Theorem gcdle2d

Description: The greatest common divisor of a positive integer and another integer is less than or equal to the positive integer. (Contributed by SN, 25-Aug-2024)

	Ref	Expression
Hypotheses	gcdle2d.m	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
	gcdle2d.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
Assertion	gcdle2d	⊢ ( 𝜑 → ( 𝑀 gcd 𝑁 ) ≤ 𝑁 )

Proof

Step	Hyp	Ref	Expression
1		gcdle2d.m	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
2		gcdle2d.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
3	2	nnzd	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
4		gcddvds	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 𝑀 gcd 𝑁 ) ∥ 𝑀 ∧ ( 𝑀 gcd 𝑁 ) ∥ 𝑁 ) )
5	1 3 4	syl2anc	⊢ ( 𝜑 → ( ( 𝑀 gcd 𝑁 ) ∥ 𝑀 ∧ ( 𝑀 gcd 𝑁 ) ∥ 𝑁 ) )
6	5	simprd	⊢ ( 𝜑 → ( 𝑀 gcd 𝑁 ) ∥ 𝑁 )
7	1 3	gcdcld	⊢ ( 𝜑 → ( 𝑀 gcd 𝑁 ) ∈ ℕ0 )
8	7	nn0zd	⊢ ( 𝜑 → ( 𝑀 gcd 𝑁 ) ∈ ℤ )
9		dvdsle	⊢ ( ( ( 𝑀 gcd 𝑁 ) ∈ ℤ ∧ 𝑁 ∈ ℕ ) → ( ( 𝑀 gcd 𝑁 ) ∥ 𝑁 → ( 𝑀 gcd 𝑁 ) ≤ 𝑁 ) )
10	8 2 9	syl2anc	⊢ ( 𝜑 → ( ( 𝑀 gcd 𝑁 ) ∥ 𝑁 → ( 𝑀 gcd 𝑁 ) ≤ 𝑁 ) )
11	6 10	mpd	⊢ ( 𝜑 → ( 𝑀 gcd 𝑁 ) ≤ 𝑁 )