Theorem dvasca

Description: The ring base set of the constructed partial vector space A are all translation group endomorphisms (for a fiducial co-atom W ). (Contributed by NM, 22-Jun-2014)

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

Proof

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