Theorem dvhsca

Description: The ring of scalars of the constructed full vector space H. (Contributed by NM, 22-Jun-2014)

	Ref	Expression
Hypotheses	dvhsca.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dvhsca.d	⊢ 𝐷 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
	dvhsca.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	dvhsca.f	⊢ 𝐹 = ( Scalar ‘ 𝑈 )
Assertion	dvhsca	⊢ ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → 𝐹 = 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		dvhsca.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		dvhsca.d	⊢ 𝐷 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
3		dvhsca.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4		dvhsca.f	⊢ 𝐹 = ( Scalar ‘ 𝑈 )
5		eqid	⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6		eqid	⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
7	1 5 6 2 3	dvhset	⊢ ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → 𝑈 = ( { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝐷 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) )
8	7	fveq2d	⊢ ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → ( Scalar ‘ 𝑈 ) = ( Scalar ‘ ( { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝐷 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) ) )
9	2	fvexi	⊢ 𝐷 ∈ V
10		eqid	⊢ ( { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝐷 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) = ( { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝐷 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ } )
11	10	lmodsca	⊢ ( 𝐷 ∈ V → 𝐷 = ( Scalar ‘ ( { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝐷 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) ) )
12	9 11	ax-mp	⊢ 𝐷 = ( Scalar ‘ ( { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝐷 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) )
13	8 4 12	3eqtr4g	⊢ ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → 𝐹 = 𝐷 )