Theorem dvdscmul 11780

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 991	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ N e. ZZ ) )
2		zmulcl 9265	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( K x. M ) e. ZZ )
3	2	3adant3 1012	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) e. ZZ )
4		zmulcl 9265	. . . . 5 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( K x. N ) e. ZZ )
5	4	3adant2 1011	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K x. N ) e. ZZ )
6	3, 5	jca 304	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) e. ZZ /\ ( K x. N ) e. ZZ ) )
7		simpr 109	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> x e. ZZ )
8		zcn 9217	. . . . . . . . 9 |- ( x e. ZZ -> x e. CC )
9		zcn 9217	. . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
10		zcn 9217	. . . . . . . . 9 |- ( M e. ZZ -> M e. CC )
11		mul12 8048	. . . . . . . . 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 1275	. . . . . . . 8 |- ( ( x e. ZZ /\ K e. ZZ /\ M e. ZZ ) -> ( x x. ( K x. M ) ) = ( K x. ( x x. M ) ) )
13	12	3coml 1205	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ x e. ZZ ) -> ( x x. ( K x. M ) ) = ( K x. ( x x. M ) ) )
14	13	3expa 1198	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ ) /\ x e. ZZ ) -> ( x x. ( K x. M ) ) = ( K x. ( x x. M ) ) )
15	14	3adantl3 1150	. . . . 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 5861	. . . . 5 |- ( ( x x. M ) = N -> ( K x. ( x x. M ) ) = ( K x. N ) )
17	15, 16	sylan9eq 2223	. . . 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 114	. . 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 11764	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M || N -> ( K x. M ) || ( K x. N ) ) )
20	19	3coml 1205	1 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M || N -> ( K x. M ) || ( K x. N ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   CCcc 7772   x. cmul 7779   ZZcz 9212   || 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-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  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-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885

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-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  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-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-dvds 11750

This theorem is referenced by:  dvdscmulr  11782  mulgcd  11971  dvdsmulgcd  11980  rpmulgcd2  12049  pcprendvds2  12245  pcpremul  12247