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Theorem modqmul1 10391
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 10338 . . . . . . 7  |-  ( ( A  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( A  mod  D )  =  ( A  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) ) )
62, 3, 4, 5syl3anc 1248 . . . . . 6  |-  ( ph  ->  ( A  mod  D
)  =  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) ) )
7 modqmul1.b . . . . . . 7  |-  ( ph  ->  B  e.  QQ )
8 modqval 10338 . . . . . . 7  |-  ( ( B  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( B  mod  D )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) ) )
97, 3, 4, 8syl3anc 1248 . . . . . 6  |-  ( ph  ->  ( B  mod  D
)  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) ) )
106, 9eqeq12d 2202 . . . . 5  |-  ( ph  ->  ( ( A  mod  D )  =  ( B  mod  D )  <->  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) ) ) )
11 oveq1 5895 . . . . 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 9648 . . . . . . . . . 10  |-  ( D  e.  QQ  ->  D  e.  CC )
143, 13syl 14 . . . . . . . . 9  |-  ( ph  ->  D  e.  CC )
15 modqmul1.c . . . . . . . . . 10  |-  ( ph  ->  C  e.  ZZ )
1615zcnd 9390 . . . . . . . . 9  |-  ( ph  ->  C  e.  CC )
174gt0ne0d 8483 . . . . . . . . . . . 12  |-  ( ph  ->  D  =/=  0 )
18 qdivcl 9657 . . . . . . . . . . . 12  |-  ( ( A  e.  QQ  /\  D  e.  QQ  /\  D  =/=  0 )  ->  ( A  /  D )  e.  QQ )
192, 3, 17, 18syl3anc 1248 . . . . . . . . . . 11  |-  ( ph  ->  ( A  /  D
)  e.  QQ )
2019flqcld 10291 . . . . . . . . . 10  |-  ( ph  ->  ( |_ `  ( A  /  D ) )  e.  ZZ )
2120zcnd 9390 . . . . . . . . 9  |-  ( ph  ->  ( |_ `  ( A  /  D ) )  e.  CC )
2214, 16, 21mulassd 7995 . . . . . . . 8  |-  ( ph  ->  ( ( D  x.  C )  x.  ( |_ `  ( A  /  D ) ) )  =  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )
2314, 16, 21mul32d 8124 . . . . . . . 8  |-  ( ph  ->  ( ( D  x.  C )  x.  ( |_ `  ( A  /  D ) ) )  =  ( ( D  x.  ( |_ `  ( A  /  D
) ) )  x.  C ) )
2422, 23eqtr3d 2222 . . . . . . 7  |-  ( ph  ->  ( D  x.  ( C  x.  ( |_ `  ( A  /  D
) ) ) )  =  ( ( D  x.  ( |_ `  ( A  /  D
) ) )  x.  C ) )
2524oveq2d 5904 . . . . . 6  |-  ( ph  ->  ( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )  =  ( ( A  x.  C )  -  (
( D  x.  ( |_ `  ( A  /  D ) ) )  x.  C ) ) )
26 qcn 9648 . . . . . . . 8  |-  ( A  e.  QQ  ->  A  e.  CC )
272, 26syl 14 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
2814, 21mulcld 7992 . . . . . . 7  |-  ( ph  ->  ( D  x.  ( |_ `  ( A  /  D ) ) )  e.  CC )
2927, 28, 16subdird 8386 . . . . . 6  |-  ( ph  ->  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  x.  C )  =  ( ( A  x.  C )  -  ( ( D  x.  ( |_ `  ( A  /  D ) ) )  x.  C ) ) )
3025, 29eqtr4d 2223 . . . . 5  |-  ( ph  ->  ( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D ) ) ) ) )  =  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  x.  C ) )
31 qdivcl 9657 . . . . . . . . . . . 12  |-  ( ( B  e.  QQ  /\  D  e.  QQ  /\  D  =/=  0 )  ->  ( B  /  D )  e.  QQ )
327, 3, 17, 31syl3anc 1248 . . . . . . . . . . 11  |-  ( ph  ->  ( B  /  D
)  e.  QQ )
