Metamath Proof Explorer


Theorem ipsvsca

Description: The scalar product operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 29-Aug-2015) (Revised by Thierry Arnoux, 16-Jun-2019)

Ref Expression
Hypothesis ipspart.a 𝐴 = ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , × ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , 𝐼 ⟩ } )
Assertion ipsvsca ( ·𝑉· = ( ·𝑠𝐴 ) )

Proof

Step Hyp Ref Expression
1 ipspart.a 𝐴 = ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , × ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , 𝐼 ⟩ } )
2 1 ipsstr 𝐴 Struct ⟨ 1 , 8 ⟩
3 vscaid ·𝑠 = Slot ( ·𝑠 ‘ ndx )
4 snsstp2 { ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ } ⊆ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , 𝐼 ⟩ }
5 ssun2 { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , 𝐼 ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , × ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , 𝐼 ⟩ } )
6 5 1 sseqtrri { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , 𝐼 ⟩ } ⊆ 𝐴
7 4 6 sstri { ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ } ⊆ 𝐴
8 2 3 7 strfv ( ·𝑉· = ( ·𝑠𝐴 ) )