Theorem enrbreq

Description: Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995) (New usage is discouraged.)

	Ref	Expression
Assertion	enrbreq	⊢ ( ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) ∧ ( 𝐶 ∈ P ∧ 𝐷 ∈ P ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ ~R ⟨ 𝐶 , 𝐷 ⟩ ↔ ( 𝐴 +P 𝐷 ) = ( 𝐵 +P 𝐶 ) ) )

Proof

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