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Metamath Proof Explorer

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 ) )