Theorem dvdscmul 12378

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 1022	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ N e. ZZ ) )
2		zmulcl 9532	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( K x. M ) e. ZZ )
3	2	3adant3 1043	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) e. ZZ )
4		zmulcl 9532	. . . . 5 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( K x. N ) e. ZZ )
5	4	3adant2 1042	. . . 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 9483	. . . . . . . . 9 |- ( x e. ZZ -> x e. CC )
9		zcn 9483	. . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
10		zcn 9483	. . . . . . . . 9 |- ( M e. ZZ -> M e. CC )
11		mul12 8307	. . . . . . . . 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 1315	. . . . . . . 8 |- ( ( x e. ZZ /\ K e. ZZ /\ M e. ZZ ) -> ( x x. ( K x. M ) ) = ( K x. ( x x. M ) ) )
13	12	3coml 1236	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ x e. ZZ ) -> ( x x. ( K x. M ) ) = ( K x. ( x x. M ) ) )
14	13	3expa 1229	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ ) /\ x e. ZZ ) -> ( x x. ( K x. M ) ) = ( K x. ( x x. M ) ) )
15	14	3adantl3 1181	. . . . 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 6025	. . . . 5 |- ( ( x x. M ) = N -> ( K x. ( x x. M ) ) = ( K x. N ) )
17	15, 16	sylan9eq 2284	. . . 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 12362	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M || N -> ( K x. M ) || ( K x. N ) ) )
20	19	3coml 1236	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 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088  (class class class)co 6017   CCcc 8029   x. cmul 8036   ZZcz 9478   || cdvds 12347

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-sub 8351  df-neg 8352  df-inn 9143  df-n0 9402  df-z 9479  df-dvds 12348

This theorem is referenced by:  dvdscmulr  12380  mulgcd  12586  dvdsmulgcd  12595  rpmulgcd2  12666  pcprendvds2  12863  pcpremul  12865  mpodvdsmulf1o  15713