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Theorem btwnzge0 9703
Description: A real bounded between an integer and its successor is nonnegative iff the integer is nonnegative. Second half of Lemma 13-4.1 of [Gleason] p. 217. (Contributed by NM, 12-Mar-2005.)
Assertion
Ref Expression
btwnzge0  |-  ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  ->  (
0  <_  A  <->  0  <_  N ) )

Proof of Theorem btwnzge0
StepHypRef Expression
1 0red 7487 . . . 4  |-  ( ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  /\  0  <_  A )  -> 
0  e.  RR )
2 simplll 500 . . . 4  |-  ( ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  /\  0  <_  A )  ->  A  e.  RR )
3 simplr 497 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  ->  N  e.  ZZ )
43zred 8866 . . . . . 6  |-  ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  ->  N  e.  RR )
54adantr 270 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  /\  0  <_  A )  ->  N  e.  RR )
6 1red 7501 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  /\  0  <_  A )  -> 
1  e.  RR )
75, 6readdcld 7515 . . . 4  |-  ( ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  /\  0  <_  A )  -> 
( N  +  1 )  e.  RR )
8 simpr 108 . . . 4  |-  ( ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  /\  0  <_  A )  -> 
0  <_  A )
9 simplrr 503 . . . 4  |-  ( ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  /\  0  <_  A )  ->  A  <  ( N  + 
1 ) )
101, 2, 7, 8, 9lelttrd 7606 . . 3  |-  ( ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  /\  0  <_  A )  -> 
0  <  ( N  +  1 ) )
11 0z 8759 . . . . 5  |-  0  e.  ZZ
12 zleltp1 8803 . . . . 5  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  <_  N  <->  0  <  ( N  + 
1 ) ) )
1311, 12mpan 415 . . . 4  |-  ( N  e.  ZZ  ->  (
0  <_  N  <->  0  <  ( N  +  1 ) ) )
1413ad3antlr 477 . . 3  |-  ( ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  /\  0  <_  A )  -> 
( 0  <_  N  <->  0  <  ( N  + 
1 ) ) )
1510, 14mpbird 165 . 2  |-  ( ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  /\  0  <_  A )  -> 
0  <_  N )
16 0red 7487 . . 3  |-  ( ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  /\  0  <_  N )  -> 
0  e.  RR )
174adantr 270 . . 3  |-  ( ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  /\  0  <_  N )  ->  N  e.  RR )
18 simplll 500 . . 3  |-  ( ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  /\  0  <_  N )  ->  A  e.  RR )
19 simpr 108 . . 3  |-  ( ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  /\  0  <_  N )  -> 
0  <_  N )
20 simplrl 502 . . 3  |-  ( ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  /\  0  <_  N )  ->  N  <_  A )
2116, 17, 18, 19, 20letrd 7605 . 2  |-  ( ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  /\  0  <_  N )  -> 
0  <_  A )
2215, 21impbida 563 1  |-  ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  ->  (
0  <_  A  <->  0  <_  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1438   class class class wbr 3845  (class class class)co 5652   RRcr 7347   0cc0 7348   1c1 7349    + caddc 7351    < clt 7520    <_ cle 7521   ZZcz 8748
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-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-addcom 7443  ax-addass 7445  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-0id 7451  ax-rnegex 7452  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-ltadd 7459
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-inn 8421  df-n0 8672  df-z 8749
This theorem is referenced by:  2tnp1ge0ge0  9704
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