Theorem muldvds1 12322

Description: If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.)

Assertion

Ref	Expression
muldvds1	|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) || N -> K || N ) )

Proof of Theorem muldvds1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zmulcl 9496	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( K x. M ) e. ZZ )
2	1	anim1i 340	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ ) /\ N e. ZZ ) -> ( ( K x. M ) e. ZZ /\ N e. ZZ ) )
3	2	3impa 1218	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) e. ZZ /\ N e. ZZ ) )
4		3simpb 1019	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ N e. ZZ ) )
5		zmulcl 9496	. . . 4 |- ( ( x e. ZZ /\ M e. ZZ ) -> ( x x. M ) e. ZZ )
6	5	ancoms 268	. . 3 |- ( ( M e. ZZ /\ x e. ZZ ) -> ( x x. M ) e. ZZ )
7	6	3ad2antl2 1184	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( x x. M ) e. ZZ )
8		zcn 9447	. . . . . . . 8 |- ( x e. ZZ -> x e. CC )
9		zcn 9447	. . . . . . . 8 |- ( K e. ZZ -> K e. CC )
10		zcn 9447	. . . . . . . 8 |- ( M e. ZZ -> M e. CC )
11		mulass 8126	. . . . . . . . 9 |- ( ( x e. CC /\ K e. CC /\ M e. CC ) -> ( ( x x. K ) x. M ) = ( x x. ( K x. M ) ) )
12		mul32 8272	. . . . . . . . 9 |- ( ( x e. CC /\ K e. CC /\ M e. CC ) -> ( ( x x. K ) x. M ) = ( ( x x. M ) x. K ) )
13	11, 12	eqtr3d 2264	. . . . . . . 8 |- ( ( x e. CC /\ K e. CC /\ M e. CC ) -> ( x x. ( K x. M ) ) = ( ( x x. M ) x. K ) )
14	8, 9, 10, 13	syl3an 1313	. . . . . . 7 |- ( ( x e. ZZ /\ K e. ZZ /\ M e. ZZ ) -> ( x x. ( K x. M ) ) = ( ( x x. M ) x. K ) )
15	14	3coml 1234	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ x e. ZZ ) -> ( x x. ( K x. M ) ) = ( ( x x. M ) x. K ) )
16	15	3expa 1227	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ ) /\ x e. ZZ ) -> ( x x. ( K x. M ) ) = ( ( x x. M ) x. K ) )
17	16	3adantl3 1179	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( x x. ( K x. M ) ) = ( ( x x. M ) x. K ) )
18	17	eqeq1d 2238	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( ( x x. ( K x. M ) ) = N <-> ( ( x x. M ) x. K ) = N ) )
19	18	biimpd 144	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( ( x x. ( K x. M ) ) = N -> ( ( x x. M ) x. K ) = N ) )
20	3, 4, 7, 19	dvds1lem 12308	1 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) || N -> K || N ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4082  (class class class)co 6000   CCcc 7993   x. cmul 8000   ZZcz 9442   || cdvds 12293

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 4201  ax-pow 4257  ax-pr 4292  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106

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 3888  df-int 3923  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-sub 8315  df-neg 8316  df-inn 9107  df-n0 9366  df-z 9443  df-dvds 12294

This theorem is referenced by:  3dvds  12370  pw2dvdseulemle  12684