Theorem lcmfdvds

Description: The least common multiple of a set of integers divides any integer which is divisible by all elements of the set. (Contributed by AV, 26-Aug-2020)

	Ref	Expression
Assertion	lcmfdvds	|- ( ( K e. ZZ /\ Z C_ ZZ /\ Z e. Fin ) -> ( A. m e. Z m || K -> ( _lcm ` Z ) || K ) )

Proof

Step	Hyp	Ref	Expression
1		breq2	|- ( k = K -> ( m || k <-> m || K ) )
2	1	ralbidv	|- ( k = K -> ( A. m e. Z m || k <-> A. m e. Z m || K ) )
3		breq2	|- ( k = K -> ( ( _lcm ` Z ) || k <-> ( _lcm ` Z ) || K ) )
4	2 3	imbi12d	|- ( k = K -> ( ( A. m e. Z m || k -> ( _lcm ` Z ) || k ) <-> ( A. m e. Z m || K -> ( _lcm ` Z ) || K ) ) )
5	4	rspccv	|- ( A. k e. ZZ ( A. m e. Z m || k -> ( _lcm ` Z ) || k ) -> ( K e. ZZ -> ( A. m e. Z m || K -> ( _lcm ` Z ) || K ) ) )
6	5	adantr	|- ( ( A. k e. ZZ ( A. m e. Z m || k -> ( _lcm ` Z ) || k ) /\ A. n e. ZZ ( _lcm ` ( Z u. { n } ) ) = ( ( _lcm ` Z ) lcm n ) ) -> ( K e. ZZ -> ( A. m e. Z m || K -> ( _lcm ` Z ) || K ) ) )
7		lcmfunsnlem	|- ( ( Z C_ ZZ /\ Z e. Fin ) -> ( A. k e. ZZ ( A. m e. Z m || k -> ( _lcm ` Z ) || k ) /\ A. n e. ZZ ( _lcm ` ( Z u. { n } ) ) = ( ( _lcm ` Z ) lcm n ) ) )
8	6 7	syl11	|- ( K e. ZZ -> ( ( Z C_ ZZ /\ Z e. Fin ) -> ( A. m e. Z m || K -> ( _lcm ` Z ) || K ) ) )
9	8	3impib	|- ( ( K e. ZZ /\ Z C_ ZZ /\ Z e. Fin ) -> ( A. m e. Z m || K -> ( _lcm ` Z ) || K ) )