Theorem mulmoddvds 11882

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 999	. . . . . . 7 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> A e. ZZ )
2		zq 9639	. . . . . . 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 1000	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> B e. ZZ )
5		simp1 998	. . . . . . 7 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> N e. NN )
6		nnq 9646	. . . . . . 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 8976	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> 0 < N )
9		modqmulmod 10402	. . . . . 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 1249	. . . . 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 2193	. . . 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 11811	. . . . . . . 8 |- ( ( N e. NN /\ A e. ZZ ) -> ( N || A <-> ( A mod N ) = 0 ) )
14	13	3adant3 1018	. . . . . . 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 5903	. . . . 5 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( ( A mod N ) x. B ) = ( 0 x. B ) )
17	16	oveq1d 5903	. . . 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 9389	. . . . . . 7 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> B e. CC )
20	19	mul02d 8362	. . . . . 6 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( 0 x. B ) = 0 )
21	20	oveq1d 5903	. . . . 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 10368	. . . . . 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 2220	. . . 4 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( ( 0 x. B ) mod N ) = 0 )
27	17, 26	eqtrd 2220	. . 3 |- ( ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) /\ N || A ) -> ( ( ( A mod N ) x. B ) mod N ) = 0 )
28	12, 27	eqtrd 2220	. 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 979   = wceq 1363   e. wcel 2158   class class class wbr 4015  (class class class)co 5888   0cc0 7824   x. cmul 7829   < clt 8005   NNcn 8932   ZZcz 9266   QQcq 9632   mod cmo 10335   || cdvds 11807

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942  ax-arch 7943

This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-n0 9190  df-z 9267  df-q 9633  df-rp 9667  df-fl 10283  df-mod 10336  df-dvds 11808

This theorem is referenced by: (None)