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Metamath Proof Explorer

Theorem dvdsmultr1

Description: If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012)

Ref Expression
Assertion dvdsmultr1 
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K || M -> K || ( M x. N ) ) )

Proof

Step Hyp Ref Expression
1 dvdsmul1 
 |-  ( ( M e. ZZ /\ N e. ZZ ) -> M || ( M x. N ) )
2 1 3adant1 
 |-  ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M || ( M x. N ) )
3 zmulcl 
 |-  ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
4 3 3adant1 
 |-  ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
5 dvdstr 
 |-  ( ( K e. ZZ /\ M e. ZZ /\ ( M x. N ) e. ZZ ) -> ( ( K || M /\ M || ( M x. N ) ) -> K || ( M x. N ) ) )
6 4 5 syld3an3 
 |-  ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ M || ( M x. N ) ) -> K || ( M x. N ) ) )
7 2 6 mpan2d 
 |-  ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K || M -> K || ( M x. N ) ) )