Theorem muldvds2 11855

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 9335	. . . 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 9335	. . . 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 9287	. . . . . . . 8 |- ( x e. ZZ -> x e. CC )
9		zcn 9287	. . . . . . . 8 |- ( K e. ZZ -> K e. CC )
10		zcn 9287	. . . . . . . 8 |- ( M e. ZZ -> M e. CC )
11		mulass 7971	. . . . . . . 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 2198	. . 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 11840	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 1364   e. wcel 2160   class class class wbr 4018  (class class class)co 5895   CCcc 7838   x. cmul 7845   ZZcz 9282   || cdvds 11825

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-cnre 7951

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-sub 8159  df-neg 8160  df-inn 8949  df-n0 9206  df-z 9283  df-dvds 11826

This theorem is referenced by: (None)