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Theorem mulmoddvds 10408
Description: If an integer is divisible by a positive integer, the product of this integer with another integer modulo the positive integer is 0. (Contributed by Alexander van der Vekens, 30-Aug-2018.)
Assertion
Ref Expression
mulmoddvds  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( N  ||  A  ->  (
( A  x.  B
)  mod  N )  =  0 ) )

Proof of Theorem mulmoddvds
StepHypRef Expression
1 simp2 940 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  A  e.  ZZ )
2 zq 8792 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  QQ )
31, 2syl 14 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  A  e.  QQ )
4 simp3 941 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  B  e.  ZZ )
5 simp1 939 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  N  e.  NN )
6 nnq 8799 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  QQ )
75, 6syl 14 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  N  e.  QQ )
85nngt0d 8149 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  0  <  N )
9 modqmulmod 9471 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  B  e.  ZZ )  /\  ( N  e.  QQ  /\  0  < 
N ) )  -> 
( ( ( A  mod  N )  x.  B )  mod  N
)  =  ( ( A  x.  B )  mod  N ) )
103, 4, 7, 8, 9syl22anc 1171 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( ( A  mod  N )  x.  B )  mod  N )  =  ( ( A  x.  B )  mod  N
) )
1110eqcomd 2087 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A  x.  B
)  mod  N )  =  ( ( ( A  mod  N )  x.  B )  mod 
N ) )
1211adantr 270 . . 3  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( ( A  x.  B )  mod 
N )  =  ( ( ( A  mod  N )  x.  B )  mod  N ) )
13 dvdsval3 10344 . . . . . . . 8  |-  ( ( N  e.  NN  /\  A  e.  ZZ )  ->  ( N  ||  A  <->  ( A  mod  N )  =  0 ) )
14133adant3 959 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( N  ||  A  <->  ( A  mod  N )  =  0 ) )
1514biimpa 290 . . . . . 6  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( A  mod  N )  =  0 )
1615oveq1d 5558 . . . . 5  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( ( A  mod  N )  x.  B )  =  ( 0  x.  B ) )
1716oveq1d 5558 . . . 4  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( ( ( A  mod  N )  x.  B )  mod 
N )  =  ( ( 0  x.  B
)  mod  N )
)
184adantr 270 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  B  e.  ZZ )
1918zcnd 8551 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  B  e.  CC )
2019mul02d 7563 . . . . . 6  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( 0  x.  B )  =  0 )
2120oveq1d 5558 . . . . 5  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( ( 0  x.  B )  mod 
N )  =  ( 0  mod  N ) )
227adantr 270 . . . . . 6  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  N  e.  QQ )
238adantr 270 . . . . . 6  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  0  <  N
)
24 q0mod 9437 . . . . . 6  |-  ( ( N  e.  QQ  /\  0  <  N )  -> 
( 0  mod  N
)  =  0 )
2522, 23, 24syl2anc 403 . . . . 5  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( 0  mod 
N )  =  0 )
2621, 25eqtrd 2114 . . . 4  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( ( 0  x.  B )  mod 
N )  =  0 )
2717, 26eqtrd 2114 . . 3  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( ( ( A  mod  N )  x.  B )  mod 
N )  =  0 )
2812, 27eqtrd 2114 . 2  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( ( A  x.  B )  mod 
N )  =  0 )
2928ex 113 1  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( N  ||  A  ->  (
( A  x.  B
)  mod  N )  =  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   class class class wbr 3793  (class class class)co 5543   0cc0 7043    x. cmul 7048    < clt 7215   NNcn 8106   ZZcz 8432   QQcq 8785    mod cmo 9404    || cdvds 10340
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156  ax-arch 7157
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-po 4059  df-iso 4060  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-n0 8356  df-z 8433  df-q 8786  df-rp 8816  df-fl 9352  df-mod 9405  df-dvds 10341
This theorem is referenced by: (None)
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