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 |
⊢ ( 1st ‘ 𝑓 ) |
26 |
22
|
cv |
⊢ 𝑔 |
27 |
26 23
|
cfv |
⊢ ( 1st ‘ 𝑔 ) |
28 |
25 27
|
ccom |
⊢ ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) |
29 |
|
vh |
⊢ ℎ |
30 |
|
c2nd |
⊢ 2nd |
31 |
24 30
|
cfv |
⊢ ( 2nd ‘ 𝑓 ) |
32 |
29
|
cv |
⊢ ℎ |
33 |
32 31
|
cfv |
⊢ ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) |
34 |
26 30
|
cfv |
⊢ ( 2nd ‘ 𝑔 ) |
35 |
32 34
|
cfv |
⊢ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) |
36 |
33 35
|
ccom |
⊢ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) |
37 |
29 13 36
|
cmpt |
⊢ ( ℎ ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) |
38 |
28 37
|
cop |
⊢ 〈 ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) 〉 |
39 |
21 22 17 17 38
|
cmpo |
⊢ ( 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 ) ) ) 〉 ) |
40 |
20 39
|
cop |
⊢ 〈 ( +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 ) ) ) 〉 ) 〉 |
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 ) ) ↦ 〈 ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) 〉 ) 〉 , 〈 ( 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 ‘ ( 1st ‘ 𝑓 ) ) |
53 |
51 31
|
ccom |
⊢ ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) |
54 |
52 53
|
cop |
⊢ 〈 ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) 〉 |
55 |
50 21 16 17 54
|
cmpo |
⊢ ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ↦ 〈 ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) 〉 ) |
56 |
49 55
|
cop |
⊢ 〈 ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ↦ 〈 ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) 〉 ) 〉 |
57 |
56
|
csn |
⊢ { 〈 ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ) ↦ 〈 ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) 〉 ) 〉 } |
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 ) ) ↦ 〈 ( ( 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 ) ) 〉 ) 〉 } ) |
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 ) ) ↦ 〈 ( ( 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 ) ) 〉 ) 〉 } ) ) |
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 ) ) ↦ 〈 ( ( 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 ) ) 〉 ) 〉 } ) ) ) |
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 ) ) ↦ 〈 ( ( 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 ) ) 〉 ) 〉 } ) ) ) |