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Theorem modqlt 10350
Description: The modulo operation is less than its second argument. (Contributed by Jim Kingdon, 18-Oct-2021.)
Assertion
Ref Expression
modqlt  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( A  mod  B )  < 
B )

Proof of Theorem modqlt
StepHypRef Expression
1 qcn 9651 . . . . . 6  |-  ( A  e.  QQ  ->  A  e.  CC )
213ad2ant1 1019 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  A  e.  CC )
3 qcn 9651 . . . . . 6  |-  ( B  e.  QQ  ->  B  e.  CC )
433ad2ant2 1020 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  B  e.  CC )
5 qre 9642 . . . . . . 7  |-  ( B  e.  QQ  ->  B  e.  RR )
653ad2ant2 1020 . . . . . 6  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  B  e.  RR )
7 simp3 1000 . . . . . 6  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  0  <  B )
86, 7gt0ap0d 8603 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  B #  0 )
92, 4, 8divcanap2d 8766 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( B  x.  ( A  /  B ) )  =  A )
109oveq1d 5905 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  (
( B  x.  ( A  /  B ) )  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B
) ) ) ) )
117gt0ne0d 8486 . . . . . 6  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  B  =/=  0 )
12 qdivcl 9660 . . . . . 6  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( A  /  B )  e.  QQ )
1311, 12syld3an3 1293 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( A  /  B )  e.  QQ )
14 qcn 9651 . . . . 5  |-  ( ( A  /  B )  e.  QQ  ->  ( A  /  B )  e.  CC )
1513, 14syl 14 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( A  /  B )  e.  CC )
1613flqcld 10294 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( |_ `  ( A  /  B ) )  e.  ZZ )
1716zcnd 9393 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( |_ `  ( A  /  B ) )  e.  CC )
184, 15, 17subdid 8388 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( B  x.  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) ) )  =  ( ( B  x.  ( A  /  B ) )  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
19 modqval 10341 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( A  mod  B )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B
) ) ) ) )
2010, 18, 193eqtr4rd 2232 . 2  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( A  mod  B )  =  ( B  x.  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) ) ) )
21 qfraclt1 10297 . . . . 5  |-  ( ( A  /  B )  e.  QQ  ->  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) )  <  1 )
2213, 21syl 14 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) )  <  1 )
234, 8dividapd 8760 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( B  /  B )  =  1 )
2422, 23breqtrrd 4045 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) )  <  ( B  /  B ) )
25 qre 9642 . . . . . 6  |-  ( ( A  /  B )  e.  QQ  ->  ( A  /  B )  e.  RR )
2613, 25syl 14 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( A  /  B )  e.  RR )
2716zred 9392 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( |_ `  ( A  /  B ) )  e.  RR )
2826, 27resubcld 8355 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) )  e.  RR )
29 ltmuldiv2 8849 . . . 4  |-  ( ( ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  e.  RR  /\  B  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( ( B  x.  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) ) )  <  B  <->  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  <  ( B  /  B ) ) )
3028, 6, 6, 7, 29syl112anc 1252 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  (
( B  x.  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) ) )  <  B  <->  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  <  ( B  /  B ) ) )
3124, 30mpbird 167 . 2  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( B  x.  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) ) )  <  B
)
3220, 31eqbrtrd 4039 1  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( A  mod  B )  < 
B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    /\ w3a 979    e. wcel 2159    =/= wne 2359   class class class wbr 4017   ` cfv 5230  (class class class)co 5890   CCcc 7826   RRcr 7827   0cc0 7828   1c1 7829    x. cmul 7833    < clt 8009    - cmin 8145    / cdiv 8646   QQcq 9636   |_cfl 10285    mod cmo 10339
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-mulrcl 7927  ax-addcom 7928  ax-mulcom 7929  ax-addass 7930  ax-mulass 7931  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-1rid 7935  ax-0id 7936  ax-rnegex 7937  ax-precex 7938  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944  ax-pre-mulgt0 7945  ax-pre-mulext 7946  ax-arch 7947
This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-id 4307  df-po 4310  df-iso 4311  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-fv 5238  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-reap 8549  df-ap 8556  df-div 8647  df-inn 8937  df-n0 9194  df-z 9271  df-q 9637  df-rp 9671  df-fl 10287  df-mod 10340
This theorem is referenced by:  modqelico  10351  zmodfz  10363  modqid2  10368  modqabs  10374  modqmuladdim  10384  modaddmodup  10404  modqsubdir  10410  divalglemnn  11940  divalgmod  11949  bezoutlemnewy  12014  bezoutlemstep  12015  eucalglt  12074  odzdvds  12262  fldivp1  12363  4sqlem6  12398  lgseisenlem1  14833
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