Theorem odrngstr

Description: Functionality of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015) (Proof shortened by AV, 15-Sep-2021)

	Ref	Expression
Hypothesis	odrngstr.w	⊢ 𝑊 = ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } ∪ { ⟨ ( TopSet ‘ ndx ) , 𝐽 ⟩ , ⟨ ( le ‘ ndx ) , ≤ ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ } )
Assertion	odrngstr	⊢ 𝑊 Struct ⟨ 1 , ; 1 2 ⟩

Proof

Step	Hyp	Ref	Expression
1		odrngstr.w	⊢ 𝑊 = ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } ∪ { ⟨ ( TopSet ‘ ndx ) , 𝐽 ⟩ , ⟨ ( le ‘ ndx ) , ≤ ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ } )
2		eqid	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ }
3	2	rngstr	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } Struct ⟨ 1 , 3 ⟩
4		9nn	⊢ 9 ∈ ℕ
5		tsetndx	⊢ ( TopSet ‘ ndx ) = 9
6		9lt10	⊢ 9 < ; 1 0
7		10nn	⊢ ; 1 0 ∈ ℕ
8		plendx	⊢ ( le ‘ ndx ) = ; 1 0
9		1nn0	⊢ 1 ∈ ℕ0
10		0nn0	⊢ 0 ∈ ℕ0
11		2nn	⊢ 2 ∈ ℕ
12		2pos	⊢ 0 < 2
13	9 10 11 12	declt	⊢ ; 1 0 < ; 1 2
14	9 11	decnncl	⊢ ; 1 2 ∈ ℕ
15		dsndx	⊢ ( dist ‘ ndx ) = ; 1 2
16	4 5 6 7 8 13 14 15	strle3	⊢ { ⟨ ( TopSet ‘ ndx ) , 𝐽 ⟩ , ⟨ ( le ‘ ndx ) , ≤ ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ } Struct ⟨ 9 , ; 1 2 ⟩
17		3lt9	⊢ 3 < 9
18	3 16 17	strleun	⊢ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } ∪ { ⟨ ( TopSet ‘ ndx ) , 𝐽 ⟩ , ⟨ ( le ‘ ndx ) , ≤ ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ } ) Struct ⟨ 1 , ; 1 2 ⟩
19	1 18	eqbrtri	⊢ 𝑊 Struct ⟨ 1 , ; 1 2 ⟩