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Metamath Proof Explorer

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 ) ) ⟩ ) ⟩ } ) ) )