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Theorem btwnzge0 9434
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 7234 . . . 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 8602 . . . . . 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 7248 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  /\  0  <_  A )  -> 
1  e.  RR )
75, 6readdcld 7262 . . . 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 7353 . . 3  |-  ( ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  /\  0  <_  A )  -> 
0  <  ( N  +  1 ) )
11 0z 8495 . . . . 5  |-  0  e.  ZZ
12 zleltp1 8539 . . . . 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 7234 . . 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 7352 . 2  |-  ( ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  /\  0  <_  N )  -> 
0  <_  A )
2215, 21impbida 561 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 1434   class class class wbr 3805  (class class class)co 5563   RRcr 7094   0cc0 7095   1c1 7096    + caddc 7098    < clt 7267    <_ cle 7268   ZZcz 8484
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-ltadd 7206
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-inn 8159  df-n0 8408  df-z 8485
This theorem is referenced by:  2tnp1ge0ge0  9435
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