Metamath Proof Explorer


Theorem algsca

Description: The set of scalars of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 29-Aug-2015)

Ref Expression
Hypothesis algpart.a 𝐴 = ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , × ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ } )
Assertion algsca ( 𝑆𝑉𝑆 = ( Scalar ‘ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 algpart.a 𝐴 = ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , × ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ } )
2 1 algstr 𝐴 Struct ⟨ 1 , 6 ⟩
3 scaid Scalar = Slot ( Scalar ‘ ndx )
4 snsspr1 { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ } ⊆ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ }
5 ssun2 { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , × ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ } )
6 5 1 sseqtrri { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ } ⊆ 𝐴
7 4 6 sstri { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ } ⊆ 𝐴
8 2 3 7 strfv ( 𝑆𝑉𝑆 = ( Scalar ‘ 𝐴 ) )