Theorem dvdsmulc 12330

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 1020	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ N e. ZZ ) )
2		zmulcl 9500	. . . . . 6 |- ( ( M e. ZZ /\ K e. ZZ ) -> ( M x. K ) e. ZZ )
3	2	3adant2 1040	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M x. K ) e. ZZ )
4		zmulcl 9500	. . . . . 6 |- ( ( N e. ZZ /\ K e. ZZ ) -> ( N x. K ) e. ZZ )
5	4	3adant1 1039	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( N x. K ) e. ZZ )
6	3, 5	jca 306	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( ( M x. K ) e. ZZ /\ ( N x. K ) e. ZZ ) )
7	6	3comr 1235	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M x. K ) e. ZZ /\ ( N x. K ) e. ZZ ) )
8		simpr 110	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> x e. ZZ )
9		zcn 9451	. . . . . . . . 9 |- ( x e. ZZ -> x e. CC )
10		zcn 9451	. . . . . . . . 9 |- ( M e. ZZ -> M e. CC )
11		zcn 9451	. . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
12		mulass 8130	. . . . . . . . 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 1313	. . . . . . . 8 |- ( ( x e. ZZ /\ M e. ZZ /\ K e. ZZ ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
14	13	3com13 1232	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ x e. ZZ ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
15	14	3expa 1227	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ ) /\ x e. ZZ ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
16	15	3adantl3 1179	. . . . 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 6008	. . . . 5 |- ( ( x x. M ) = N -> ( ( x x. M ) x. K ) = ( N x. K ) )
18	16, 17	sylan9req 2283	. . . 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 115	. . 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 12313	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M || N -> ( M x. K ) || ( N x. K ) ) )
21	20	3coml 1234	1 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M || N -> ( M x. K ) || ( N x. K ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4083  (class class class)co 6001   CCcc 7997   x. cmul 8004   ZZcz 9446   || cdvds 12298

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-sub 8319  df-neg 8320  df-inn 9111  df-n0 9370  df-z 9447  df-dvds 12299

This theorem is referenced by:  dvdsmulcr  12332  coprmdvds2  12615  mulgcddvds  12616  rpmulgcd2  12617  pcpremul  12816  znrrg  14624  mpodvdsmulf1o  15664