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Theorem modqaddmulmod 10422
Description: The sum of a rational number and the product of a second rational number modulo a modulus and an integer equals the sum of the rational number and the product of the other rational number and the integer modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
Assertion
Ref Expression
modqaddmulmod  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( A  +  ( ( B  mod  M )  x.  C ) )  mod 
M )  =  ( ( A  +  ( B  x.  C ) )  mod  M ) )

Proof of Theorem modqaddmulmod
StepHypRef Expression
1 simpl1 1002 . . . . 5  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  A  e.  QQ )
2 qcn 9664 . . . . 5  |-  ( A  e.  QQ  ->  A  e.  CC )
31, 2syl 14 . . . 4  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  A  e.  CC )
4 simpl2 1003 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  B  e.  QQ )
5 simprl 529 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  M  e.  QQ )
6 simprr 531 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  0  <  M )
74, 5, 6modqcld 10359 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( B  mod  M )  e.  QQ )
8 simpl3 1004 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  C  e.  ZZ )
9 zq 9656 . . . . . . 7  |-  ( C  e.  ZZ  ->  C  e.  QQ )
108, 9syl 14 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  C  e.  QQ )
11 qmulcl 9667 . . . . . 6  |-  ( ( ( B  mod  M
)  e.  QQ  /\  C  e.  QQ )  ->  ( ( B  mod  M )  x.  C )  e.  QQ )
127, 10, 11syl2anc 411 . . . . 5  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( B  mod  M )  x.  C )  e.  QQ )
13 qcn 9664 . . . . 5  |-  ( ( ( B  mod  M
)  x.  C )  e.  QQ  ->  (
( B  mod  M
)  x.  C )  e.  CC )
1412, 13syl 14 . . . 4  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( B  mod  M )  x.  C )  e.  CC )
153, 14addcomd 8138 . . 3  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( A  +  ( ( B  mod  M )  x.  C ) )  =  ( ( ( B  mod  M )  x.  C )  +  A
) )
1615oveq1d 5911 . 2  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( A  +  ( ( B  mod  M )  x.  C ) )  mod 
M )  =  ( ( ( ( B  mod  M )  x.  C )  +  A
)  mod  M )
)
1793ad2ant3 1022 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  ->  C  e.  QQ )
1817adantr 276 . . . 4  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  C  e.  QQ )
197, 18, 11syl2anc 411 . . 3  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( B  mod  M )  x.  C )  e.  QQ )
20 qmulcl 9667 . . . . 5  |-  ( ( B  e.  QQ  /\  C  e.  QQ )  ->  ( B  x.  C
)  e.  QQ )
214, 18, 20syl2anc 411 . . . 4  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( B  x.  C )  e.  QQ )
2221, 5, 6modqcld 10359 . . 3  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( B  x.  C )  mod  M )  e.  QQ )
23 modqmulmod 10420 . . . . 5  |-  ( ( ( B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( ( B  mod  M )  x.  C )  mod  M
)  =  ( ( B  x.  C )  mod  M ) )
24233adantl1 1155 . . . 4  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( (
( B  mod  M
)  x.  C )  mod  M )  =  ( ( B  x.  C )  mod  M
) )
25 modqabs2 10389 . . . . 5  |-  ( ( ( B  x.  C
)  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  ->  (
( ( B  x.  C )  mod  M
)  mod  M )  =  ( ( B  x.  C )  mod 
M ) )
2621, 5, 6, 25syl3anc 1249 . . . 4  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( (
( B  x.  C
)  mod  M )  mod  M )  =  ( ( B  x.  C
)  mod  M )
)
2724, 26eqtr4d 2225 . . 3  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( (
( B  mod  M
)  x.  C )  mod  M )  =  ( ( ( B  x.  C )  mod 
M )  mod  M
) )
2819, 22, 1, 5, 6, 27modqadd1 10392 . 2  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( (
( ( B  mod  M )  x.  C )  +  A )  mod 
M )  =  ( ( ( ( B  x.  C )  mod 
M )  +  A
)  mod  M )
)
29 modqaddmod 10394 . . . 4  |-  ( ( ( ( B  x.  C )  e.  QQ  /\  A  e.  QQ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( ( ( B  x.  C )  mod  M )  +  A )  mod  M
)  =  ( ( ( B  x.  C
)  +  A )  mod  M ) )
3021, 1, 5, 6, 29syl22anc 1250 . . 3  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( (
( ( B  x.  C )  mod  M
)  +  A )  mod  M )  =  ( ( ( B  x.  C )  +  A )  mod  M
) )
31 qcn 9664 . . . . . 6  |-  ( ( B  x.  C )  e.  QQ  ->  ( B  x.  C )  e.  CC )
3221, 31syl 14 . . . . 5  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( B  x.  C )  e.  CC )
3332, 3addcomd 8138 . . . 4  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( B  x.  C )  +  A )  =  ( A  +  ( B  x.  C ) ) )
3433oveq1d 5911 . . 3  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( (
( B  x.  C
)  +  A )  mod  M )  =  ( ( A  +  ( B  x.  C
) )  mod  M
) )
3530, 34eqtrd 2222 . 2  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( (
( ( B  x.  C )  mod  M
)  +  A )  mod  M )  =  ( ( A  +  ( B  x.  C
) )  mod  M
) )
3616, 28, 353eqtrd 2226 1  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( A  +  ( ( B  mod  M )  x.  C ) )  mod 
M )  =  ( ( A  +  ( B  x.  C ) )  mod  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 980    = wceq 1364    e. wcel 2160   class class class wbr 4018  (class class class)co 5896   CCcc 7839   0cc0 7841    + caddc 7844    x. cmul 7846    < clt 8022   ZZcz 9283   QQcq 9649    mod cmo 10353
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959  ax-arch 7960
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-n0 9207  df-z 9284  df-q 9650  df-rp 9684  df-fl 10301  df-mod 10354
This theorem is referenced by:  modprm0  12286  modprmn0modprm0  12288
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