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Theorem flltdivnn0lt 9711
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 943 . . . . . . 7  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  K  e.  NN0 )
21nn0zd 8866 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  K  e.  ZZ )
3 simp3 945 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  L  e.  NN )
4 znq 9109 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  L  e.  NN )  ->  ( K  /  L
)  e.  QQ )
54flqcld 9684 . . . . . 6  |-  ( ( K  e.  ZZ  /\  L  e.  NN )  ->  ( |_ `  ( K  /  L ) )  e.  ZZ )
62, 3, 5syl2anc 403 . . . . 5  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  ( |_ `  ( K  /  L ) )  e.  ZZ )
76adantr 270 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( |_ `  ( K  /  L
) )  e.  ZZ )
87zred 8868 . . 3  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( |_ `  ( K  /  L
) )  e.  RR )
92adantr 270 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  K  e.  ZZ )
103adantr 270 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  L  e.  NN )
11 qre 9110 . . . . 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 403 . . 3  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( K  /  L )  e.  RR )
14 simp2 944 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  N  e.  NN0 )
1514nn0zd 8866 . . . . 5  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  N  e.  ZZ )
1615adantr 270 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  N  e.  ZZ )
17 znq 9109 . . . . 5  |-  ( ( N  e.  ZZ  /\  L  e.  NN )  ->  ( N  /  L
)  e.  QQ )
18 qre 9110 . . . . 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 403 . . 3  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( N  /  L )  e.  RR )
21 fldivnn0le 9710 . . . . 5  |-  ( ( K  e.  NN0  /\  L  e.  NN )  ->  ( |_ `  ( K  /  L ) )  <_  ( K  /  L ) )
22213adant2 962 . . . 4  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  ( |_ `  ( K  /  L ) )  <_ 
( K  /  L
) )
2322adantr 270 . . 3  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( |_ `  ( K  /  L
) )  <_  ( K  /  L ) )
24 simpr 108 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  K  <  N
)
25 nn0re 8682 . . . . . . 7  |-  ( K  e.  NN0  ->  K  e.  RR )
26 nn0re 8682 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  RR )
27 nnre 8429 . . . . . . . 8  |-  ( L  e.  NN  ->  L  e.  RR )
28 nngt0 8447 . . . . . . . 8  |-  ( L  e.  NN  ->  0  <  L )
2927, 28jca 300 . . . . . . 7  |-  ( L  e.  NN  ->  ( L  e.  RR  /\  0  <  L ) )
3025, 26, 293anim123i 1128 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  ( K  e.  RR  /\  N  e.  RR  /\  ( L  e.  RR  /\  0  <  L ) ) )
3130adantr 270 . . . . 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 8329 . . . . 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 145 . . 3  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( K  /  L )  <  ( N  /  L ) )
358, 13, 20, 23, 34lelttrd 7608 . 2  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( |_ `  ( K  /  L
) )  <  ( N  /  L ) )
3635ex 113 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 102    <-> wb 103    /\ w3a 924    e. wcel 1438   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   RRcr 7349   0cc0 7350    < clt 7522    <_ cle 7523    / cdiv 8139   NNcn 8422   NN0cn0 8673   ZZcz 8750   QQcq 9104   |_cfl 9675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463  ax-arch 7464
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-po 4123  df-iso 4124  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-n0 8674  df-z 8751  df-q 9105  df-rp 9135  df-fl 9677
This theorem is referenced by: (None)
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