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Theorem modqmul1 10101
Description: Multiplication property of the modulo operation. Note that the multiplier  C must be an integer. (Contributed by Jim Kingdon, 24-Oct-2021.)
Hypotheses
Ref Expression
modqmul1.a  |-  ( ph  ->  A  e.  QQ )
modqmul1.b  |-  ( ph  ->  B  e.  QQ )
modqmul1.c  |-  ( ph  ->  C  e.  ZZ )
modqmul1.d  |-  ( ph  ->  D  e.  QQ )
modqmul1.dgt0  |-  ( ph  ->  0  <  D )
modqmul1.ab  |-  ( ph  ->  ( A  mod  D
)  =  ( B  mod  D ) )
Assertion
Ref Expression
modqmul1  |-  ( ph  ->  ( ( A  x.  C )  mod  D
)  =  ( ( B  x.  C )  mod  D ) )

Proof of Theorem modqmul1
StepHypRef Expression
1 modqmul1.ab . 2  |-  ( ph  ->  ( A  mod  D
)  =  ( B  mod  D ) )
2 modqmul1.a . . . . . . 7  |-  ( ph  ->  A  e.  QQ )
3 modqmul1.d . . . . . . 7  |-  ( ph  ->  D  e.  QQ )
4 modqmul1.dgt0 . . . . . . 7  |-  ( ph  ->  0  <  D )
5 modqval 10048 . . . . . . 7  |-  ( ( A  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( A  mod  D )  =  ( A  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) ) )
62, 3, 4, 5syl3anc 1199 . . . . . 6  |-  ( ph  ->  ( A  mod  D
)  =  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) ) )
7 modqmul1.b . . . . . . 7  |-  ( ph  ->  B  e.  QQ )
8 modqval 10048 . . . . . . 7  |-  ( ( B  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( B  mod  D )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) ) )
97, 3, 4, 8syl3anc 1199 . . . . . 6  |-  ( ph  ->  ( B  mod  D
)  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) ) )
106, 9eqeq12d 2130 . . . . 5  |-  ( ph  ->  ( ( A  mod  D )  =  ( B  mod  D )  <->  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) ) ) )
11 oveq1 5747 . . . . 5  |-  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  ->  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  x.  C
)  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) )  x.  C ) )
1210, 11syl6bi 162 . . . 4  |-  ( ph  ->  ( ( A  mod  D )  =  ( B  mod  D )  -> 
( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  x.  C )  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  x.  C
) ) )
13 qcn 9378 . . . . . . . . . 10  |-  ( D  e.  QQ  ->  D  e.  CC )
143, 13syl 14 . . . . . . . . 9  |-  ( ph  ->  D  e.  CC )
15 modqmul1.c . . . . . . . . . 10  |-  ( ph  ->  C  e.  ZZ )
1615zcnd 9128 . . . . . . . . 9  |-  ( ph  ->  C  e.  CC )
174gt0ne0d 8238 . . . . . . . . . . . 12  |-  ( ph  ->  D  =/=  0 )
18 qdivcl 9387 . . . . . . . . . . . 12  |-  ( ( A  e.  QQ  /\  D  e.  QQ  /\  D  =/=  0 )  ->  ( A  /  D )  e.  QQ )
192, 3, 17, 18syl3anc 1199 . . . . . . . . . . 11  |-  ( ph  ->  ( A  /  D
)  e.  QQ )
2019flqcld 10001 . . . . . . . . . 10  |-  ( ph  ->  ( |_ `  ( A  /  D ) )  e.  ZZ )
2120zcnd 9128 . . . . . . . . 9  |-  ( ph  ->  ( |_ `  ( A  /  D ) )  e.  CC )
2214, 16, 21mulassd 7753 . . . . . . . 8  |-  ( ph  ->  ( ( D  x.  C )  x.  ( |_ `  ( A  /  D ) ) )  =  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )
2314, 16, 21mul32d 7879 . . . . . . . 8  |-  ( ph  ->  ( ( D  x.  C )  x.  ( |_ `  ( A  /  D ) ) )  =  ( ( D  x.  ( |_ `  ( A  /  D
) ) )  x.  C ) )
