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Theorem flltdivnn0lt 10324
Description: The floor function of a division of a nonnegative integer by a positive integer is less than the division of a greater dividend by the same positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
Assertion
Ref Expression
flltdivnn0lt  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  ( K  <  N  ->  ( |_ `  ( K  /  L ) )  < 
( N  /  L
) ) )

Proof of Theorem flltdivnn0lt
StepHypRef Expression
1 simp1 999 . . . . . . 7  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  K  e.  NN0 )
21nn0zd 9393 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  K  e.  ZZ )
3 simp3 1001 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  L  e.  NN )
4 znq 9644 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  L  e.  NN )  ->  ( K  /  L
)  e.  QQ )
54flqcld 10297 . . . . . 6  |-  ( ( K  e.  ZZ  /\  L  e.  NN )  ->  ( |_ `  ( K  /  L ) )  e.  ZZ )
62, 3, 5syl2anc 411 . . . . 5  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  ( |_ `  ( K  /  L ) )  e.  ZZ )
76adantr 276 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( |_ `  ( K  /  L
) )  e.  ZZ )
87zred 9395 . . 3  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( |_ `  ( K  /  L
) )  e.  RR )
92adantr 276 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  K  e.  ZZ )
103adantr 276 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  L  e.  NN )
11 qre 9645 . . . . 5  |-  ( ( K  /  L )  e.  QQ  ->  ( K  /  L )  e.  RR )
124, 11syl 14 . . . 4  |-  ( ( K  e.  ZZ  /\  L  e.  NN )  ->  ( K  /  L
)  e.  RR )
139, 10, 12syl2anc 411 . . 3  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( K  /  L )  e.  RR )
14 simp2 1000 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  N  e.  NN0 )
1514nn0zd 9393 . . . . 5  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  N  e.  ZZ )
1615adantr 276 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  N  e.  ZZ )
17 znq 9644 . . . . 5  |-  ( ( N  e.  ZZ  /\  L  e.  NN )  ->  ( N  /  L
)  e.  QQ )
18 qre 9645 . . . . 5  |-  ( ( N  /  L )  e.  QQ  ->  ( N  /  L )  e.  RR )
1917, 18syl 14 . . . 4  |-  ( ( N  e.  ZZ  /\  L  e.  NN )  ->  ( N  /  L
)  e.  RR )
2016, 10, 19syl2anc 411 . . 3  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( N  /  L )  e.  RR )
21 fldivnn0le 10323 . . . . 5  |-  ( ( K  e.  NN0  /\  L  e.  NN )  ->  ( |_ `  ( K  /  L ) )  <_  ( K  /  L ) )
22213adant2 1018 . . . 4  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  ( |_ `  ( K  /  L ) )  <_ 
( K  /  L
) )
2322adantr 276 . . 3  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( |_ `  ( K  /  L
) )  <_  ( K  /  L ) )
24 simpr 110 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  K  <  N
)
25 nn0re 9205 . . . . . . 7  |-  ( K  e.  NN0  ->  K  e.  RR )
26 nn0re 9205 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  RR )
27 nnre 8946 . . . . . . . 8  |-  ( L  e.  NN  ->  L  e.  RR )
28 nngt0 8964 . . . . . . . 8  |-  ( L  e.  NN  ->  0  <  L )
2927, 28jca 306 . . . . . . 7  |-  ( L  e.  NN  ->  ( L  e.  RR  /\  0  <  L ) )
3025, 26, 293anim123i 1186 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  ( K  e.  RR  /\  N  e.  RR  /\  ( L  e.  RR  /\  0  <  L ) ) )
3130adantr 276 . . . . 5  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( K  e.  RR  /\  N  e.  RR  /\  ( L  e.  RR  /\  0  <  L ) ) )
32 ltdiv1 8845 . . . . 5  |-  ( ( K  e.  RR  /\  N  e.  RR  /\  ( L  e.  RR  /\  0  <  L ) )  -> 
( K  <  N  <->  ( K  /  L )  <  ( N  /  L ) ) )
3331, 32syl 14 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( K  < 
N  <->  ( K  /  L )  <  ( N  /  L ) ) )
3424, 33mpbid 147 . . 3  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( K  /  L )  <  ( N  /  L ) )
358, 13, 20, 23, 34lelttrd 8102 . 2  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( |_ `  ( K  /  L
) )  <  ( N  /  L ) )
3635ex 115 1  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  ( K  <  N  ->  ( |_ `  ( K  /  L ) )  < 
( N  /  L
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    e. wcel 2160   class class class wbr 4018   ` cfv 5232  (class class class)co 5892   RRcr 7830   0cc0 7831    < clt 8012    <_ cle 8013    / cdiv 8649   NNcn 8939   NN0cn0 9196   ZZcz 9273   QQcq 9639   |_cfl 10288
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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7922  ax-resscn 7923  ax-1cn 7924  ax-1re 7925  ax-icn 7926  ax-addcl 7927  ax-addrcl 7928  ax-mulcl 7929  ax-mulrcl 7930  ax-addcom 7931  ax-mulcom 7932  ax-addass 7933  ax-mulass 7934  ax-distr 7935  ax-i2m1 7936  ax-0lt1 7937  ax-1rid 7938  ax-0id 7939  ax-rnegex 7940  ax-precex 7941  ax-cnre 7942  ax-pre-ltirr 7943  ax-pre-ltwlin 7944  ax-pre-lttrn 7945  ax-pre-apti 7946  ax-pre-ltadd 7947  ax-pre-mulgt0 7948  ax-pre-mulext 7949  ax-arch 7950
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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-po 4311  df-iso 4312  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-pnf 8014  df-mnf 8015  df-xr 8016  df-ltxr 8017  df-le 8018  df-sub 8150  df-neg 8151  df-reap 8552  df-ap 8559  df-div 8650  df-inn 8940  df-n0 9197  df-z 9274  df-q 9640  df-rp 9674  df-fl 10290
This theorem is referenced by: (None)
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