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Theorem modmulconst 10921
Description: Constant multiplication in a modulo operation, see theorem 5.3 in [ApostolNT] p. 108. (Contributed by AV, 21-Jul-2021.)
Assertion
Ref Expression
modmulconst  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  ( ( A  mod  M )  =  ( B  mod  M
)  <->  ( ( C  x.  A )  mod  ( C  x.  M
) )  =  ( ( C  x.  B
)  mod  ( C  x.  M ) ) ) )

Proof of Theorem modmulconst
StepHypRef Expression
1 nnz 8739 . . . . 5  |-  ( M  e.  NN  ->  M  e.  ZZ )
21adantl 271 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  M  e.  ZZ )
3 zsubcl 8761 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B
)  e.  ZZ )
433adant3 963 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( A  -  B )  e.  ZZ )
54adantr 270 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  ( A  -  B )  e.  ZZ )
6 nnz 8739 . . . . . . 7  |-  ( C  e.  NN  ->  C  e.  ZZ )
7 nnne0 8422 . . . . . . 7  |-  ( C  e.  NN  ->  C  =/=  0 )
86, 7jca 300 . . . . . 6  |-  ( C  e.  NN  ->  ( C  e.  ZZ  /\  C  =/=  0 ) )
983ad2ant3 966 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( C  e.  ZZ  /\  C  =/=  0 ) )
109adantr 270 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  ( C  e.  ZZ  /\  C  =/=  0 ) )
11 dvdscmulr 10918 . . . . 5  |-  ( ( M  e.  ZZ  /\  ( A  -  B
)  e.  ZZ  /\  ( C  e.  ZZ  /\  C  =/=  0 ) )  ->  ( ( C  x.  M )  ||  ( C  x.  ( A  -  B )
)  <->  M  ||  ( A  -  B ) ) )
1211bicomd 139 . . . 4  |-  ( ( M  e.  ZZ  /\  ( A  -  B
)  e.  ZZ  /\  ( C  e.  ZZ  /\  C  =/=  0 ) )  ->  ( M  ||  ( A  -  B
)  <->  ( C  x.  M )  ||  ( C  x.  ( A  -  B ) ) ) )
132, 5, 10, 12syl3anc 1174 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  ( M  ||  ( A  -  B
)  <->  ( C  x.  M )  ||  ( C  x.  ( A  -  B ) ) ) )
14 zcn 8725 . . . . . . . 8  |-  ( A  e.  ZZ  ->  A  e.  CC )
15 zcn 8725 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
16 nncn 8402 . . . . . . . 8  |-  ( C  e.  NN  ->  C  e.  CC )
1714, 15, 163anim123i 1128 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC ) )
18 3anrot 929 . . . . . . 7  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  <->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC ) )
1917, 18sylibr 132 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( C  e.  CC  /\  A  e.  CC  /\  B  e.  CC ) )
20 subdi 7842 . . . . . 6  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  ( C  x.  ( A  -  B ) )  =  ( ( C  x.  A )  -  ( C  x.  B )
) )
2119, 20syl 14 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( C  x.  ( A  -  B ) )  =  ( ( C  x.  A )  -  ( C  x.  B )
) )
2221adantr 270 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  ( C  x.  ( A  -  B
) )  =  ( ( C  x.  A
)  -  ( C  x.  B ) ) )
2322breq2d 3849 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  ( ( C  x.  M )  ||  ( C  x.  ( A  -  B )
)  <->  ( C  x.  M )  ||  (
( C  x.  A
)  -  ( C  x.  B ) ) ) )
2413, 23bitrd 186 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  ( M  ||  ( A  -  B
)  <->  ( C  x.  M )  ||  (
( C  x.  A
)  -  ( C  x.  B ) ) ) )
25 simpr 108 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  M  e.  NN )
26 simp1 943 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  A  e.  ZZ )
2726adantr 270 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  A  e.  ZZ )
28 simp2 944 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  B  e.  ZZ )
2928adantr 270 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  B  e.  ZZ )
30 moddvds 10898 . . 3  |-  ( ( M  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A  mod  M
)  =  ( B  mod  M )  <->  M  ||  ( A  -  B )
) )
3125, 27, 29, 30syl3anc 1174 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  ( ( A  mod  M )  =  ( B  mod  M
)  <->  M  ||  ( A  -  B ) ) )
32 simpl3 948 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  C  e.  NN )
3332, 25nnmulcld 8442 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  ( C  x.  M )  e.  NN )
3463ad2ant3 966 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  C  e.  ZZ )
3534, 26zmulcld 8844 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( C  x.  A )  e.  ZZ )
3635adantr 270 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  ( C  x.  A )  e.  ZZ )
3734, 28zmulcld 8844 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( C  x.  B )  e.  ZZ )
3837adantr 270 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  ( C  x.  B )  e.  ZZ )
39 moddvds 10898 . . 3  |-  ( ( ( C  x.  M
)  e.  NN  /\  ( C  x.  A
)  e.  ZZ  /\  ( C  x.  B
)  e.  ZZ )  ->  ( ( ( C  x.  A )  mod  ( C  x.  M ) )  =  ( ( C  x.  B )  mod  ( C  x.  M )
)  <->  ( C  x.  M )  ||  (
( C  x.  A
)  -  ( C  x.  B ) ) ) )
4033, 36, 38, 39syl3anc 1174 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  ( ( ( C  x.  A )  mod  ( C  x.  M ) )  =  ( ( C  x.  B )  mod  ( C  x.  M )
)  <->  ( C  x.  M )  ||  (
( C  x.  A
)  -  ( C  x.  B ) ) ) )
4124, 31, 403bitr4d 218 1  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  ( ( A  mod  M )  =  ( B  mod  M
)  <->  ( ( C  x.  A )  mod  ( C  x.  M
) )  =  ( ( C  x.  B
)  mod  ( C  x.  M ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438    =/= wne 2255   class class class wbr 3837  (class class class)co 5634   CCcc 7327   0cc0 7329    x. cmul 7334    - cmin 7632   NNcn 8394   ZZcz 8720    mod cmo 9694    || cdvds 10889
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-po 4114  df-iso 4115  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-n0 8644  df-z 8721  df-q 9074  df-rp 9104  df-fl 9642  df-mod 9695  df-dvds 10890
This theorem is referenced by: (None)
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