Theorem mulmoddvds 11801

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

Step	Hyp	Ref	Expression
1		simp2 988	. . . . . . 7 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> A e. ZZ )
2		zq 9564	. . . . . . 7 |- ( A e. ZZ -> A e. QQ )
3	1, 2	syl 14	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> A e. QQ )
4		simp3 989	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> B e. ZZ )
5		simp1 987	. . . . . . 7 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> N e. NN )
6		nnq 9571	. . . . . . 7 |- ( N e. NN -> N e. QQ )
7	5, 6	syl 14	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> N e. QQ )
8	5	nngt0d 8901	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> 0 < N )
9		modqmulmod 10324	. . . . . 6 |- ( ( ( A e. QQ /\ B e. ZZ ) /\ ( N e. QQ /\ 0 < N ) ) -> ( ( ( A mod N ) x. B ) mod N ) = ( ( A x. B ) mod N ) )
10	3, 4, 7, 8, 9	syl22anc 1229	. . . . 5 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> ( ( ( A mod N ) x. B ) mod N ) = ( ( A x. B ) mod N ) )
11	10	eqcomd 2171	. . . 4 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> ( ( A x. B ) mod N ) = ( ( ( A mod N ) x. B ) mod N ) )
12	11	adantr 274	. . 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 11731	. . . . . . . 8 |- ( ( N e. NN /\ A e. ZZ ) -> ( N || A <-> ( A mod N ) = 0 ) )
14	13	3adant3 1007	. . . . . . 7 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> ( N || A <-> ( A mod N ) = 0 ) )
15	14	biimpa 294	. . . . . 6 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( A mod N ) = 0 )
16	15	oveq1d 5857	. . . . 5 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( ( A mod N ) x. B ) = ( 0 x. B ) )
17	16	oveq1d 5857	. . . 4 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( ( ( A mod N ) x. B ) mod N ) = ( ( 0 x. B ) mod N ) )
18	4	adantr 274	. . . . . . . 8 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> B e. ZZ )
19	18	zcnd 9314	. . . . . . 7 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> B e. CC )
20	19	mul02d 8290	. . . . . 6 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( 0 x. B ) = 0 )
21	20	oveq1d 5857	. . . . 5 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( ( 0 x. B ) mod N ) = ( 0 mod N ) )
22	7	adantr 274	. . . . . 6 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> N e. QQ )
23	8	adantr 274	. . . . . 6 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> 0 < N )
24		q0mod 10290	. . . . . 6 |- ( ( N e. QQ /\ 0 < N ) -> ( 0 mod N ) = 0 )
25	22, 23, 24	syl2anc 409	. . . . 5 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( 0 mod N ) = 0 )
26	21, 25	eqtrd 2198	. . . 4 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( ( 0 x. B ) mod N ) = 0 )
27	17, 26	eqtrd 2198	. . 3 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( ( ( A mod N ) x. B ) mod N ) = 0 )
28	12, 27	eqtrd 2198	. 2 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( ( A x. B ) mod N ) = 0 )
29	28	ex 114	1 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> ( N || A -> ( ( A x. B ) mod N ) = 0 ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   0cc0 7753   x. cmul 7758   < clt 7933   NNcn 8857   ZZcz 9191   QQcq 9557   mod cmo 10257   || cdvds 11727

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-n0 9115  df-z 9192  df-q 9558  df-rp 9590  df-fl 10205  df-mod 10258  df-dvds 11728

This theorem is referenced by: (None)