Metamath Proof Explorer

Theorem 1nqenq

Description: The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996) (Revised by Mario Carneiro, 28-Apr-2013) (New usage is discouraged.)

		Ref	Expression
	Assertion	1nqenq	$⊢ A \in 𝑵 \to 1_{𝑸} ~_{𝑸} ⟨A, A⟩$

Proof

Step	Hyp	Ref	Expression
1		enqer	$⊢ ~_{𝑸} Er (𝑵 \times 𝑵)$
2	1	a1i	$⊢ A \in 𝑵 \to ~_{𝑸} Er (𝑵 \times 𝑵)$
3		mulidpi	$⊢ A \in 𝑵 \to A \cdot_{𝑵} 1_{𝑜} = A$
4	3 3	opeq12d	$⊢ A \in 𝑵 \to ⟨A \cdot_{𝑵} 1_{𝑜}, A \cdot_{𝑵} 1_{𝑜}⟩ = ⟨A, A⟩$
5		1pi	$⊢ 1_{𝑜} \in 𝑵$
6		mulcanenq	$⊢ (A \in 𝑵 \land 1_{𝑜} \in 𝑵 \land 1_{𝑜} \in 𝑵) \to ⟨A \cdot_{𝑵} 1_{𝑜}, A \cdot_{𝑵} 1_{𝑜}⟩ ~_{𝑸} ⟨1_{𝑜}, 1_{𝑜}⟩$
7	5 5 6	mp3an23	$⊢ A \in 𝑵 \to ⟨A \cdot_{𝑵} 1_{𝑜}, A \cdot_{𝑵} 1_{𝑜}⟩ ~_{𝑸} ⟨1_{𝑜}, 1_{𝑜}⟩$
8		df-1nq	$⊢ 1_{𝑸} = ⟨1_{𝑜}, 1_{𝑜}⟩$
9	7 8	breqtrrdi	$⊢ A \in 𝑵 \to ⟨A \cdot_{𝑵} 1_{𝑜}, A \cdot_{𝑵} 1_{𝑜}⟩ ~_{𝑸} 1_{𝑸}$
10	4 9	eqbrtrrd	$⊢ A \in 𝑵 \to ⟨A, A⟩ ~_{𝑸} 1_{𝑸}$
11	2 10	ersym	$⊢ A \in 𝑵 \to 1_{𝑸} ~_{𝑸} ⟨A, A⟩$