Metamath Proof Explorer


Theorem odrngmulr

Description: The multiplication operation 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 odrngmulr ( ·𝑉· = ( .r𝑊 ) )

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 mulrid .r = Slot ( .r ‘ ndx )
4 snsstp3 { ⟨ ( .r ‘ ndx ) , · ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ }
5 ssun1 { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } ∪ { ⟨ ( TopSet ‘ ndx ) , 𝐽 ⟩ , ⟨ ( le ‘ ndx ) , ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ } )
6 5 1 sseqtrri { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } ⊆ 𝑊
7 4 6 sstri { ⟨ ( .r ‘ ndx ) , · ⟩ } ⊆ 𝑊
8 2 3 7 strfv ( ·𝑉· = ( .r𝑊 ) )