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Theorem modmulconst 11452
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 9041 . . . . 5  |-  ( M  e.  NN  ->  M  e.  ZZ )
21adantl 275 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  M  e.  ZZ )
3 zsubcl 9063 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B
)  e.  ZZ )
433adant3 986 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( A  -  B )  e.  ZZ )
54adantr 274 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  ( A  -  B )  e.  ZZ )
6 nnz 9041 . . . . . . 7  |-  ( C  e.  NN  ->  C  e.  ZZ )
7 nnne0 8716 . . . . . . 7  |-  ( C  e.  NN  ->  C  =/=  0 )
86, 7jca 304 . . . . . 6  |-  ( C  e.  NN  ->  ( C  e.  ZZ  /\  C  =/=  0 ) )
983ad2ant3 989 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( C  e.  ZZ  /\  C  =/=  0 ) )
109adantr 274 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  ( C  e.  ZZ  /\  C  =/=  0 ) )
11 dvdscmulr 11449 . . . . 5  |-  ( ( M  e.  ZZ  /\  ( A  -  B
)  e.  ZZ  /\  ( C  e.  ZZ  /\  C  =/=  0 ) )  ->  ( ( C  x.  M )  ||  ( C  x.  ( A  -  B )
)  <->  M  ||  ( A  -  B ) ) )
1211bicomd 140 . . . 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 1201 . . 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 9027 . . . . . . . 8  |-  ( A  e.  ZZ  ->  A  e.  CC )
15 zcn 9027 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
16 nncn 8696 . . . . . . . 8  |-  ( C  e.  NN  ->  C  e.  CC )
1714, 15, 163anim123i 1151 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC ) )
18 3anrot 952 . . . . . . 7  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  <->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC ) )
1917, 18sylibr 133 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( C  e.  CC  /\  A  e.  CC  /\  B  e.  CC ) )
20 subdi 8115 . . . . . 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 274 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  ( C  x.  ( A  -  B
) )  =  ( ( C  x.  A
)  -  ( C  x.  B ) ) )
2322breq2d 3911 . . 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 187 . 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 109 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  M  e.  NN )
26 simp1 966 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  A  e.  ZZ )
2726adantr 274 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  A  e.  ZZ )
28 simp2 967 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  B  e.  ZZ )
2928adantr 274 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  B  e.  ZZ )
30 moddvds 11429 . . 3  |-  ( ( M  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A  mod  M
)  =  ( B  mod  M )  <->  M  ||  ( A  -  B )
) )
3125, 27, 29, 30syl3anc 1201 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  ( ( A  mod  M )  =  ( B  mod  M
)  <->  M  ||  ( A  -  B ) ) )
32 simpl3 971 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  C  e.  NN )
3332, 25nnmulcld 8737 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  ( C  x.  M )  e.  NN )
3463ad2ant3 989 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  C  e.  ZZ )
3534, 26zmulcld 9147 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( C  x.  A )  e.  ZZ )
3635adantr 274 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  ( C  x.  A )  e.  ZZ )
3734, 28zmulcld 9147 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( C  x.  B )  e.  ZZ )
3837adantr 274 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  M  e.  NN )  ->  ( C  x.  B )  e.  ZZ )
39 moddvds 11429 . . 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 1201 . 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 219 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 103    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465    =/= wne 2285   class class class wbr 3899  (class class class)co 5742   CCcc 7586   0cc0 7588    x. cmul 7593    - cmin 7901   NNcn 8688   ZZcz 9022    mod cmo 10063    || cdvds 11420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-po 4188  df-iso 4189  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-n0 8946  df-z 9023  df-q 9380  df-rp 9410  df-fl 10011  df-mod 10064  df-dvds 11421
This theorem is referenced by: (None)
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