Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemn4.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
cdlemn4.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
cdlemn4.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
cdlemn4.p |
⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
cdlemn4.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
cdlemn4.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
cdlemn4.o |
⊢ 𝑂 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) |
8 |
|
cdlemn4.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
cdlemn4.f |
⊢ 𝐹 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑄 ) |
10 |
|
cdlemn4.g |
⊢ 𝐺 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑅 ) |
11 |
|
cdlemn4.j |
⊢ 𝐽 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑄 ) = 𝑅 ) |
12 |
|
cdlemn4.s |
⊢ + = ( +g ‘ 𝑈 ) |
13 |
|
simp1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
14 |
2 3 5 4
|
lhpocnel2 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
15 |
13 14
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
16 |
|
simp2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
17 |
2 3 5 6 9
|
ltrniotacl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 ) |
18 |
13 15 16 17
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 ) |
19 |
|
eqid |
⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
20 |
5 6 19
|
tendoidcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( I ↾ 𝑇 ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) |
21 |
13 20
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( I ↾ 𝑇 ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) |
22 |
2 3 5 6 11
|
ltrniotacl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → 𝐽 ∈ 𝑇 ) |
23 |
1 5 6 19 7
|
tendo0cl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑂 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) |
24 |
13 23
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → 𝑂 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) |
25 |
|
eqid |
⊢ ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 ) |
26 |
|
eqid |
⊢ ( +g ‘ ( Scalar ‘ 𝑈 ) ) = ( +g ‘ ( Scalar ‘ 𝑈 ) ) |
27 |
5 6 19 8 25 12 26
|
dvhopvadd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( I ↾ 𝑇 ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ∧ ( 𝐽 ∈ 𝑇 ∧ 𝑂 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) → ( 〈 𝐹 , ( I ↾ 𝑇 ) 〉 + 〈 𝐽 , 𝑂 〉 ) = 〈 ( 𝐹 ∘ 𝐽 ) , ( ( I ↾ 𝑇 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑂 ) 〉 ) |
28 |
13 18 21 22 24 27
|
syl122anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( 〈 𝐹 , ( I ↾ 𝑇 ) 〉 + 〈 𝐽 , 𝑂 〉 ) = 〈 ( 𝐹 ∘ 𝐽 ) , ( ( I ↾ 𝑇 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑂 ) 〉 ) |
29 |
5 6
|
ltrncom |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐽 ∈ 𝑇 ) → ( 𝐹 ∘ 𝐽 ) = ( 𝐽 ∘ 𝐹 ) ) |
30 |
13 18 22 29
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( 𝐹 ∘ 𝐽 ) = ( 𝐽 ∘ 𝐹 ) ) |
31 |
2 3 4 5 6 9 10 11
|
cdlemn3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( 𝐽 ∘ 𝐹 ) = 𝐺 ) |
32 |
30 31
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( 𝐹 ∘ 𝐽 ) = 𝐺 ) |
33 |
|
eqid |
⊢ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) |
34 |
5 33 8 25
|
dvhsca |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( Scalar ‘ 𝑈 ) = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) |
35 |
34
|
fveq2d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 0g ‘ ( Scalar ‘ 𝑈 ) ) = ( 0g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
36 |
|
eqid |
⊢ ( 0g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = ( 0g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) |
37 |
1 5 6 33 7 36
|
erng0g |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 0g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑂 ) |
38 |
35 37
|
eqtrd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 0g ‘ ( Scalar ‘ 𝑈 ) ) = 𝑂 ) |
39 |
13 38
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( 0g ‘ ( Scalar ‘ 𝑈 ) ) = 𝑂 ) |
40 |
39
|
oveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( ( I ↾ 𝑇 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) ( 0g ‘ ( Scalar ‘ 𝑈 ) ) ) = ( ( I ↾ 𝑇 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑂 ) ) |
41 |
5 33
|
erngdv |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ∈ DivRing ) |
42 |
|
drnggrp |
⊢ ( ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ∈ DivRing → ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ∈ Grp ) |
43 |
41 42
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ∈ Grp ) |
44 |
34 43
|
eqeltrd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( Scalar ‘ 𝑈 ) ∈ Grp ) |
45 |
13 44
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( Scalar ‘ 𝑈 ) ∈ Grp ) |
46 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑈 ) ) = ( Base ‘ ( Scalar ‘ 𝑈 ) ) |
47 |
5 19 8 25 46
|
dvhbase |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( Base ‘ ( Scalar ‘ 𝑈 ) ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) |
48 |
13 47
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( Base ‘ ( Scalar ‘ 𝑈 ) ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) |
49 |
21 48
|
eleqtrrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( I ↾ 𝑇 ) ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ) |
50 |
|
eqid |
⊢ ( 0g ‘ ( Scalar ‘ 𝑈 ) ) = ( 0g ‘ ( Scalar ‘ 𝑈 ) ) |
51 |
46 26 50
|
grprid |
⊢ ( ( ( Scalar ‘ 𝑈 ) ∈ Grp ∧ ( I ↾ 𝑇 ) ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ) → ( ( I ↾ 𝑇 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) ( 0g ‘ ( Scalar ‘ 𝑈 ) ) ) = ( I ↾ 𝑇 ) ) |
52 |
45 49 51
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( ( I ↾ 𝑇 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) ( 0g ‘ ( Scalar ‘ 𝑈 ) ) ) = ( I ↾ 𝑇 ) ) |
53 |
40 52
|
eqtr3d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( ( I ↾ 𝑇 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑂 ) = ( I ↾ 𝑇 ) ) |
54 |
32 53
|
opeq12d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → 〈 ( 𝐹 ∘ 𝐽 ) , ( ( I ↾ 𝑇 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑂 ) 〉 = 〈 𝐺 , ( I ↾ 𝑇 ) 〉 ) |
55 |
28 54
|
eqtr2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → 〈 𝐺 , ( I ↾ 𝑇 ) 〉 = ( 〈 𝐹 , ( I ↾ 𝑇 ) 〉 + 〈 𝐽 , 𝑂 〉 ) ) |