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Theorem modqlt 9993
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 9322 . . . . . 6  |-  ( A  e.  QQ  ->  A  e.  CC )
213ad2ant1 983 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  A  e.  CC )
3 qcn 9322 . . . . . 6  |-  ( B  e.  QQ  ->  B  e.  CC )
433ad2ant2 984 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  B  e.  CC )
5 qre 9313 . . . . . . 7  |-  ( B  e.  QQ  ->  B  e.  RR )
653ad2ant2 984 . . . . . 6  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  B  e.  RR )
7 simp3 964 . . . . . 6  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  0  <  B )
86, 7gt0ap0d 8303 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  B #  0 )
92, 4, 8divcanap2d 8459 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( B  x.  ( A  /  B ) )  =  A )
109oveq1d 5741 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  (
( B  x.  ( A  /  B ) )  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B
) ) ) ) )
117gt0ne0d 8187 . . . . . 6  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  B  =/=  0 )
12 qdivcl 9331 . . . . . 6  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( A  /  B )  e.  QQ )
1311, 12syld3an3 1242 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( A  /  B )  e.  QQ )
14 qcn 9322 . . . . 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 9937 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( |_ `  ( A  /  B ) )  e.  ZZ )
1716zcnd 9072 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( |_ `  ( A  /  B ) )  e.  CC )
184, 15, 17subdid 8089 . . 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 9984 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( A  mod  B )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B
) ) ) ) )
2010, 18, 193eqtr4rd 2156 . 2  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( A  mod  B )  =  ( B  x.  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) ) ) )
21 qfraclt1 9940 . . . . 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 8453 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( B  /  B )  =  1 )
2422, 23breqtrrd 3919 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) )  <  ( B  /  B ) )
25 qre 9313 . . . . . 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 9071 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( |_ `  ( A  /  B ) )  e.  RR )
2826, 27resubcld 8056 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) )  e.  RR )
29 ltmuldiv2 8537 . . . 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 1201 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  (
( B  x.  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) ) )  <  B  <->  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  <  ( B  /  B ) ) )
3124, 30mpbird 166 . 2  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( B  x.  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) ) )  <  B
)
3220, 31eqbrtrd 3913 1  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( A  mod  B )  < 
B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    /\ w3a 943    e. wcel 1461    =/= wne 2280   class class class wbr 3893   ` cfv 5079  (class class class)co 5726   CCcc 7539   RRcr 7540   0cc0 7541   1c1 7542    x. cmul 7546    < clt 7718    - cmin 7850    / cdiv 8339   QQcq 9307   |_cfl 9928    mod cmo 9982
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 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-mulrcl 7638  ax-addcom 7639  ax-mulcom 7640  ax-addass 7641  ax-mulass 7642  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-1rid 7646  ax-0id 7647  ax-rnegex 7648  ax-precex 7649  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-apti 7654  ax-pre-ltadd 7655  ax-pre-mulgt0 7656  ax-pre-mulext 7657  ax-arch 7658
This theorem depends on definitions:  df-bi 116  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rmo 2396  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-po 4176  df-iso 4177  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-reap 8249  df-ap 8256  df-div 8340  df-inn 8625  df-n0 8876  df-z 8953  df-q 9308  df-rp 9338  df-fl 9930  df-mod 9983
This theorem is referenced by:  modqelico  9994  zmodfz  10006  modqid2  10011  modqabs  10017  modqmuladdim  10027  modaddmodup  10047  modqsubdir  10053  divalglemnn  11457  divalgmod  11466  bezoutlemnewy  11524  bezoutlemstep  11525  eucalglt  11578
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