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Theorem btwnzge0 10660
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 8275 . . . 4 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> 0 e. RR )
2 simplll 535 . . . 4 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> A e. RR )
3 simplr 529 . . . . . . 7 |- ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) -> N e. ZZ )
43zred 9700 . . . . . 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 8289 . . . . 5 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> 1 e. RR )
75, 6readdcld 8303 . . . 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 538 . . . 4 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> A < ( N + 1 ) )
101, 2, 7, 8, 9lelttrd 8398 . . 3 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> 0 < ( N + 1 ) )
11 0z 9588 . . . . 5 |- 0 e. ZZ
12 zleltp1 9633 . . . . 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 8275 . . 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 535 . . 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 537 . . 3 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ N ) -> N <_ A )
2116, 17, 18, 19, 20letrd 8397 . 2 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ N ) -> 0 <_ A )
2215, 21impbida 600 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 2203   class class class wbr 4109  (class class class)co 6050   RRcr 8126   0cc0 8127   1c1 8128   + caddc 8130   < clt 8308   <_ cle 8309   ZZcz 9577
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243
This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578
This theorem is referenced by:  2tnp1ge0ge0  10661
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