Theorem muldvds2 12243

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 9461	. . . 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 1197	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) e. ZZ /\ N e. ZZ ) )
4		3simpc 999	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ N e. ZZ ) )
5		zmulcl 9461	. . . 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 1162	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( x x. K ) e. ZZ )
8		zcn 9412	. . . . . . . 8 |- ( x e. ZZ -> x e. CC )
9		zcn 9412	. . . . . . . 8 |- ( K e. ZZ -> K e. CC )
10		zcn 9412	. . . . . . . 8 |- ( M e. ZZ -> M e. CC )
11		mulass 8091	. . . . . . . 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 1292	. . . . . . 7 |- ( ( x e. ZZ /\ K e. ZZ /\ M e. ZZ ) -> ( ( x x. K ) x. M ) = ( x x. ( K x. M ) ) )
13	12	3coml 1213	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ x e. ZZ ) -> ( ( x x. K ) x. M ) = ( x x. ( K x. M ) ) )
14	13	3expa 1206	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ ) /\ x e. ZZ ) -> ( ( x x. K ) x. M ) = ( x x. ( K x. M ) ) )
15	14	3adantl3 1158	. . . 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 2216	. . 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 12228	1 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) || N -> M || N ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2178   class class class wbr 4059  (class class class)co 5967   CCcc 7958   x. cmul 7965   ZZcz 9407   || cdvds 12213

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408  df-dvds 12214

This theorem is referenced by: (None)