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Theorem modqlt 10136
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 9452 . . . . . 6  |-  ( A  e.  QQ  ->  A  e.  CC )
213ad2ant1 1003 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  A  e.  CC )
3 qcn 9452 . . . . . 6  |-  ( B  e.  QQ  ->  B  e.  CC )
433ad2ant2 1004 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  B  e.  CC )
5 qre 9443 . . . . . . 7  |-  ( B  e.  QQ  ->  B  e.  RR )
653ad2ant2 1004 . . . . . 6  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  B  e.  RR )
7 simp3 984 . . . . . 6  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  0  <  B )
86, 7gt0ap0d 8414 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  B #  0 )
92, 4, 8divcanap2d 8575 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( B  x.  ( A  /  B ) )  =  A )
109oveq1d 5796 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  (
( B  x.  ( A  /  B ) )  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B
) ) ) ) )
117gt0ne0d 8297 . . . . . 6  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  B  =/=  0 )
12 qdivcl 9461 . . . . . 6  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( A  /  B )  e.  QQ )
1311, 12syld3an3 1262 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( A  /  B )  e.  QQ )
14 qcn 9452 . . . . 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 10080 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( |_ `  ( A  /  B ) )  e.  ZZ )
1716zcnd 9197 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( |_ `  ( A  /  B ) )  e.  CC )
184, 15, 17subdid 8199 . . 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 10127 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( A  mod  B )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B
) ) ) ) )
2010, 18, 193eqtr4rd 2184 . 2  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( A  mod  B )  =  ( B  x.  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) ) ) )
21 qfraclt1 10083 . . . . 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 8569 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( B  /  B )  =  1 )
2422, 23breqtrrd 3963 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) )  <  ( B  /  B ) )
25 qre 9443 . . . . . 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 9196 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( |_ `  ( A  /  B ) )  e.  RR )
2826, 27resubcld 8166 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) )  e.  RR )
29 ltmuldiv2 8656 . . . 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 1221 . . 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 3957 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 963    e. wcel 1481    =/= wne 2309   class class class wbr 3936   ` cfv 5130  (class class class)co 5781   CCcc 7641   RRcr 7642   0cc0 7643   1c1 7644    x. cmul 7648    < clt 7823    - cmin 7956    / cdiv 8455   QQcq 9437   |_cfl 10071    mod cmo 10125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-mulrcl 7742  ax-addcom 7743  ax-mulcom 7744  ax-addass 7745  ax-mulass 7746  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-1rid 7750  ax-0id 7751  ax-rnegex 7752  ax-precex 7753  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759  ax-pre-mulgt0 7760  ax-pre-mulext 7761  ax-arch 7762
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-po 4225  df-iso 4226  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-reap 8360  df-ap 8367  df-div 8456  df-inn 8744  df-n0 9001  df-z 9078  df-q 9438  df-rp 9470  df-fl 10073  df-mod 10126
This theorem is referenced by:  modqelico  10137  zmodfz  10149  modqid2  10154  modqabs  10160  modqmuladdim  10170  modaddmodup  10190  modqsubdir  10196  divalglemnn  11649  divalgmod  11658  bezoutlemnewy  11718  bezoutlemstep  11719  eucalglt  11772
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