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 ) ) ⟩ } ) ) )