Theorem muldvds2 12047

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

Assertion

Ref	Expression
muldvds2	|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) || N -> M || N ) )

Proof of Theorem muldvds2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zmulcl 9408	. . . 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 1196	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) e. ZZ /\ N e. ZZ ) )
4		3simpc 998	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ N e. ZZ ) )
5		zmulcl 9408	. . . 4 |- ( ( x e. ZZ /\ K e. ZZ ) -> ( x x. K ) e. ZZ )
6	5	ancoms 268	. . 3 |- ( ( K e. ZZ /\ x e. ZZ ) -> ( x x. K ) e. ZZ )
7	6	3ad2antl1 1161	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( x x. K ) e. ZZ )
8		zcn 9359	. . . . . . . 8 |- ( x e. ZZ -> x e. CC )
9		zcn 9359	. . . . . . . 8 |- ( K e. ZZ -> K e. CC )
10		zcn 9359	. . . . . . . 8 |- ( M e. ZZ -> M e. CC )
11		mulass 8038	. . . . . . . 8 |- ( ( x e. CC /\ K e. CC /\ M e. CC ) -> ( ( x x. K ) x. M ) = ( x x. ( K x. M ) ) )
12	8, 9, 10, 11	syl3an 1291	. . . . . . 7 |- ( ( x e. ZZ /\ K e. ZZ /\ M e. ZZ ) -> ( ( x x. K ) x. M ) = ( x x. ( K x. M ) ) )
13	12	3coml 1212	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ x e. ZZ ) -> ( ( x x. K ) x. M ) = ( x x. ( K x. M ) ) )
14	13	3expa 1205	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ ) /\ x e. ZZ ) -> ( ( x x. K ) x. M ) = ( x x. ( K x. M ) ) )
15	14	3adantl3 1157	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( ( x x. K ) x. M ) = ( x x. ( K x. M ) ) )
16	15	eqeq1d 2213	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( ( ( x x. K ) x. M ) = N <-> ( x x. ( K x. M ) ) = N ) )
17	16	biimprd 158	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( ( x x. ( K x. M ) ) = N -> ( ( x x. K ) x. M ) = N ) )
18	3, 4, 7, 17	dvds1lem 12032	1 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) || N -> M || N ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1372   e. wcel 2175   class class class wbr 4043  (class class class)co 5934   CCcc 7905   x. cmul 7912   ZZcz 9354   || cdvds 12017

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-setind 4583  ax-cnex 7998  ax-resscn 7999  ax-1cn 8000  ax-1re 8001  ax-icn 8002  ax-addcl 8003  ax-addrcl 8004  ax-mulcl 8005  ax-mulrcl 8006  ax-addcom 8007  ax-mulcom 8008  ax-addass 8009  ax-mulass 8010  ax-distr 8011  ax-i2m1 8012  ax-1rid 8014  ax-0id 8015  ax-rnegex 8016  ax-cnre 8018

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-iota 5229  df-fun 5270  df-fv 5276  df-riota 5889  df-ov 5937  df-oprab 5938  df-mpo 5939  df-sub 8227  df-neg 8228  df-inn 9019  df-n0 9278  df-z 9355  df-dvds 12018

This theorem is referenced by: (None)