Theorem dvdsmulc 10223

Description: Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)

Assertion

Ref	Expression
dvdsmulc	|- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M || N -> ( M x. K ) || ( N x. K ) ) )

Proof of Theorem dvdsmulc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		3simpc 937	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ N e. ZZ ) )
2		zmulcl 8404	. . . . . 6 |- ( ( M e. ZZ /\ K e. ZZ ) -> ( M x. K ) e. ZZ )
3	2	3adant2 957	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M x. K ) e. ZZ )
4		zmulcl 8404	. . . . . 6 |- ( ( N e. ZZ /\ K e. ZZ ) -> ( N x. K ) e. ZZ )
5	4	3adant1 956	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( N x. K ) e. ZZ )
6	3, 5	jca 300	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( ( M x. K ) e. ZZ /\ ( N x. K ) e. ZZ ) )
7	6	3comr 1146	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M x. K ) e. ZZ /\ ( N x. K ) e. ZZ ) )
8		simpr 108	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> x e. ZZ )
9		zcn 8356	. . . . . . . . 9 |- ( x e. ZZ -> x e. CC )
10		zcn 8356	. . . . . . . . 9 |- ( M e. ZZ -> M e. CC )
11		zcn 8356	. . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
12		mulass 7104	. . . . . . . . 9 |- ( ( x e. CC /\ M e. CC /\ K e. CC ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
13	9, 10, 11, 12	syl3an 1211	. . . . . . . 8 |- ( ( x e. ZZ /\ M e. ZZ /\ K e. ZZ ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
14	13	3com13 1143	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ x e. ZZ ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
15	14	3expa 1138	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ ) /\ x e. ZZ ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
16	15	3adantl3 1096	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
17		oveq1 5539	. . . . 5 |- ( ( x x. M ) = N -> ( ( x x. M ) x. K ) = ( N x. K ) )
18	16, 17	sylan9req 2134	. . . 4 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) /\ ( x x. M ) = N ) -> ( x x. ( M x. K ) ) = ( N x. K ) )
19	18	ex 113	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( ( x x. M ) = N -> ( x x. ( M x. K ) ) = ( N x. K ) ) )
20	1, 7, 8, 19	dvds1lem 10206	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M || N -> ( M x. K ) || ( N x. K ) ) )
21	20	3coml 1145	1 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M || N -> ( M x. K ) || ( N x. K ) ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   = wceq 1284   e. wcel 1433   class class class wbr 3785  (class class class)co 5532   CCcc 6979   x. cmul 6986   ZZcz 8351   || cdvds 10195

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-dvds 10196

This theorem is referenced by:  dvdsmulcr  10225  coprmdvds2  10475  mulgcddvds  10476  rpmulgcd2  10477