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Theorem mulmoddvds 11203
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 945 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  A  e.  ZZ )
2 zq 9172 . . . . . . 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 946 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  B  e.  ZZ )
5 simp1 944 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  N  e.  NN )
6 nnq 9179 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  QQ )
75, 6syl 14 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  N  e.  QQ )
85nngt0d 8527 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  0  <  N )
9 modqmulmod 9857 . . . . . 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 1176 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( ( A  mod  N )  x.  B )  mod  N )  =  ( ( A  x.  B )  mod  N
) )
1110eqcomd 2094 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A  x.  B
)  mod  N )  =  ( ( ( A  mod  N )  x.  B )  mod 
N ) )
1211adantr 271 . . 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 11139 . . . . . . . 8  |-  ( ( N  e.  NN  /\  A  e.  ZZ )  ->  ( N  ||  A  <->  ( A  mod  N )  =  0 ) )
14133adant3 964 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( N  ||  A  <->  ( A  mod  N )  =  0 ) )
1514biimpa 291 . . . . . 6  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( A  mod  N )  =  0 )
1615oveq1d 5681 . . . . 5  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( ( A  mod  N )  x.  B )  =  ( 0  x.  B ) )
1716oveq1d 5681 . . . 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 271 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  B  e.  ZZ )
1918zcnd 8930 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  B  e.  CC )
2019mul02d 7931 . . . . . 6  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( 0  x.  B )  =  0 )
2120oveq1d 5681 . . . . 5  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( ( 0  x.  B )  mod 
N )  =  ( 0  mod  N ) )
227adantr 271 . . . . . 6  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  N  e.  QQ )
238adantr 271 . . . . . 6  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  0  <  N
)
24 q0mod 9823 . . . . . 6  |-  ( ( N  e.  QQ  /\  0  <  N )  -> 
( 0  mod  N
)  =  0 )
2522, 23, 24syl2anc 404 . . . . 5  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( 0  mod 
N )  =  0 )
2621, 25eqtrd 2121 . . . 4  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( ( 0  x.  B )  mod 
N )  =  0 )
2717, 26eqtrd 2121 . . 3  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( ( ( A  mod  N )  x.  B )  mod 
N )  =  0 )
2812, 27eqtrd 2121 . 2  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( ( A  x.  B )  mod 
N )  =  0 )
2928ex 114 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 103    <-> wb 104    /\ w3a 925    = wceq 1290    e. wcel 1439   class class class wbr 3851  (class class class)co 5666   0cc0 7411    x. cmul 7416    < clt 7583   NNcn 8483   ZZcz 8811   QQcq 9165    mod cmo 9790    || cdvds 11135
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 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-mulrcl 7505  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-precex 7516  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522  ax-pre-mulgt0 7523  ax-pre-mulext 7524  ax-arch 7525
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 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-po 4132  df-iso 4133  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-reap 8113  df-ap 8120  df-div 8201  df-inn 8484  df-n0 8735  df-z 8812  df-q 9166  df-rp 9196  df-fl 9738  df-mod 9791  df-dvds 11136
This theorem is referenced by: (None)
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