Metamath Proof Explorer

Theorem dvafset

Description: The constructed partial vector space A for a lattice K . (Contributed by NM, 8-Oct-2013) (Revised by Mario Carneiro, 22-Jun-2014)

Ref Expression
Hypothesis dvaset.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion dvafset ( 𝐾 ∈ 𝑉 → ( DVecA ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ ( { ⟨ ( Base ‘ ndx ) , ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑠 ‘ 𝑓 ) ) ⟩ } ) ) )

Proof

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