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Theorem modqmul1 10380
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 10327 . . . . . . 7  |-  ( ( A  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( A  mod  D )  =  ( A  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) ) )
62, 3, 4, 5syl3anc 1238 . . . . . 6  |-  ( ph  ->  ( A  mod  D
)  =  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) ) )
7 modqmul1.b . . . . . . 7  |-  ( ph  ->  B  e.  QQ )
8 modqval 10327 . . . . . . 7  |-  ( ( B  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( B  mod  D )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) ) )
97, 3, 4, 8syl3anc 1238 . . . . . 6  |-  ( ph  ->  ( B  mod  D
)  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) ) )
106, 9eqeq12d 2192 . . . . 5  |-  ( ph  ->  ( ( A  mod  D )  =  ( B  mod  D )  <->  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) ) ) )
11 oveq1 5885 . . . . 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, 11biimtrdi 163 . . . 4  |-  ( ph  ->  ( ( A  mod  D )  =  ( B  mod  D )  -> 
( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  x.  C )  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  x.  C
) ) )
13 qcn 9637 . . . . . . . . . 10  |-  ( D  e.  QQ  ->  D  e.  CC )
143, 13syl 14 . . . . . . . . 9  |-  ( ph  ->  D  e.  CC )
15 modqmul1.c . . . . . . . . . 10  |-  ( ph  ->  C  e.  ZZ )
1615zcnd 9379 . . . . . . . . 9  |-  ( ph  ->  C  e.  CC )
174gt0ne0d 8472 . . . . . . . . . . . 12  |-  ( ph  ->  D  =/=  0 )
18 qdivcl 9646 . . . . . . . . . . . 12  |-  ( ( A  e.  QQ  /\  D  e.  QQ  /\  D  =/=  0 )  ->  ( A  /  D )  e.  QQ )
192, 3, 17, 18syl3anc 1238 . . . . . . . . . . 11  |-  ( ph  ->  ( A  /  D
)  e.  QQ )
2019flqcld 10280 . . . . . . . . . 10  |-  ( ph  ->  ( |_ `  ( A  /  D ) )  e.  ZZ )
2120zcnd 9379 . . . . . . . . 9  |-  ( ph  ->  ( |_ `  ( A  /  D ) )  e.  CC )
2214, 16, 21mulassd 7984 . . . . . . . 8  |-  ( ph  ->  ( ( D  x.  C )  x.  ( |_ `  ( A  /  D ) ) )  =  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )
2314, 16, 21mul32d 8113 . . . . . . . 8  |-  ( ph  ->  ( ( D  x.  C )  x.  ( |_ `  ( A  /  D ) ) )  =  ( ( D  x.  ( |_ `  ( A  /  D
) ) )  x.  C ) )
2422, 23eqtr3d 2212 . . . . . . 7  |-  ( ph  ->  ( D  x.  ( C  x.  ( |_ `  ( A  /  D
) ) ) )  =  ( ( D  x.  ( |_ `  ( A  /  D
) ) )  x.  C ) )
2524oveq2d 5894 . . . . . 6  |-  ( ph  ->  ( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )  =  ( ( A  x.  C )  -  (
( D  x.  ( |_ `  ( A  /  D ) ) )  x.  C ) ) )
26 qcn 9637 . . . . . . . 8  |-  ( A  e.  QQ  ->  A  e.  CC )
272, 26syl 14 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
2814, 21mulcld 7981 . . . . . . 7  |-  ( ph  ->  ( D  x.  ( |_ `  ( A  /  D ) ) )  e.  CC )
2927, 28, 16subdird 8375 . . . . . 6  |-  ( ph  ->  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  x.  C )  =  ( ( A  x.  C )  -  ( ( D  x.  ( |_ `  ( A  /  D ) ) )  x.  C ) ) )
3025, 29eqtr4d 2213 . . . . 5  |-  ( ph  ->  ( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )  =  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  x.  C ) )
31 qdivcl 9646 . . . . . . . . . . . 12  |-  ( ( B  e.  QQ  /\  D  e.  QQ  /\  D  =/=  0 )  ->  ( B  /  D )  e.  QQ )
327, 3, 17, 31syl3anc 1238 . . . . . . . . . . 11  |-  ( ph  ->  ( B  /  D
)  e.  QQ )
3332flqcld 10280 . . . . . . . . . 10  |-  ( ph  ->  ( |_ `  ( B  /  D ) )  e.  ZZ )
