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Theorem mulmoddvds 13919
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 zre 10880 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  RR )
2 id 22 . . . . . . 7  |-  ( B  e.  ZZ  ->  B  e.  ZZ )
3 nnrp 11241 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  RR+ )
41, 2, 33anim123i 1181 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  ( A  e.  RR  /\  B  e.  ZZ  /\  N  e.  RR+ ) )
543comr 1204 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  e.  RR  /\  B  e.  ZZ  /\  N  e.  RR+ ) )
65adantr 465 . . . 4  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( A  e.  RR  /\  B  e.  ZZ  /\  N  e.  RR+ ) )
7 modmulmod 12032 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  ZZ  /\  N  e.  RR+ )  ->  (
( ( A  mod  N )  x.  B )  mod  N )  =  ( ( A  x.  B )  mod  N
) )
87eqcomd 2475 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  ZZ  /\  N  e.  RR+ )  ->  (
( A  x.  B
)  mod  N )  =  ( ( ( A  mod  N )  x.  B )  mod 
N ) )
96, 8syl 16 . . 3  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( ( A  x.  B )  mod 
N )  =  ( ( ( A  mod  N )  x.  B )  mod  N ) )
10 dvdsval3 13867 . . . . . . . 8  |-  ( ( N  e.  NN  /\  A  e.  ZZ )  ->  ( N  ||  A  <->  ( A  mod  N )  =  0 ) )
11103adant3 1016 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( N  ||  A  <->  ( A  mod  N )  =  0 ) )
1211biimpa 484 . . . . . 6  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( A  mod  N )  =  0 )
1312oveq1d 6310 . . . . 5  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( ( A  mod  N )  x.  B )  =  ( 0  x.  B ) )
1413oveq1d 6310 . . . 4  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( ( ( A  mod  N )  x.  B )  mod 
N )  =  ( ( 0  x.  B
)  mod  N )
)
15 zcn 10881 . . . . . . . . . 10  |-  ( B  e.  ZZ  ->  B  e.  CC )
1615mul02d 9789 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  (
0  x.  B )  =  0 )
1716adantl 466 . . . . . . . 8  |-  ( ( N  e.  NN  /\  B  e.  ZZ )  ->  ( 0  x.  B
)  =  0 )
1817oveq1d 6310 . . . . . . 7  |-  ( ( N  e.  NN  /\  B  e.  ZZ )  ->  ( ( 0  x.  B )  mod  N
)  =  ( 0  mod  N ) )
19 0mod 12007 . . . . . . . . 9  |-  ( N  e.  RR+  ->  ( 0  mod  N )  =  0 )
203, 19syl 16 . . . . . . . 8  |-  ( N  e.  NN  ->  (
0  mod  N )  =  0 )
2120adantr 465 . . . . . . 7  |-  ( ( N  e.  NN  /\  B  e.  ZZ )  ->  ( 0  mod  N
)  =  0 )
2218, 21eqtrd 2508 . . . . . 6  |-  ( ( N  e.  NN  /\  B  e.  ZZ )  ->  ( ( 0  x.  B )  mod  N
)  =  0 )
23223adant2 1015 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( 0  x.  B
)  mod  N )  =  0 )
2423adantr 465 . . . 4  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( ( 0  x.  B )  mod 
N )  =  0 )
2514, 24eqtrd 2508 . . 3  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( ( ( A  mod  N )  x.  B )  mod 
N )  =  0 )
269, 25eqtrd 2508 . 2  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( ( A  x.  B )  mod 
N )  =  0 )
2726ex 434 1  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( N  ||  A  ->  (
( A  x.  B
)  mod  N )  =  0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4453  (class class class)co 6295   RRcr 9503   0cc0 9504    x. cmul 9509   NNcn 10548   ZZcz 10876   RR+crp 11232    mod cmo 11976    || cdivides 13863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fl 11909  df-mod 11977  df-dvds 13864
This theorem is referenced by:  numclwwlk5  24944  numclwwlk7  24946
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