Metamath Proof Explorer


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 ) → ( ( 𝐾𝑀𝐾𝑁 ) → 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 ) ) → ( ( 𝐾𝑀𝐾𝑁 ) → 𝐾 ≤ ( 𝑀 gcd 𝑁 ) ) )
10 4 8 9 syl2anc ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( ( 𝐾𝑀𝐾𝑁 ) → 𝐾 ≤ ( 𝑀 gcd 𝑁 ) ) )