This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem gcdn0cl

Description: Closure of the gcd operator. (Contributed by Paul Chapman, 21-Mar-2011)

Ref Expression
Assertion gcdn0cl ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ¬ ( 𝑀 = 0 ∧ 𝑁 = 0 ) ) → ( 𝑀 gcd 𝑁 ) ∈ ℕ )

Proof

Step Hyp Ref Expression
1 gcdn0val ⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ¬ ( 𝑀 = 0 ∧ 𝑁 = 0 ) ) → ( 𝑀 gcd 𝑁 ) = sup ( { 𝑛 ∈ ℤ ∣ ( 𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁 ) } , ℝ , < ) )
2 eqid ⊢ { 𝑛 ∈ ℤ ∣ ∀ 𝑧 ∈ { 𝑀 , 𝑁 } 𝑛 ∥ 𝑧 } = { 𝑛 ∈ ℤ ∣ ∀ 𝑧 ∈ { 𝑀 , 𝑁 } 𝑛 ∥ 𝑧 }
3 eqid ⊢ { 𝑛 ∈ ℤ ∣ ( 𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁 ) } = { 𝑛 ∈ ℤ ∣ ( 𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁 ) }
4 2 3 gcdcllem3 ⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ¬ ( 𝑀 = 0 ∧ 𝑁 = 0 ) ) → ( sup ( { 𝑛 ∈ ℤ ∣ ( 𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁 ) } , ℝ , < ) ∈ ℕ ∧ ( sup ( { 𝑛 ∈ ℤ ∣ ( 𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁 ) } , ℝ , < ) ∥ 𝑀 ∧ sup ( { 𝑛 ∈ ℤ ∣ ( 𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁 ) } , ℝ , < ) ∥ 𝑁 ) ∧ ( ( 𝐾 ∈ ℤ ∧ 𝐾 ∥ 𝑀 ∧ 𝐾 ∥ 𝑁 ) → 𝐾 ≤ sup ( { 𝑛 ∈ ℤ ∣ ( 𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁 ) } , ℝ , < ) ) ) )
5 4 simp1d ⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ¬ ( 𝑀 = 0 ∧ 𝑁 = 0 ) ) → sup ( { 𝑛 ∈ ℤ ∣ ( 𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁 ) } , ℝ , < ) ∈ ℕ )
6 1 5 eqeltrd ⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ¬ ( 𝑀 = 0 ∧ 𝑁 = 0 ) ) → ( 𝑀 gcd 𝑁 ) ∈ ℕ )