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Theorem modqaddmulmod 9861
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 947 . . . . 5  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  A  e.  QQ )
2 qcn 9182 . . . . 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 948 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  B  e.  QQ )
5 simprl 499 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  M  e.  QQ )
6 simprr 500 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  0  <  M )
74, 5, 6modqcld 9798 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( B  mod  M )  e.  QQ )
8 simpl3 949 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  C  e.  ZZ )
9 zq 9174 . . . . . . 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 9185 . . . . . 6  |-  ( ( ( B  mod  M
)  e.  QQ  /\  C  e.  QQ )  ->  ( ( B  mod  M )  x.  C )  e.  QQ )
127, 10, 11syl2anc 404 . . . . 5  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( B  mod  M )  x.  C )  e.  QQ )
13 qcn 9182 . . . . 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 7696 . . 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 5683 . 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 967 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  ->  C  e.  QQ )
1817adantr 271 . . . 4  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  C  e.  QQ )
197, 18, 11syl2anc 404 . . 3  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( B  mod  M )  x.  C )  e.  QQ )
20 qmulcl 9185 . . . . 5  |-  ( ( B  e.  QQ  /\  C  e.  QQ )  ->  ( B  x.  C
)  e.  QQ )
214, 18, 20syl2anc 404 . . . 4  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( B  x.  C )  e.  QQ )
2221, 5, 6modqcld 9798 . . 3  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( B  x.  C )  mod  M )  e.  QQ )
23 modqmulmod 9859 . . . . 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 1100 . . . 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 9828 . . . . 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 1175 . . . 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 2124 . . 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 9831 . 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 9833 . . . 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 1176 . . 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 9182 . . . . . 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 7696 . . . 4  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( B  x.  C )  +  A )  =  ( A  +  ( B  x.  C ) ) )
3433oveq1d 5683 . . 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 2121 . 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 2125 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 103    /\ w3a 925    = wceq 1290    e. wcel 1439   class class class wbr 3853  (class class class)co 5668   CCcc 7411   0cc0 7413    + caddc 7416    x. cmul 7418    < clt 7585   ZZcz 8813   QQcq 9167    mod cmo 9792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-cnex 7499  ax-resscn 7500  ax-1cn 7501  ax-1re 7502  ax-icn 7503  ax-addcl 7504  ax-addrcl 7505  ax-mulcl 7506  ax-mulrcl 7507  ax-addcom 7508  ax-mulcom 7509  ax-addass 7510  ax-mulass 7511  ax-distr 7512  ax-i2m1 7513  ax-0lt1 7514  ax-1rid 7515  ax-0id 7516  ax-rnegex 7517  ax-precex 7518  ax-cnre 7519  ax-pre-ltirr 7520  ax-pre-ltwlin 7521  ax-pre-lttrn 7522  ax-pre-apti 7523  ax-pre-ltadd 7524  ax-pre-mulgt0 7525  ax-pre-mulext 7526  ax-arch 7527
This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2624  df-sbc 2844  df-csb 2937  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-iun 3740  df-br 3854  df-opab 3908  df-mpt 3909  df-id 4131  df-po 4134  df-iso 4135  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-fv 5038  df-riota 5624  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-1st 5927  df-2nd 5928  df-pnf 7587  df-mnf 7588  df-xr 7589  df-ltxr 7590  df-le 7591  df-sub 7718  df-neg 7719  df-reap 8115  df-ap 8122  df-div 8203  df-inn 8486  df-n0 8737  df-z 8814  df-q 9168  df-rp 9198  df-fl 9740  df-mod 9793
This theorem is referenced by: (None)
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