Theorem lmodsca

Description: The set of scalars 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	lmodsca	⊢ ( 𝐹 ∈ 𝑋 → 𝐹 = ( Scalar ‘ 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		scaid	⊢ Scalar = Slot ( Scalar ‘ ndx )
4		snsstp3	⊢ { ⟨ ( Scalar ‘ ndx ) , 𝐹 ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝐹 ⟩ }
5		ssun1	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝐹 ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝐹 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ } )
6	5 1	sseqtrri	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝐹 ⟩ } ⊆ 𝑊
7	4 6	sstri	⊢ { ⟨ ( Scalar ‘ ndx ) , 𝐹 ⟩ } ⊆ 𝑊
8	2 3 7	strfv	⊢ ( 𝐹 ∈ 𝑋 → 𝐹 = ( Scalar ‘ 𝑊 ) )