Theorem algvsca

Description: The scalar product operation 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	algvsca	⊢ ( · ∈ 𝑉 → · = ( ·𝑠 ‘ 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		vscaid	⊢ ·𝑠 = Slot ( ·𝑠 ‘ ndx )
4		snsspr2	⊢ { ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ } ⊆ { ⟨ ( 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	⊢ { ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ } ⊆ 𝐴
8	2 3 7	strfv	⊢ ( · ∈ 𝑉 → · = ( ·𝑠 ‘ 𝐴 ) )