2422, 23eqtr3d 2150 . . . . . . 7  |-  ( ph  ->  ( D  x.  ( C  x.  ( |_ `  ( A  /  D
) ) ) )  =  ( ( D  x.  ( |_ `  ( A  /  D
) ) )  x.  C ) )
2524oveq2d 5756 . . . . . 6  |-  ( ph  ->  ( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )  =  ( ( A  x.  C )  -  (
( D  x.  ( |_ `  ( A  /  D ) ) )  x.  C ) ) )
26 qcn 9378 . . . . . . . 8  |-  ( A  e.  QQ  ->  A  e.  CC )
272, 26syl 14 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
2814, 21mulcld 7750 . . . . . . 7  |-  ( ph  ->  ( D  x.  ( |_ `  ( A  /  D ) ) )  e.  CC )
2927, 28, 16subdird 8141 . . . . . 6  |-  ( ph  ->  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  x.  C )  =  ( ( A  x.  C )  -  ( ( D  x.  ( |_ `  ( A  /  D ) ) )  x.  C ) ) )
3025, 29eqtr4d 2151 . . . . 5  |-  ( ph  ->  ( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )  =  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  x.  C ) )
31 qdivcl 9387 . . . . . . . . . . . 12  |-  ( ( B  e.  QQ  /\  D  e.  QQ  /\  D  =/=  0 )  ->  ( B  /  D )  e.  QQ )
327, 3, 17, 31syl3anc 1199 . . . . . . . . . . 11  |-  ( ph  ->  ( B  /  D
)  e.  QQ )
3332flqcld 10001 . . . . . . . . . 10  |-  ( ph  ->  ( |_ `  ( B  /  D ) )  e.  ZZ )
3433zcnd 9128 . . . . . . . . 9  |-  ( ph  ->  ( |_ `  ( B  /  D ) )  e.  CC )
3514, 16, 34mulassd 7753 . . . . . . . 8  |-  ( ph  ->  ( ( D  x.  C )  x.  ( |_ `  ( B  /  D ) ) )  =  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )
3614, 16, 34mul32d 7879 . . . . . . . 8  |-  ( ph  ->  ( ( D  x.  C )  x.  ( |_ `  ( B  /  D ) ) )  =  ( ( D  x.  ( |_ `  ( B  /  D
) ) )  x.  C ) )
3735, 36eqtr3d 2150 . . . . . . 7  |-  ( ph  ->  ( D  x.  ( C  x.  ( |_ `  ( B  /  D
) ) ) )  =  ( ( D  x.  ( |_ `  ( B  /  D
) ) )  x.  C ) )
3837oveq2d 5756 . . . . . 6  |-  ( ph  ->  ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  =  ( ( B  x.  C )  -  (
( D  x.  ( |_ `  ( B  /  D ) ) )  x.  C ) ) )
39 qcn 9378 . . . . . . . 8  |-  ( B  e.  QQ  ->  B  e.  CC )
407, 39syl 14 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
4114, 34mulcld 7750 . . . . . . 7  |-  ( ph  ->  ( D  x.  ( |_ `  ( B  /  D ) ) )  e.  CC )
4240, 41, 16subdird 8141 . . . . . 6  |-  ( ph  ->  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  x.  C )  =  ( ( B  x.  C )  -  ( ( D  x.  ( |_ `  ( B  /  D ) ) )  x.  C ) ) )
4338, 42eqtr4d 2151 . . . . 5  |-  ( ph  ->  ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  x.  C ) )
4430, 43eqeq12d 2130 . . . 4  |-  ( ph  ->  ( ( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D
) ) ) ) )  =  ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  <->  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  x.  C
)  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) )  x.  C ) ) )
4512, 44sylibrd 168 . . 3  |-  ( ph  ->  ( ( A  mod  D )  =  ( B  mod  D )  -> 
( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )  =  ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) ) ) )
46 oveq1 5747 . . . 4  |-  ( ( ( A  x.  C
)  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )  =  ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  -> 
( ( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D
