Theorem enqbreq

Description: Equivalence relation for positive fractions in terms of positive integers. (Contributed by NM, 27-Aug-1995) (New usage is discouraged.)

	Ref	Expression
Assertion	enqbreq	⊢ ( ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) ∧ ( 𝐶 ∈ N ∧ 𝐷 ∈ N ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ ~Q ⟨ 𝐶 , 𝐷 ⟩ ↔ ( 𝐴 ·N 𝐷 ) = ( 𝐵 ·N 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-enq	⊢ ~Q = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ ( N × N ) ∧ 𝑦 ∈ ( N × N ) ) ∧ ∃ 𝑧 ∃ 𝑤 ∃ 𝑣 ∃ 𝑢 ( ( 𝑥 = ⟨ 𝑧 , 𝑤 ⟩ ∧ 𝑦 = ⟨ 𝑣 , 𝑢 ⟩ ) ∧ ( 𝑧 ·N 𝑢 ) = ( 𝑤 ·N 𝑣 ) ) ) }
2	1	ecopoveq	⊢ ( ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) ∧ ( 𝐶 ∈ N ∧ 𝐷 ∈ N ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ ~Q ⟨ 𝐶 , 𝐷 ⟩ ↔ ( 𝐴 ·N 𝐷 ) = ( 𝐵 ·N 𝐶 ) ) )