Theorem ncoprmgcdgt1b

Description: Two positive integers are not coprime, i.e. there is an integer greater than 1 which divides both integers, iff their greatest common divisor is greater than 1. (Contributed by AV, 9-Aug-2020)

	Ref	Expression
Assertion	ncoprmgcdgt1b	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ∃ 𝑖 ∈ ( ℤ≥ ‘ 2 ) ( 𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐵 ) ↔ 1 < ( 𝐴 gcd 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ncoprmgcdne1b	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ∃ 𝑖 ∈ ( ℤ≥ ‘ 2 ) ( 𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐵 ) ↔ ( 𝐴 gcd 𝐵 ) ≠ 1 ) )
2		gcdnncl	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 gcd 𝐵 ) ∈ ℕ )
3		nngt1ne1	⊢ ( ( 𝐴 gcd 𝐵 ) ∈ ℕ → ( 1 < ( 𝐴 gcd 𝐵 ) ↔ ( 𝐴 gcd 𝐵 ) ≠ 1 ) )
4	2 3	syl	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( 1 < ( 𝐴 gcd 𝐵 ) ↔ ( 𝐴 gcd 𝐵 ) ≠ 1 ) )
5	1 4	bitr4d	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ∃ 𝑖 ∈ ( ℤ≥ ‘ 2 ) ( 𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐵 ) ↔ 1 < ( 𝐴 gcd 𝐵 ) ) )