Metamath Proof Explorer


Definition df-dvech

Description: Define constructed full vector space H. (Contributed by NM, 17-Oct-2013)

Ref Expression
Assertion 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 ) ) ↦ ⟨ ( ( 1st𝑓 ) ∘ ( 1st𝑔 ) ) , ( ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( ( 2nd𝑓 ) ‘ ) ∘ ( ( 2nd𝑔 ) ‘ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st𝑓 ) ) , ( 𝑠 ∘ ( 2nd𝑓 ) ) ⟩ ) ⟩ } ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cdvh DVecH
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 ctendo TEndo
15 5 14 cfv ( TEndo ‘ 𝑘 )
16 12 15 cfv ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 )
17 13 16 cxp ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) )
18 9 17 cop ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ⟩
19 cplusg +g
20 8 19 cfv ( +g ‘ ndx )
21 vf 𝑓
22 vg 𝑔
23 c1st 1st
24 21 cv 𝑓
25 24 23 cfv ( 1st𝑓 )
26 22 cv 𝑔
27 26 23 cfv ( 1st𝑔 )
28 25 27 ccom ( ( 1st𝑓 ) ∘ ( 1st𝑔 ) )
29 vh
30 c2nd 2nd
31 24 30 cfv ( 2nd𝑓 )
32 29 cv
33 32 31 cfv ( ( 2nd𝑓 ) ‘ )
34 26 30 cfv ( 2nd𝑔 )
35 32 34 cfv ( ( 2nd𝑔 ) ‘ )
36 33 35 ccom ( ( ( 2nd𝑓 ) ‘ ) ∘ ( ( 2nd𝑔 ) ‘ ) )
37 29 13 36 cmpt ( ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( ( 2nd𝑓 ) ‘ ) ∘ ( ( 2nd𝑔 ) ‘ ) ) )
38 28 37 cop ⟨ ( ( 1st𝑓 ) ∘ ( 1st𝑔 ) ) , ( ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( ( 2nd𝑓 ) ‘ ) ∘ ( ( 2nd𝑔 ) ‘ ) ) ) ⟩
39 21 22 17 17 38 cmpo ( 𝑓 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ↦ ⟨ ( ( 1st𝑓 ) ∘ ( 1st𝑔 ) ) , ( ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( ( 2nd𝑓 ) ‘ ) ∘ ( ( 2nd𝑔 ) ‘ ) ) ) ⟩ )
40 20 39 cop ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ↦ ⟨ ( ( 1st𝑓 ) ∘ ( 1st𝑔 ) ) , ( ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( ( 2nd𝑓 ) ‘ ) ∘ ( ( 2nd𝑔 ) ‘ ) ) ) ⟩ ) ⟩
41 csca Scalar
42 8 41 cfv ( Scalar ‘ ndx )
43 cedring EDRing
44 5 43 cfv ( EDRing ‘ 𝑘 )
45 12 44 cfv ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 )
46 42 45 cop ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 ) ⟩
47 18 40 46 ctp { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ↦ ⟨ ( ( 1st𝑓 ) ∘ ( 1st𝑔 ) ) , ( ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( ( 2nd𝑓 ) ‘ ) ∘ ( ( 2nd𝑔 ) ‘ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 ) ⟩ }
48 cvsca ·𝑠
49 8 48 cfv ( ·𝑠 ‘ ndx )
50 vs 𝑠
51 50 cv 𝑠
52 25 51 cfv ( 𝑠 ‘ ( 1st𝑓 ) )
53 51 31 ccom ( 𝑠 ∘ ( 2nd𝑓 ) )
54 52 53 cop ⟨ ( 𝑠 ‘ ( 1st𝑓 ) ) , ( 𝑠 ∘ ( 2nd𝑓 ) ) ⟩
55 50 21 16 17 54 cmpo ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st𝑓 ) ) , ( 𝑠 ∘ ( 2nd𝑓 ) ) ⟩ )
56 49 55 cop ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st𝑓 ) ) , ( 𝑠 ∘ ( 2nd𝑓 ) ) ⟩ ) ⟩
57 56 csn { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st𝑓 ) ) , ( 𝑠 ∘ ( 2nd𝑓 ) ) ⟩ ) ⟩ }
58 47 57 cun ( { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ↦ ⟨ ( ( 1st𝑓 ) ∘ ( 1st𝑔 ) ) , ( ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( ( 2nd𝑓 ) ‘ ) ∘ ( ( 2nd𝑔 ) ‘ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st𝑓 ) ) , ( 𝑠 ∘ ( 2nd𝑓 ) ) ⟩ ) ⟩ } )
59 3 6 58 cmpt ( 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 ) ) ↦ ⟨ ( ( 1st𝑓 ) ∘ ( 1st𝑔 ) ) , ( ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( ( 2nd𝑓 ) ‘ ) ∘ ( ( 2nd𝑔 ) ‘ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st𝑓 ) ) , ( 𝑠 ∘ ( 2nd𝑓 ) ) ⟩ ) ⟩ } ) )
60 1 2 59 cmpt ( 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 ) ) ↦ ⟨ ( ( 1st𝑓 ) ∘ ( 1st𝑔 ) ) , ( ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( ( 2nd𝑓 ) ‘ ) ∘ ( ( 2nd𝑔 ) ‘ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st𝑓 ) ) , ( 𝑠 ∘ ( 2nd𝑓 ) ) ⟩ ) ⟩ } ) ) )
61 0 60 wceq 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 ) ) ↦ ⟨ ( ( 1st𝑓 ) ∘ ( 1st𝑔 ) ) , ( ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( ( 2nd𝑓 ) ‘ ) ∘ ( ( 2nd𝑔 ) ‘ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝑘 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st𝑓 ) ) , ( 𝑠 ∘ ( 2nd𝑓 ) ) ⟩ ) ⟩ } ) ) )