3433zcnd 9379 . . . . . . . . 9  |-  ( ph  ->  ( |_ `  ( B  /  D ) )  e.  CC )
3514, 16, 34mulassd 7984 . . . . . . . 8  |-  ( ph  ->  ( ( D  x.  C )  x.  ( |_ `  ( B  /  D ) ) )  =  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )
3614, 16, 34mul32d 8113 . . . . . . . 8  |-  ( ph  ->  ( ( D  x.  C )  x.  ( |_ `  ( B  /  D ) ) )  =  ( ( D  x.  ( |_ `  ( B  /  D
) ) )  x.  C ) )
3735, 36eqtr3d 2212 . . . . . . 7  |-  ( ph  ->  ( D  x.  ( C  x.  ( |_ `  ( B  /  D
) ) ) )  =  ( ( D  x.  ( |_ `  ( B  /  D
) ) )  x.  C ) )
3837oveq2d 5894 . . . . . 6  |-  ( ph  ->  ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  =  ( ( B  x.  C )  -  (
( D  x.  ( |_ `  ( B  /  D ) ) )  x.  C ) ) )
39 qcn 9637 . . . . . . . 8  |-  ( B  e.  QQ  ->  B  e.  CC )
407, 39syl 14 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
4114, 34mulcld 7981 . . . . . . 7  |-  ( ph  ->  ( D  x.  ( |_ `  ( B  /  D ) ) )  e.  CC )
4240, 41, 16subdird 8375 . . . . . 6  |-  ( ph  ->  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  x.  C )  =  ( ( B  x.  C )  -  ( ( D  x.  ( |_ `  ( B  /  D ) ) )  x.  C ) ) )
4338, 42eqtr4d 2213 . . . . 5  |-  ( ph  ->  ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  x.  C ) )
4430, 43eqeq12d 2192 . . . 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 169 . . 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 5885 . . . 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 9629 . . . . . . . 8  |-  ( C  e.  ZZ  ->  C  e.  QQ )
4815, 47syl 14 . . . . . . 7  |-  ( ph  ->  C  e.  QQ )
49 qmulcl 9640 . . . . . . 7  |-  ( ( A  e.  QQ  /\  C  e.  QQ )  ->  ( A  x.  C
)  e.  QQ )
502, 48, 49syl2anc 411 . . . . . 6  |-  ( ph  ->  ( A  x.  C
)  e.  QQ )
5115, 20zmulcld 9384 . . . . . 6  |-  ( ph  ->  ( C  x.  ( |_ `  ( A  /  D ) ) )  e.  ZZ )
52 modqcyc2 10363 . . . . . 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 1239 . . . . 5  |-  ( ph  ->  ( ( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D
) ) ) ) )  mod  D )  =  ( ( A  x.  C )  mod 
D ) )
54 qmulcl 9640 . . . . . . 7  |-  ( ( B  e.  QQ  /\  C  e.  QQ )  ->  ( B  x.  C
)  e.  QQ )
557, 48, 54syl2anc 411 . . . . . 6  |-  ( ph  ->  ( B  x.  C
)  e.  QQ )
5615, 33zmulcld 9384 . . . . . 6  |-  ( ph  ->  ( C  x.  ( |_ `  ( B  /  D ) ) )  e.  ZZ )
57 modqcyc2 10363 . . . . . 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 1239 . . . . 5  |-  ( ph  ->  ( ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D
) ) ) ) )  mod  D )  =  ( ( B  x.  C )  mod 
D ) )
5953, 58eqeq12d 2192 . . . 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, 59imbitrid 154 . . 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 1353    e. wcel 2148    =/= wne 2347   class class class wbr 4005   ` cfv 5218  (class class class)co 5878   CCcc 7812   0cc0 7814    x. cmul 7819    < clt 7995    - cmin 8131    / cdiv 8632   ZZcz 9256   QQcq 9622   |_cfl 10271    mod cmo 10325
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930  ax-pre-mulgt0 7931  ax-pre-mulext 7932  ax-arch 7933
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-reap 8535  df-ap 8542  df-div 8633  df-inn 8923  df-n0 9180  df-z 9257  df-q 9623  df-rp 9657  df-fl 10273  df-mod 10326
This theorem is referenced by:  modqmul12d  10381  modqnegd  10382  modqmulmod  10392  eulerthlema  12233  fermltl  12237  odzdvds  12248  lgsdir2lem4  14620  lgsdirprm  14623
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