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Theorem modqlt 9415
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 8800 . . . . . 6  |-  ( A  e.  QQ  ->  A  e.  CC )
213ad2ant1 960 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  A  e.  CC )
3 qcn 8800 . . . . . 6  |-  ( B  e.  QQ  ->  B  e.  CC )
433ad2ant2 961 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  B  e.  CC )
5 qre 8791 . . . . . . 7  |-  ( B  e.  QQ  ->  B  e.  RR )
653ad2ant2 961 . . . . . 6  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  B  e.  RR )
7 simp3 941 . . . . . 6  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  0  <  B )
86, 7gt0ap0d 7795 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  B #  0 )
92, 4, 8divcanap2d 7946 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( B  x.  ( A  /  B ) )  =  A )
109oveq1d 5558 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  (
( B  x.  ( A  /  B ) )  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B
) ) ) ) )
117gt0ne0d 7680 . . . . . 6  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  B  =/=  0 )
12 qdivcl 8809 . . . . . 6  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( A  /  B )  e.  QQ )
1311, 12syld3an3 1215 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( A  /  B )  e.  QQ )
14 qcn 8800 . . . . 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 9359 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( |_ `  ( A  /  B ) )  e.  ZZ )
1716zcnd 8551 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( |_ `  ( A  /  B ) )  e.  CC )
184, 15, 17subdid 7585 . . 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 9406 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( A  mod  B )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B
) ) ) ) )
2010, 18, 193eqtr4rd 2125 . 2  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( A  mod  B )  =  ( B  x.  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) ) ) )
21 qfraclt1 9362 . . . . 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 7941 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( B  /  B )  =  1 )
2422, 23breqtrrd 3819 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) )  <  ( B  /  B ) )
25 qre 8791 . . . . . 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 8550 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( |_ `  ( A  /  B ) )  e.  RR )
2826, 27resubcld 7552 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) )  e.  RR )
29 ltmuldiv2 8020 . . . 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 1174 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  (
( B  x.  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) ) )  <  B  <->  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  <  ( B  /  B ) ) )
3124, 30mpbird 165 . 2  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( B  x.  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) ) )  <  B
)
3220, 31eqbrtrd 3813 1  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( A  mod  B )  < 
B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    /\ w3a 920    e. wcel 1434    =/= wne 2246   class class class wbr 3793   ` cfv 4932  (class class class)co 5543   CCcc 7041   RRcr 7042   0cc0 7043   1c1 7044    x. cmul 7048    < clt 7215    - cmin 7346    / cdiv 7827   QQcq 8785   |_cfl 9350    mod cmo 9404
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156  ax-arch 7157
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-po 4059  df-iso 4060  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-n0 8356  df-z 8433  df-q 8786  df-rp 8816  df-fl 9352  df-mod 9405
This theorem is referenced by:  modqelico  9416  zmodfz  9428  modqid2  9433  modqabs  9439  modqmuladdim  9449  modaddmodup  9469  modqsubdir  9475  divalglemnn  10462  divalgmod  10471  bezoutlemnewy  10529  bezoutlemstep  10530  eucalglt  10583
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