Theorem dvdscmul 12369

Description: Multiplication by a constant maintains the divides relation. Theorem 1.1(d) in [ApostolNT] p. 14 (multiplication property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)

Assertion

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

Proof of Theorem dvdscmul

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 9523	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( K x. M ) e. ZZ )
3	2	3adant3 1041	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) e. ZZ )
4		zmulcl 9523	. . . . 5 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( K x. N ) e. ZZ )
5	4	3adant2 1040	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K x. N ) e. ZZ )
6	3, 5	jca 306	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) e. ZZ /\ ( K x. N ) e. ZZ ) )
7		simpr 110	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> x e. ZZ )
8		zcn 9474	. . . . . . . . 9 |- ( x e. ZZ -> x e. CC )
9		zcn 9474	. . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
10		zcn 9474	. . . . . . . . 9 |- ( M e. ZZ -> M e. CC )
11		mul12 8298	. . . . . . . . 9 |- ( ( x e. CC /\ K e. CC /\ M e. CC ) -> ( x x. ( K x. M ) ) = ( K x. ( x x. M ) ) )
12	8, 9, 10, 11	syl3an 1313	. . . . . . . 8 |- ( ( x e. ZZ /\ K e. ZZ /\ M e. ZZ ) -> ( x x. ( K x. M ) ) = ( K x. ( x x. M ) ) )
13	12	3coml 1234	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ x e. ZZ ) -> ( x x. ( K x. M ) ) = ( K x. ( x x. M ) ) )
14	13	3expa 1227	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ ) /\ x e. ZZ ) -> ( x x. ( K x. M ) ) = ( K x. ( x x. M ) ) )
15	14	3adantl3 1179	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( x x. ( K x. M ) ) = ( K x. ( x x. M ) ) )
16		oveq2 6021	. . . . 5 |- ( ( x x. M ) = N -> ( K x. ( x x. M ) ) = ( K x. N ) )
17	15, 16	sylan9eq 2282	. . . 4 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) /\ ( x x. M ) = N ) -> ( x x. ( K x. M ) ) = ( K x. N ) )
18	17	ex 115	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( ( x x. M ) = N -> ( x x. ( K x. M ) ) = ( K x. N ) ) )
19	1, 6, 7, 18	dvds1lem 12353	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M || N -> ( K x. M ) || ( K x. N ) ) )
20	19	3coml 1234	1 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M || N -> ( K x. M ) || ( K x. N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4086  (class class class)co 6013   CCcc 8020   x. cmul 8027   ZZcz 9469   || cdvds 12338

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 4205  ax-pow 4262  ax-pr 4297  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133

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 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-sub 8342  df-neg 8343  df-inn 9134  df-n0 9393  df-z 9470  df-dvds 12339

This theorem is referenced by:  dvdscmulr  12371  mulgcd  12577  dvdsmulgcd  12586  rpmulgcd2  12657  pcprendvds2  12854  pcpremul  12856  mpodvdsmulf1o  15704