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Theorem btwnzge0 10369
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 8020 . . . 4 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> 0 e. RR )
2 simplll 533 . . . 4 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> A e. RR )
3 simplr 528 . . . . . . 7 |- ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) -> N e. ZZ )
43zred 9439 . . . . . 6 |- ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) -> N e. RR )
54adantr 276 . . . . 5 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> N e. RR )
6 1red 8034 . . . . 5 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> 1 e. RR )
75, 6readdcld 8049 . . . 4 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> ( N + 1 ) e. RR )
8 simpr 110 . . . 4 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> 0 <_ A )
9 simplrr 536 . . . 4 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> A < ( N + 1 ) )
101, 2, 7, 8, 9lelttrd 8144 . . 3 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> 0 < ( N + 1 ) )
11 0z 9328 . . . . 5 |- 0 e. ZZ
12 zleltp1 9372 . . . . 5 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( 0 <_ N <-> 0 < ( N + 1 ) ) )
1311, 12mpan 424 . . . 4 |- ( N e. ZZ -> ( 0 <_ N <-> 0 < ( N + 1 ) ) )
1413ad3antlr 493 . . 3 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> ( 0 <_ N <-> 0 < ( N + 1 ) ) )
1510, 14mpbird 167 . 2 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> 0 <_ N )
16 0red 8020 . . 3 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ N ) -> 0 e. RR )
174adantr 276 . . 3 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ N ) -> N e. RR )
18 simplll 533 . . 3 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ N ) -> A e. RR )
19 simpr 110 . . 3 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ N ) -> 0 <_ N )
20 simplrl 535 . . 3 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ N ) -> N <_ A )
2116, 17, 18, 19, 20letrd 8143 . 2 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ N ) -> 0 <_ A )
2215, 21impbida 596 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 104   <-> wb 105   e. wcel 2164   class class class wbr 4029  (class class class)co 5918   RRcr 7871   0cc0 7872   1c1 7873   + caddc 7875   < clt 8054   <_ cle 8055   ZZcz 9317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-z 9318
This theorem is referenced by:  2tnp1ge0ge0  10370
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