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Metamath Proof Explorer

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 ) )