Metamath Proof Explorer


Definition df-dveca

Description: Define constructed partial vector space A. (Contributed by NM, 8-Oct-2013)

Ref Expression
Assertion df-dveca DVecA = ( 𝑘 ∈ V ↦ ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ ( { ⟨ ( Base ‘ ndx ) , ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑓𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑠𝑓 ) ) ⟩ } ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cdveca DVecA
1 vk 𝑘
2 cvv V
3 vw 𝑤
4 clh LHyp
5 1 cv 𝑘
6 5 4 cfv ( LHyp ‘ 𝑘 )
7 cbs Base
8 cnx ndx
9 8 7 cfv ( Base ‘ ndx )
10 cltrn LTrn
11 5 10 cfv ( LTrn ‘ 𝑘 )
12 3 cv 𝑤
13 12 11 cfv ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 )
14 9 13 cop ⟨ ( Base ‘ ndx ) , ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ⟩
15 cplusg +g
16 8 15 cfv ( +g ‘ ndx )
17 vf 𝑓
18 vg 𝑔
19 17 cv 𝑓
20 18 cv 𝑔
21 19 20 ccom ( 𝑓𝑔 )
22 17 18 13 13 21 cmpo ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑓𝑔 ) )
23 16 22 cop ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑓𝑔 ) ) ⟩
24 csca Scalar
25 8 24 cfv ( Scalar ‘ ndx )
26 cedring EDRing
27 5 26 cfv ( EDRing ‘ 𝑘 )
28 12 27 cfv ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 )
29 25 28 cop ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 ) ⟩
30 14 23 29 ctp { ⟨ ( Base ‘ ndx ) , ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑓𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 ) ⟩ }
31 cvsca ·𝑠
32 8 31 cfv ( ·𝑠 ‘ ndx )
33 vs 𝑠
34 ctendo TEndo
35 5 34 cfv ( TEndo ‘ 𝑘 )
36 12 35 cfv ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 )
37 33 cv 𝑠
38 19 37 cfv ( 𝑠𝑓 )
39 33 17 36 13 38 cmpo ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑠𝑓 ) )
40 32 39 cop ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑠𝑓 ) ) ⟩
41 40 csn { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑠𝑓 ) ) ⟩ }
42 30 41 cun ( { ⟨ ( Base ‘ ndx ) , ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑓𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑠𝑓 ) ) ⟩ } )
43 3 6 42 cmpt ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ ( { ⟨ ( Base ‘ ndx ) , ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑓𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑠𝑓 ) ) ⟩ } ) )
44 1 2 43 cmpt ( 𝑘 ∈ V ↦ ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ ( { ⟨ ( Base ‘ ndx ) , ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑓𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑠𝑓 ) ) ⟩ } ) ) )
45 0 44 wceq DVecA = ( 𝑘 ∈ V ↦ ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ ( { ⟨ ( Base ‘ ndx ) , ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) , 𝑔 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑓𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑠𝑓 ) ) ⟩ } ) ) )