Theorem dvhfset

Description: The constructed full vector space H for a lattice K . (Contributed by NM, 17-Oct-2013) (Revised by Mario Carneiro, 22-Jun-2014)

	Ref	Expression
Hypothesis	dvhset.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion	dvhfset	⊢ ( 𝐾 ∈ 𝑉 → ( DVecH ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ ( { ⟨ ( 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 ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) ) )

Proof

Step	Hyp	Ref	Expression
1		dvhset.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		elex	⊢ ( 𝐾 ∈ 𝑉 → 𝐾 ∈ V )
3		fveq2	⊢ ( 𝑘 = 𝐾 → ( LHyp ‘ 𝑘 ) = ( LHyp ‘ 𝐾 ) )
4	3 1	eqtr4di	⊢ ( 𝑘 = 𝐾 → ( LHyp ‘ 𝑘 ) = 𝐻 )
5		fveq2	⊢ ( 𝑘 = 𝐾 → ( LTrn ‘ 𝑘 ) = ( LTrn ‘ 𝐾 ) )
6	5	fveq1d	⊢ ( 𝑘 = 𝐾 → ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) )
7		fveq2	⊢ ( 𝑘 = 𝐾 → ( TEndo ‘ 𝑘 ) = ( TEndo ‘ 𝐾 ) )
8	7	fveq1d	⊢ ( 𝑘 = 𝐾 → ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) )
9	6 8	xpeq12d	⊢ ( 𝑘 = 𝐾 → ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) = ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) )
10	9	opeq2d	⊢ ( 𝑘 = 𝐾 → ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ⟩ = ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ⟩ )
11	6	mpteq1d	⊢ ( 𝑘 = 𝐾 → ( ℎ ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) = ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) )
12	11	opeq2d	⊢ ( 𝑘 = 𝐾 → ⟨ ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ = ⟨ ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ )
13	9 9 12	mpoeq123dv	⊢ ( 𝑘 = 𝐾 → ( 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 ) ) ) ⟩ ) = ( 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 ) ) ) ⟩ ) )
14	13	opeq2d	⊢ ( 𝑘 = 𝐾 → ⟨ ( +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 ) ) ) ⟩ ) ⟩ = ⟨ ( +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 ) ) ) ⟩ ) ⟩ )
15		fveq2	⊢ ( 𝑘 = 𝐾 → ( EDRing ‘ 𝑘 ) = ( EDRing ‘ 𝐾 ) )
16	15	fveq1d	⊢ ( 𝑘 = 𝐾 → ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 ) = ( ( EDRing ‘ 𝐾 ) ‘ 𝑤 ) )
17	16	opeq2d	⊢ ( 𝑘 = 𝐾 → ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 ) ⟩ = ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑤 ) ⟩ )
18	10 14 17	tpeq123d	⊢ ( 𝑘 = 𝐾 → { ⟨ ( 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 ) , ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 ) ⟩ } = { ⟨ ( 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 ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑤 ) ⟩ } )
19		eqidd	⊢ ( 𝑘 = 𝐾 → ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ = ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ )
20	8 9 19	mpoeq123dv	⊢ ( 𝑘 = 𝐾 → ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) = ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) )
21	20	opeq2d	⊢ ( 𝑘 = 𝐾 → ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ = ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ )
22	21	sneqd	⊢ ( 𝑘 = 𝐾 → { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ } = { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ } )
23	18 22	uneq12d	⊢ ( 𝑘 = 𝐾 → ( { ⟨ ( 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 ) , ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·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 ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) )
24	4 23	mpteq12dv	⊢ ( 𝑘 = 𝐾 → ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ ( { ⟨ ( 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 ) , ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) ) = ( 𝑤 ∈ 𝐻 ↦ ( { ⟨ ( 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 ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) ) )
25		df-dvech	⊢ DVecH = ( 𝑘 ∈ V ↦ ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ ( { ⟨ ( 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 ) , ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) ) )
26	24 25 1	mptfvmpt	⊢ ( 𝐾 ∈ V → ( DVecH ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ ( { ⟨ ( 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 ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) ) )
27	2 26	syl	⊢ ( 𝐾 ∈ 𝑉 → ( DVecH ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ ( { ⟨ ( 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 ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) ) )