Theorem rngstr

Description: A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013) (Revised by Mario Carneiro, 29-Aug-2015)

	Ref	Expression
Hypothesis	rngfn.r	⊢ 𝑅 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ }
Assertion	rngstr	⊢ 𝑅 Struct ⟨ 1 , 3 ⟩

Proof

Step	Hyp	Ref	Expression
1		rngfn.r	⊢ 𝑅 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ }
2		1nn	⊢ 1 ∈ ℕ
3		basendx	⊢ ( Base ‘ ndx ) = 1
4		1lt2	⊢ 1 < 2
5		2nn	⊢ 2 ∈ ℕ
6		plusgndx	⊢ ( +g ‘ ndx ) = 2
7		2lt3	⊢ 2 < 3
8		3nn	⊢ 3 ∈ ℕ
9		mulrndx	⊢ ( .r ‘ ndx ) = 3
10	2 3 4 5 6 7 8 9	strle3	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } Struct ⟨ 1 , 3 ⟩
11	1 10	eqbrtri	⊢ 𝑅 Struct ⟨ 1 , 3 ⟩