Theorem mulmoddvds 11823

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 993	. . . . . . 7 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> A e. ZZ )
2		zq 9585	. . . . . . 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 994	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> B e. ZZ )
5		simp1 992	. . . . . . 7 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> N e. NN )
6		nnq 9592	. . . . . . 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 8922	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> 0 < N )
9		modqmulmod 10345	. . . . . 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 1234	. . . . 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 2176	. . . 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 11753	. . . . . . . 8 |- ( ( N e. NN /\ A e. ZZ ) -> ( N || A <-> ( A mod N ) = 0 ) )
14	13	3adant3 1012	. . . . . . 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 5868	. . . . 5 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( ( A mod N ) x. B ) = ( 0 x. B ) )
17	16	oveq1d 5868	. . . 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 9335	. . . . . . 7 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> B e. CC )
20	19	mul02d 8311	. . . . . 6 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( 0 x. B ) = 0 )
21	20	oveq1d 5868	. . . . 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 10311	. . . . . 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 2203	. . . 4 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( ( 0 x. B ) mod N ) = 0 )
27	17, 26	eqtrd 2203	. . 3 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( ( ( A mod N ) x. B ) mod N ) = 0 )
28	12, 27	eqtrd 2203	. 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 973   = wceq 1348   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   0cc0 7774   x. cmul 7779   < clt 7954   NNcn 8878   ZZcz 9212   QQcq 9578   mod cmo 10278   || cdvds 11749

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-n0 9136  df-z 9213  df-q 9579  df-rp 9611  df-fl 10226  df-mod 10279  df-dvds 11750

This theorem is referenced by: (None)