Theorem mulmoddvds 11550

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 982	. . . . . . 7 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> A e. ZZ )
2		zq 9411	. . . . . . 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 983	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> B e. ZZ )
5		simp1 981	. . . . . . 7 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> N e. NN )
6		nnq 9418	. . . . . . 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 8757	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> 0 < N )
9		modqmulmod 10155	. . . . . 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 1217	. . . . 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 2143	. . . 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 11486	. . . . . . . 8 |- ( ( N e. NN /\ A e. ZZ ) -> ( N || A <-> ( A mod N ) = 0 ) )
14	13	3adant3 1001	. . . . . . 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 5782	. . . . 5 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( ( A mod N ) x. B ) = ( 0 x. B ) )
17	16	oveq1d 5782	. . . 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 9167	. . . . . . 7 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> B e. CC )
20	19	mul02d 8147	. . . . . 6 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( 0 x. B ) = 0 )
21	20	oveq1d 5782	. . . . 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 10121	. . . . . 6 |- ( ( N e. QQ /\ 0 < N ) -> ( 0 mod N ) = 0 )
25	22, 23, 24	syl2anc 408	. . . . 5 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( 0 mod N ) = 0 )
26	21, 25	eqtrd 2170	. . . 4 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( ( 0 x. B ) mod N ) = 0 )
27	17, 26	eqtrd 2170	. . 3 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( ( ( A mod N ) x. B ) mod N ) = 0 )
28	12, 27	eqtrd 2170	. 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 962   = wceq 1331   e. wcel 1480   class class class wbr 3924  (class class class)co 5767   0cc0 7613   x. cmul 7618   < clt 7793   NNcn 8713   ZZcz 9047   QQcq 9404   mod cmo 10088   || cdvds 11482

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-po 4213  df-iso 4214  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-n0 8971  df-z 9048  df-q 9405  df-rp 9435  df-fl 10036  df-mod 10089  df-dvds 11483

This theorem is referenced by: (None)