stoweidlem42  29683  stoweidlem51  29692  stoweidlem59  29700  stirlinglem5  29719
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