Theorem nndvdslegcd

Description: A positive integer which divides both positive operands of the gcd operator is bounded by it. (Contributed by AV, 9-Aug-2020)

	Ref	Expression
Assertion	nndvdslegcd	⊢ ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( ( 𝐾 ∥ 𝑀 ∧ 𝐾 ∥ 𝑁 ) → 𝐾 ≤ ( 𝑀 gcd 𝑁 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nnz	⊢ ( 𝐾 ∈ ℕ → 𝐾 ∈ ℤ )
2		nnz	⊢ ( 𝑀 ∈ ℕ → 𝑀 ∈ ℤ )
3		nnz	⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℤ )
4	1 2 3	3anim123i	⊢ ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) )
5		nnne0	⊢ ( 𝑀 ∈ ℕ → 𝑀 ≠ 0 )
6	5	neneqd	⊢ ( 𝑀 ∈ ℕ → ¬ 𝑀 = 0 )
7	6	3ad2ant2	⊢ ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ¬ 𝑀 = 0 )
8	7	intnanrd	⊢ ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ¬ ( 𝑀 = 0 ∧ 𝑁 = 0 ) )
9		dvdslegcd	⊢ ( ( ( 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ¬ ( 𝑀 = 0 ∧ 𝑁 = 0 ) ) → ( ( 𝐾 ∥ 𝑀 ∧ 𝐾 ∥ 𝑁 ) → 𝐾 ≤ ( 𝑀 gcd 𝑁 ) ) )
10	4 8 9	syl2anc	⊢ ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( ( 𝐾 ∥ 𝑀 ∧ 𝐾 ∥ 𝑁 ) → 𝐾 ≤ ( 𝑀 gcd 𝑁 ) ) )