Theorem gcddvdslcm

Description: The greatest common divisor of two numbers divides their least common multiple. (Contributed by Steve Rodriguez, 20-Jan-2020)

	Ref	Expression
Assertion	gcddvdslcm	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 gcd 𝑁 ) ∥ ( 𝑀 lcm 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		gcdcl	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 gcd 𝑁 ) ∈ ℕ0 )
2	1	nn0zd	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 gcd 𝑁 ) ∈ ℤ )
3		simpl	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → 𝑀 ∈ ℤ )
4		lcmcl	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 lcm 𝑁 ) ∈ ℕ0 )
5	4	nn0zd	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 lcm 𝑁 ) ∈ ℤ )
6		gcddvds	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 𝑀 gcd 𝑁 ) ∥ 𝑀 ∧ ( 𝑀 gcd 𝑁 ) ∥ 𝑁 ) )
7	6	simpld	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 gcd 𝑁 ) ∥ 𝑀 )
8		dvdslcm	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ ( 𝑀 lcm 𝑁 ) ∧ 𝑁 ∥ ( 𝑀 lcm 𝑁 ) ) )
9	8	simpld	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → 𝑀 ∥ ( 𝑀 lcm 𝑁 ) )
10	2 3 5 7 9	dvdstrd	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 gcd 𝑁 ) ∥ ( 𝑀 lcm 𝑁 ) )