Theorem muldvds1 12561

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 9677	. . . 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 1225	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) e. ZZ /\ N e. ZZ ) )
4		3simpb 1026	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ N e. ZZ ) )
5		zmulcl 9677	. . . 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 1191	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( x x. M ) e. ZZ )
8		zcn 9628	. . . . . . . 8 |- ( x e. ZZ -> x e. CC )
9		zcn 9628	. . . . . . . 8 |- ( K e. ZZ -> K e. CC )
10		zcn 9628	. . . . . . . 8 |- ( M e. ZZ -> M e. CC )
11		mulass 8300	. . . . . . . . 9 |- ( ( x e. CC /\ K e. CC /\ M e. CC ) -> ( ( x x. K ) x. M ) = ( x x. ( K x. M ) ) )
12		mul32 8446	. . . . . . . . 9 |- ( ( x e. CC /\ K e. CC /\ M e. CC ) -> ( ( x x. K ) x. M ) = ( ( x x. M ) x. K ) )
13	11, 12	eqtr3d 2273	. . . . . . . 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 1320	. . . . . . 7 |- ( ( x e. ZZ /\ K e. ZZ /\ M e. ZZ ) -> ( x x. ( K x. M ) ) = ( ( x x. M ) x. K ) )
15	14	3coml 1241	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ x e. ZZ ) -> ( x x. ( K x. M ) ) = ( ( x x. M ) x. K ) )
16	15	3expa 1234	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ ) /\ x e. ZZ ) -> ( x x. ( K x. M ) ) = ( ( x x. M ) x. K ) )
17	16	3adantl3 1186	. . . 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 2247	. . 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 12547	1 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) || N -> K || N ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4125  (class class class)co 6075   CCcc 8167   x. cmul 8174   ZZcz 9623   || cdvds 12532

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-dvds 12533

This theorem is referenced by:  3dvds  12609  pw2dvdseulemle  12923