Theorem lmodvsca

Description: The scalar product operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013) (Revised by Mario Carneiro, 29-Aug-2015)

	Ref	Expression
Hypothesis	lmodstr.w	⊢ 𝑊 = ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝐹 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ } )
Assertion	lmodvsca	⊢ ( · ∈ 𝑋 → · = ( ·𝑠 ‘ 𝑊 ) )

Proof

Step	Hyp	Ref	Expression
1		lmodstr.w	⊢ 𝑊 = ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝐹 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ } )
2	1	lmodstr	⊢ 𝑊 Struct ⟨ 1 , 6 ⟩
3		vscaid	⊢ ·𝑠 = Slot ( ·𝑠 ‘ ndx )
4		ssun2	⊢ { ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝐹 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ } )
5	4 1	sseqtrri	⊢ { ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ } ⊆ 𝑊
6	2 3 5	strfv	⊢ ( · ∈ 𝑋 → · = ( ·𝑠 ‘ 𝑊 ) )