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Theorem mulmoddvds 30244
Description: If an integer is divisible by a positive number, the product of this integer with another integer modulo the positive number 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 10649 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  RR )
2 id 22 . . . . . . 7  |-  ( B  e.  ZZ  ->  B  e.  ZZ )
3 nnrp 10999 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  RR+ )
41, 2, 33anim123i 1173 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  ( A  e.  RR  /\  B  e.  ZZ  /\  N  e.  RR+ ) )
543comr 1195 . . . . 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 11763 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  ZZ  /\  N  e.  RR+ )  ->  (
( ( A  mod  N )  x.  B )  mod  N )  =  ( ( A  x.  B )  mod  N
) )
87eqcomd 2447 . . . 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 13538 . . . . . . . 8  |-  ( ( N  e.  NN  /\  A  e.  ZZ )  ->  ( N  ||  A  <->  ( A  mod  N )  =  0 ) )
11103adant3 1008 . . . . . . 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 6105 . . . . 5  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( ( A  mod  N )  x.  B )  =  ( 0  x.  B ) )
1413oveq1d 6105 . . . 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 10650 . . . . . . . . . 10  |-  ( B  e.  ZZ  ->  B  e.  CC )
1615mul02d 9566 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  (
0  x.  B )  =  0 )
1716adantl 466 . . . . . . . 8  |-  ( ( N  e.  NN  /\  B  e.  ZZ )  ->  ( 0  x.  B
)  =  0 )
1817oveq1d 6105 . . . . . . 7  |-  ( ( N  e.  NN  /\  B  e.  ZZ )  ->  ( ( 0  x.  B )  mod  N
)  =  ( 0  mod  N ) )
19 0mod 11738 . . . . . . . . 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 2474 . . . . . 6  |-  ( ( N  e.  NN  /\  B  e.  ZZ )  ->  ( ( 0  x.  B )  mod  N
)  =  0 )
23223adant2 1007 . . . . 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 2474 . . 3  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  N  ||  A )  ->  ( ( ( A  mod  N )  x.  B )  mod 
N )  =  0 )
269, 25eqtrd 2474 . 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 965    = wceq 1369    e. wcel 1756   class class class wbr 4291  (class class class)co 6090   RRcr 9280   0cc0 9281    x. cmul 9286   NNcn 10321   ZZcz 10645   RR+crp 10990    mod cmo 11707    || cdivides 13534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6831  df-rdg 6865  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-n0 10579  df-z 10646  df-uz 10861  df-rp 10991  df-fl 11641  df-mod 11708  df-dvds 13535
This theorem is referenced by:  numclwwlk5  30703  numclwwlk7  30705
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