Theorem lcmfsn

Description: The least common multiple of a singleton is its absolute value. (Contributed by AV, 22-Aug-2020)

	Ref	Expression
Assertion	lcmfsn	|- ( M e. ZZ -> ( _lcm ` { M } ) = ( abs ` M ) )

Proof

Step	Hyp	Ref	Expression
1		dfsn2	|- { M } = { M , M }
2	1	a1i	|- ( M e. ZZ -> { M } = { M , M } )
3	2	fveq2d	|- ( M e. ZZ -> ( _lcm ` { M } ) = ( _lcm ` { M , M } ) )
4		lcmfpr	|- ( ( M e. ZZ /\ M e. ZZ ) -> ( _lcm ` { M , M } ) = ( M lcm M ) )
5	4	anidms	|- ( M e. ZZ -> ( _lcm ` { M , M } ) = ( M lcm M ) )
6		lcmid	|- ( M e. ZZ -> ( M lcm M ) = ( abs ` M ) )
7	3 5 6	3eqtrd	|- ( M e. ZZ -> ( _lcm ` { M } ) = ( abs ` M ) )