Metamath Proof Explorer


Theorem rngmulr

Description: The multiplicative operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013) (Revised by Mario Carneiro, 30-Apr-2015)

Ref Expression
Hypothesis rngfn.r 𝑅 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ }
Assertion rngmulr ( ·𝑉· = ( .r𝑅 ) )

Proof

Step Hyp Ref Expression
1 rngfn.r 𝑅 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ }
2 1 rngstr 𝑅 Struct ⟨ 1 , 3 ⟩
3 mulrid .r = Slot ( .r ‘ ndx )
4 snsstp3 { ⟨ ( .r ‘ ndx ) , · ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ }
5 4 1 sseqtrri { ⟨ ( .r ‘ ndx ) , · ⟩ } ⊆ 𝑅
6 2 3 5 strfv ( ·𝑉· = ( .r𝑅 ) )