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Metamath Proof Explorer

Theorem gcd1

Description: The gcd of a number with 1 is 1. Theorem 1.4(d)1 in ApostolNT p. 16. (Contributed by Mario Carneiro, 19-Feb-2014)

Ref Expression
Assertion gcd1 ( 𝑀 ∈ ℤ → ( 𝑀 gcd 1 ) = 1 )

Proof

Step Hyp Ref Expression
1 1z ⊢ 1 ∈ ℤ
2 gcddvds ⊢ ( ( 𝑀 ∈ ℤ ∧ 1 ∈ ℤ ) → ( ( 𝑀 gcd 1 ) ∥ 𝑀 ∧ ( 𝑀 gcd 1 ) ∥ 1 ) )
3 1 2 mpan2 ⊢ ( 𝑀 ∈ ℤ → ( ( 𝑀 gcd 1 ) ∥ 𝑀 ∧ ( 𝑀 gcd 1 ) ∥ 1 ) )
4 3 simprd ⊢ ( 𝑀 ∈ ℤ → ( 𝑀 gcd 1 ) ∥ 1 )
5 ax-1ne0 ⊢ 1 ≠ 0
6 simpr ⊢ ( ( 𝑀 = 0 ∧ 1 = 0 ) → 1 = 0 )
7 6 necon3ai ⊢ ( 1 ≠ 0 → ¬ ( 𝑀 = 0 ∧ 1 = 0 ) )
8 5 7 ax-mp ⊢ ¬ ( 𝑀 = 0 ∧ 1 = 0 )
9 gcdn0cl ⊢ ( ( ( 𝑀 ∈ ℤ ∧ 1 ∈ ℤ ) ∧ ¬ ( 𝑀 = 0 ∧ 1 = 0 ) ) → ( 𝑀 gcd 1 ) ∈ ℕ )
10 8 9 mpan2 ⊢ ( ( 𝑀 ∈ ℤ ∧ 1 ∈ ℤ ) → ( 𝑀 gcd 1 ) ∈ ℕ )
11 1 10 mpan2 ⊢ ( 𝑀 ∈ ℤ → ( 𝑀 gcd 1 ) ∈ ℕ )
12 11 nnzd ⊢ ( 𝑀 ∈ ℤ → ( 𝑀 gcd 1 ) ∈ ℤ )
13 1nn ⊢ 1 ∈ ℕ
14 dvdsle ⊢ ( ( ( 𝑀 gcd 1 ) ∈ ℤ ∧ 1 ∈ ℕ ) → ( ( 𝑀 gcd 1 ) ∥ 1 → ( 𝑀 gcd 1 ) ≤ 1 ) )
15 12 13 14 sylancl ⊢ ( 𝑀 ∈ ℤ → ( ( 𝑀 gcd 1 ) ∥ 1 → ( 𝑀 gcd 1 ) ≤ 1 ) )
16 4 15 mpd ⊢ ( 𝑀 ∈ ℤ → ( 𝑀 gcd 1 ) ≤ 1 )
17 nnle1eq1 ⊢ ( ( 𝑀 gcd 1 ) ∈ ℕ → ( ( 𝑀 gcd 1 ) ≤ 1 ↔ ( 𝑀 gcd 1 ) = 1 ) )
18 11 17 syl ⊢ ( 𝑀 ∈ ℤ → ( ( 𝑀 gcd 1 ) ≤ 1 ↔ ( 𝑀 gcd 1 ) = 1 ) )
19 16 18 mpbid ⊢ ( 𝑀 ∈ ℤ → ( 𝑀 gcd 1 ) = 1 )