ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  btwnzge0 Unicode version
Theorem btwnzge0 10235
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 7900 . . . 4 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> 0 e. RR )
2 simplll 523 . . . 4 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> A e. RR )
3 simplr 520 . . . . . . 7 |- ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) -> N e. ZZ )
43zred 9313 . . . . . 6 |- ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) -> N e. RR )
54adantr 274 . . . . 5 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> N e. RR )
6 1red 7914 . . . . 5 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> 1 e. RR )
75, 6readdcld 7928 . . . 4 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> ( N + 1 ) e. RR )
8 simpr 109 . . . 4 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> 0 <_ A )
9 simplrr 526 . . . 4 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> A < ( N + 1 ) )
101, 2, 7, 8, 9lelttrd 8023 . . 3 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> 0 < ( N + 1 ) )
11 0z 9202 . . . . 5 |- 0 e. ZZ
12 zleltp1 9246 . . . . 5 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( 0 <_ N <-> 0 < ( N + 1 ) ) )
1311, 12mpan 421 . . . 4 |- ( N e. ZZ -> ( 0 <_ N <-> 0 < ( N + 1 ) ) )
1413ad3antlr 485 . . 3 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> ( 0 <_ N <-> 0 < ( N + 1 ) ) )
1510, 14mpbird 166 . 2 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> 0 <_ N )
16 0red 7900 . . 3 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ N ) -> 0 e. RR )
174adantr 274 . . 3 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ N ) -> N e. RR )
18 simplll 523 . . 3 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ N ) -> A e. RR )
19 simpr 109 . . 3 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ N ) -> 0 <_ N )
20 simplrl 525 . . 3 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ N ) -> N <_ A )
2116, 17, 18, 19, 20letrd 8022 . 2 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ N ) -> 0 <_ A )
2215, 21impbida 586 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 103   <-> wb 104   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   RRcr 7752   0cc0 7753   1c1 7754   + caddc 7756   < clt 7933   <_ cle 7934   ZZcz 9191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869
This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192
This theorem is referenced by:  2tnp1ge0ge0  10236
  Copyright terms: Public domain W3C validator