3332flqcld 10291 . . . . . . . . . 10  |-  ( ph  ->  ( |_ `  ( B  /  D ) )  e.  ZZ )
3433zcnd 9390 . . . . . . . . 9  |-  ( ph  ->  ( |_ `  ( B  /  D ) )  e.  CC )
3514, 16, 34mulassd 7995 . . . . . . . 8  |-  ( ph  ->  ( ( D  x.  C )  x.  ( |_ `  ( B  /  D ) ) )  =  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )
3614, 16, 34mul32d 8124 . . . . . . . 8  |-  ( ph  ->  ( ( D  x.  C )  x.  ( |_ `  ( B  /  D ) ) )  =  ( ( D  x.  ( |_ `  ( B  /  D
) ) )  x.  C ) )
3735, 36eqtr3d 2222 . . . . . . 7  |-  ( ph  ->  ( D  x.  ( C  x.  ( |_ `  ( B  /  D
) ) ) )  =  ( ( D  x.  ( |_ `  ( B  /  D
) ) )  x.  C ) )
3837oveq2d 5904 . . . . . 6  |-  ( ph  ->  ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  =  ( ( B  x.  C )  -  (
( D  x.  ( |_ `  ( B  /  D ) ) )  x.  C ) ) )
39 qcn 9648 . . . . . . . 8  |-  ( B  e.  QQ  ->  B  e.  CC )
407, 39syl 14 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
4114, 34mulcld 7992 . . . . . . 7  |-  ( ph  ->  ( D  x.  ( |_ `  ( B  /  D ) ) )  e.  CC )
4240, 41, 16subdird 8386 . . . . . 6  |-  ( ph  ->  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  x.  C )  =  ( ( B  x.  C )  -  ( ( D  x.  ( |_ `  ( B  /  D ) ) )  x.  C ) ) )
4338, 42eqtr4d 2223 . . . . 5  |-  ( ph  ->  ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D ) ) ) ) )  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  x.  C ) )
4430, 43eqeq12d 2202 . . . 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 5895 . . . 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 9640 . . . . . . . 8  |-  ( C  e.  ZZ  ->  C  e.  QQ )
4815, 47syl 14 . . . . . . 7  |-  ( ph  ->  C  e.  QQ )
49 qmulcl 9651 . . . . . . 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 9395 . . . . . 6  |-  ( ph  ->  ( C  x.  ( |_ `  ( A  /  D ) ) )  e.  ZZ )
52 modqcyc2 10374 . . . . . 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 1249 . . . . 5  |-  ( ph  ->  ( ( ( A  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( A  /  D
) ) ) ) )  mod  D )  =  ( ( A  x.  C )  mod 
D ) )
54 qmulcl 9651 . . . . . . 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 9395 . . . . . 6  |-  ( ph  ->  ( C  x.  ( |_ `  ( B  /  D ) ) )  e.  ZZ )
57 modqcyc2 10374 . . . . . 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 1249 . . . . 5  |-  ( ph  ->  ( ( ( B  x.  C )  -  ( D  x.  ( C  x.  ( |_ `  ( B  /  D
) ) ) ) )  mod  D )  =  ( ( B  x.  C )  mod 
D ) )
5953, 58eqeq12d 2202 . . . 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 1363    e. wcel 2158    =/= wne 2357   class class class wbr 4015   ` cfv 5228  (class class class)co 5888   CCcc 7823   0cc0 7825    x. cmul 7830    < clt 8006    - cmin 8142    / cdiv 8643   ZZcz 9267   QQcq 9633   |_cfl 10282    mod cmo 10336
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944
This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-n0 9191  df-z 9268  df-q 9634  df-rp 9668  df-fl 10284  df-mod 10337
This theorem is referenced by:  modqmul12d  10392  modqnegd  10393  modqmulmod  10403  eulerthlema  12244  fermltl  12248  odzdvds  12259  lgsdir2lem4  14785  lgsdirprm  14788
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