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Theorem btwnzge0 10256
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 7921 . . . 4 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> 0 e. RR )
2 simplll 528 . . . 4 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> A e. RR )
3 simplr 525 . . . . . . 7 |- ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) -> N e. ZZ )
43zred 9334 . . . . . 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 7935 . . . . 5 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> 1 e. RR )
75, 6readdcld 7949 . . . 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 531 . . . 4 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> A < ( N + 1 ) )
101, 2, 7, 8, 9lelttrd 8044 . . 3 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> 0 < ( N + 1 ) )
11 0z 9223 . . . . 5 |- 0 e. ZZ
12 zleltp1 9267 . . . . 5 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( 0 <_ N <-> 0 < ( N + 1 ) ) )
1311, 12mpan 422 . . . 4 |- ( N e. ZZ -> ( 0 <_ N <-> 0 < ( N + 1 ) ) )
1413ad3antlr 490 . . 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 7921 . . 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 528 . . 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 530 . . 3 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ N ) -> N <_ A )
2116, 17, 18, 19, 20letrd 8043 . 2 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ N ) -> 0 <_ A )
2215, 21impbida 591 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 2141   class class class wbr 3989  (class class class)co 5853   RRcr 7773   0cc0 7774   1c1 7775   + caddc 7777   < clt 7954   <_ cle 7955   ZZcz 9212
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213
This theorem is referenced by:  2tnp1ge0ge0  10257
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