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Theorem btwnzge0 10684
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 8291 . . . 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 9718 . . . . . 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 8305 . . . . 5 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> 1 e. RR )
75, 6readdcld 8319 . . . 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 8414 . . 3 |- ( ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) /\ 0 <_ A ) -> 0 < ( N + 1 ) )
11 0z 9605 . . . . 5 |- 0 e. ZZ
12 zleltp1 9650 . . . . 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 8291 . . 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 8413 . 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 2205   class class class wbr 4114  (class class class)co 6058   RRcr 8142   0cc0 8143   1c1 8144   + caddc 8146   < clt 8324   <_ cle 8325   ZZcz 9594
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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259
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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595
This theorem is referenced by:  2tnp1ge0ge0  10685
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