Theorem mulmoddvds 11888

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 1000	. . . . . . 7 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> A e. ZZ )
2		zq 9645	. . . . . . 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 1001	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> B e. ZZ )
5		simp1 999	. . . . . . 7 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> N e. NN )
6		nnq 9652	. . . . . . 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 8982	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> 0 < N )
9		modqmulmod 10408	. . . . . 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 1250	. . . . 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 2195	. . . 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 276	. . 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 11817	. . . . . . . 8 |- ( ( N e. NN /\ A e. ZZ ) -> ( N || A <-> ( A mod N ) = 0 ) )
14	13	3adant3 1019	. . . . . . 7 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> ( N || A <-> ( A mod N ) = 0 ) )
15	14	biimpa 296	. . . . . 6 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( A mod N ) = 0 )
16	15	oveq1d 5906	. . . . 5 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( ( A mod N ) x. B ) = ( 0 x. B ) )
17	16	oveq1d 5906	. . . 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 276	. . . . . . . 8 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> B e. ZZ )
19	18	zcnd 9395	. . . . . . 7 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> B e. CC )
20	19	mul02d 8368	. . . . . 6 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( 0 x. B ) = 0 )
21	20	oveq1d 5906	. . . . 5 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( ( 0 x. B ) mod N ) = ( 0 mod N ) )
22	7	adantr 276	. . . . . 6 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> N e. QQ )
23	8	adantr 276	. . . . . 6 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> 0 < N )
24		q0mod 10374	. . . . . 6 |- ( ( N e. QQ /\ 0 < N ) -> ( 0 mod N ) = 0 )
25	22, 23, 24	syl2anc 411	. . . . 5 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( 0 mod N ) = 0 )
26	21, 25	eqtrd 2222	. . . 4 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( ( 0 x. B ) mod N ) = 0 )
27	17, 26	eqtrd 2222	. . 3 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( ( ( A mod N ) x. B ) mod N ) = 0 )
28	12, 27	eqtrd 2222	. 2 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( ( A x. B ) mod N ) = 0 )
29	28	ex 115	1 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> ( N || A -> ( ( A x. B ) mod N ) = 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2160   class class class wbr 4018  (class class class)co 5891   0cc0 7830   x. cmul 7835   < clt 8011   NNcn 8938   ZZcz 9272   QQcq 9638   mod cmo 10341   || cdvds 11813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-mulrcl 7929  ax-addcom 7930  ax-mulcom 7931  ax-addass 7932  ax-mulass 7933  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-1rid 7937  ax-0id 7938  ax-rnegex 7939  ax-precex 7940  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-apti 7945  ax-pre-ltadd 7946  ax-pre-mulgt0 7947  ax-pre-mulext 7948  ax-arch 7949

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-po 4311  df-iso 4312  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-reap 8551  df-ap 8558  df-div 8649  df-inn 8939  df-n0 9196  df-z 9273  df-q 9639  df-rp 9673  df-fl 10289  df-mod 10342  df-dvds 11814

This theorem is referenced by: (None)