Theorem enqer

Description: The equivalence relation for positive fractions is an equivalence relation. Proposition 9-2.1 of Gleason p. 117. (Contributed by NM, 27-Aug-1995) (Revised by Mario Carneiro, 6-Jul-2015) (New usage is discouraged.)

	Ref	Expression
Assertion	enqer	⊢ ~Q Er ( N × N )

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		mulcompi	⊢ ( 𝑥 ·N 𝑦 ) = ( 𝑦 ·N 𝑥 )
3		mulclpi	⊢ ( ( 𝑥 ∈ N ∧ 𝑦 ∈ N ) → ( 𝑥 ·N 𝑦 ) ∈ N )
4		mulasspi	⊢ ( ( 𝑥 ·N 𝑦 ) ·N 𝑧 ) = ( 𝑥 ·N ( 𝑦 ·N 𝑧 ) )
5		mulcanpi	⊢ ( ( 𝑥 ∈ N ∧ 𝑦 ∈ N ) → ( ( 𝑥 ·N 𝑦 ) = ( 𝑥 ·N 𝑧 ) ↔ 𝑦 = 𝑧 ) )
6	5	biimpd	⊢ ( ( 𝑥 ∈ N ∧ 𝑦 ∈ N ) → ( ( 𝑥 ·N 𝑦 ) = ( 𝑥 ·N 𝑧 ) → 𝑦 = 𝑧 ) )
7	1 2 3 4 6	ecopover	⊢ ~Q Er ( N × N )