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 ∈ ℤ → ( lcm ‘ { 𝑀 } ) = ( abs ‘ 𝑀 ) )

Proof

Step	Hyp	Ref	Expression
1		dfsn2	⊢ { 𝑀 } = { 𝑀 , 𝑀 }
2	1	a1i	⊢ ( 𝑀 ∈ ℤ → { 𝑀 } = { 𝑀 , 𝑀 } )
3	2	fveq2d	⊢ ( 𝑀 ∈ ℤ → ( lcm ‘ { 𝑀 } ) = ( lcm ‘ { 𝑀 , 𝑀 } ) )
4		lcmfpr	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ ) → ( lcm ‘ { 𝑀 , 𝑀 } ) = ( 𝑀 lcm 𝑀 ) )
5	4	anidms	⊢ ( 𝑀 ∈ ℤ → ( lcm ‘ { 𝑀 , 𝑀 } ) = ( 𝑀 lcm 𝑀 ) )
6		lcmid	⊢ ( 𝑀 ∈ ℤ → ( 𝑀 lcm 𝑀 ) = ( abs ‘ 𝑀 ) )
7	3 5 6	3eqtrd	⊢ ( 𝑀 ∈ ℤ → ( lcm ‘ { 𝑀 } ) = ( abs ‘ 𝑀 ) )