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Theorem flltdivnn0lt 9970
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 964 . . . . . . 7  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  K  e.  NN0 )
21nn0zd 9075 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  K  e.  ZZ )
3 simp3 966 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  L  e.  NN )
4 znq 9318 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  L  e.  NN )  ->  ( K  /  L
)  e.  QQ )
54flqcld 9943 . . . . . 6  |-  ( ( K  e.  ZZ  /\  L  e.  NN )  ->  ( |_ `  ( K  /  L ) )  e.  ZZ )
62, 3, 5syl2anc 406 . . . . 5  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  ( |_ `  ( K  /  L ) )  e.  ZZ )
76adantr 272 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( |_ `  ( K  /  L
) )  e.  ZZ )
87zred 9077 . . 3  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( |_ `  ( K  /  L
) )  e.  RR )
92adantr 272 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  K  e.  ZZ )
103adantr 272 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  L  e.  NN )
11 qre 9319 . . . . 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 406 . . 3  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( K  /  L )  e.  RR )
14 simp2 965 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  N  e.  NN0 )
1514nn0zd 9075 . . . . 5  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  N  e.  ZZ )
1615adantr 272 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  N  e.  ZZ )
17 znq 9318 . . . . 5  |-  ( ( N  e.  ZZ  /\  L  e.  NN )  ->  ( N  /  L
)  e.  QQ )
18 qre 9319 . . . . 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 406 . . 3  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( N  /  L )  e.  RR )
21 fldivnn0le 9969 . . . . 5  |-  ( ( K  e.  NN0  /\  L  e.  NN )  ->  ( |_ `  ( K  /  L ) )  <_  ( K  /  L ) )
22213adant2 983 . . . 4  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  ( |_ `  ( K  /  L ) )  <_ 
( K  /  L
) )
2322adantr 272 . . 3  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( |_ `  ( K  /  L
) )  <_  ( K  /  L ) )
24 simpr 109 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  K  <  N
)
25 nn0re 8890 . . . . . . 7  |-  ( K  e.  NN0  ->  K  e.  RR )
26 nn0re 8890 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  RR )
27 nnre 8637 . . . . . . . 8  |-  ( L  e.  NN  ->  L  e.  RR )
28 nngt0 8655 . . . . . . . 8  |-  ( L  e.  NN  ->  0  <  L )
2927, 28jca 302 . . . . . . 7  |-  ( L  e.  NN  ->  ( L  e.  RR  /\  0  <  L ) )
3025, 26, 293anim123i 1149 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  ( K  e.  RR  /\  N  e.  RR  /\  ( L  e.  RR  /\  0  <  L ) ) )
3130adantr 272 . . . . 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 8536 . . . . 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 146 . . 3  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( K  /  L )  <  ( N  /  L ) )
358, 13, 20, 23, 34lelttrd 7810 . 2  |-  ( ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  /\  K  <  N )  ->  ( |_ `  ( K  /  L
) )  <  ( N  /  L ) )
3635ex 114 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 103    <-> wb 104    /\ w3a 945    e. wcel 1463   class class class wbr 3895   ` cfv 5081  (class class class)co 5728   RRcr 7546   0cc0 7547    < clt 7724    <_ cle 7725    / cdiv 8345   NNcn 8630   NN0cn0 8881   ZZcz 8958   QQcq 9313   |_cfl 9934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663  ax-arch 7664
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-po 4178  df-iso 4179  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-n0 8882  df-z 8959  df-q 9314  df-rp 9344  df-fl 9936
This theorem is referenced by: (None)
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