Theorem odrngle

Description: The order of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015)

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

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	1	odrngstr	⊢ 𝑊 Struct ⟨ 1 , ; 1 2 ⟩
3		pleid	⊢ le = Slot ( le ‘ ndx )
4		snsstp2	⊢ { ⟨ ( le ‘ ndx ) , ≤ ⟩ } ⊆ { ⟨ ( TopSet ‘ ndx ) , 𝐽 ⟩ , ⟨ ( le ‘ ndx ) , ≤ ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ }
5		ssun2	⊢ { ⟨ ( TopSet ‘ ndx ) , 𝐽 ⟩ , ⟨ ( le ‘ ndx ) , ≤ ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } ∪ { ⟨ ( TopSet ‘ ndx ) , 𝐽 ⟩ , ⟨ ( le ‘ ndx ) , ≤ ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ } )
6	5 1	sseqtrri	⊢ { ⟨ ( TopSet ‘ ndx ) , 𝐽 ⟩ , ⟨ ( le ‘ ndx ) , ≤ ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ } ⊆ 𝑊
7	4 6	sstri	⊢ { ⟨ ( le ‘ ndx ) , ≤ ⟩ } ⊆ 𝑊
8	2 3 7	strfv	⊢ ( ≤ ∈ 𝑉 → ≤ = ( le ‘ 𝑊 ) )