) ) ) ) )  mod  D )  =  ( ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  mod  D
) )
47 zq 9370 . . . . . . . 8  |-  ( C  e.  ZZ  ->  C  e.  QQ )
4815, 47syl 14 . . . . . . 7  |-  ( ph  ->  C  e.  QQ )
49 qmulcl 9381 . . . . . . 7  |-  ( ( A  e.  QQ  /\  C  e.  QQ )  ->  ( A  x.  C
)  e.  QQ )
502, 48, 49syl2anc 406 . . . . . 6  |-  ( ph  ->  ( A  x.  C
)  e.  QQ )
5115, 20zmulcld 9133 . . . . . 6  |-  ( ph  ->  ( C  x.  ( |_ `  ( A  /  D ) ) )  e.  ZZ )
52 modqcyc2 10084 . . . . . 6  |-  ( ( ( ( A  x.  C )  e.  QQ  /\  ( C  x.  ( |_ `  ( A  /  D ) ) )  e.  ZZ )  /\  ( D  e.  QQ  /\  0  <  D ) )  ->  ( (
( A  x.  C
)  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )  mod 
D )  =  ( ( A  x.  C
)  mod  D )
)
5350, 51, 3, 4, 52syl22anc 1200 . . . . 5  |-  ( ph  ->  ( ( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D
) ) ) ) )  mod  D )  =  ( ( A  x.  C )  mod 
D ) )
54 qmulcl 9381 . . . . . . 7  |-  ( ( B  e.  QQ  /\  C  e.  QQ )  ->  ( B  x.  C
)  e.  QQ )
557, 48, 54syl2anc 406 . . . . . 6  |-  ( ph  ->  ( B  x.  C
)  e.  QQ )
5615, 33zmulcld 9133 . . . . . 6  |-  ( ph  ->  ( C  x.  ( |_ `  ( B  /  D ) ) )  e.  ZZ )
57 modqcyc2 10084 . . . . . 6  |-  ( ( ( ( B  x.  C )  e.  QQ  /\  ( C  x.  ( |_ `  ( B  /  D ) ) )  e.  ZZ )  /\  ( D  e.  QQ  /\  0  <  D ) )  ->  ( (
( B  x.  C
)  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  mod 
D )  =  ( ( B  x.  C
)  mod  D )
)
5855, 56, 3, 4, 57syl22anc 1200 . . . . 5  |-  ( ph  ->  ( ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D
) ) ) ) )  mod  D )  =  ( ( B  x.  C )  mod 
D ) )
5953, 58eqeq12d 2130 . . . 4  |-  ( ph  ->  ( ( ( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )  mod  D
)  =  ( ( ( B  x.  C
)  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  mod 
D )  <->  ( ( A  x.  C )  mod  D )  =  ( ( B  x.  C
)  mod  D )
) )
6046, 59syl5ib 153 . . 3  |-  ( ph  ->  ( ( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D
) ) ) ) )  =  ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  ->  (
( A  x.  C
)  mod  D )  =  ( ( B  x.  C )  mod 
D ) ) )
6145, 60syld 45 . 2  |-  ( ph  ->  ( ( A  mod  D )  =  ( B  mod  D )  -> 
( ( A  x.  C )  mod  D
)  =  ( ( B  x.  C )  mod  D ) ) )
621, 61mpd 13 1  |-  ( ph  ->  ( ( A  x.  C )  mod  D
)  =  ( ( B  x.  C )  mod  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    e. wcel 1463    =/= wne 2283   class class class wbr 3897   ` cfv 5091  (class class class)co 5740   CCcc 7582   0cc0 7584    x. cmul 7589    < clt 7764    - cmin 7897    / cdiv 8395   ZZcz 9008   QQcq 9363   |_cfl 9992    mod cmo 10046
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 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 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703
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 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-po 4186  df-iso 4187  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-n0 8932  df-z 9009  df-q 9364  df-rp 9394  df-fl 9994  df-mod 10047
This theorem is referenced by:  modqmul12d  10102  modqnegd  10103  modqmulmod  10113
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