Step |
Hyp |
Ref |
Expression |
1 |
|
dihjatcclem.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dihjatcclem.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dihjatcclem.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
dihjatcclem.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
5 |
|
dihjatcclem.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
6 |
|
dihjatcclem.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
7 |
|
dihjatcclem.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dihjatcclem.s |
⊢ ⊕ = ( LSSum ‘ 𝑈 ) |
9 |
|
dihjatcclem.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
10 |
|
dihjatcclem.v |
⊢ 𝑉 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) |
11 |
|
dihjatcclem.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
12 |
|
dihjatcclem.p |
⊢ ( 𝜑 → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
13 |
|
dihjatcclem.q |
⊢ ( 𝜑 → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
14 |
|
dihjatcc.w |
⊢ 𝐶 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) |
15 |
|
dihjatcc.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
16 |
|
dihjatcc.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
17 |
|
dihjatcc.e |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
18 |
|
dihjatcc.g |
⊢ 𝐺 = ( ℩ 𝑑 ∈ 𝑇 ( 𝑑 ‘ 𝐶 ) = 𝑃 ) |
19 |
|
dihjatcc.dd |
⊢ 𝐷 = ( ℩ 𝑑 ∈ 𝑇 ( 𝑑 ‘ 𝐶 ) = 𝑄 ) |
20 |
2 6 3 14
|
lhpocnel2 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊 ) ) |
21 |
11 20
|
syl |
⊢ ( 𝜑 → ( 𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊 ) ) |
22 |
2 6 3 15 18
|
ltrniotacl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐺 ∈ 𝑇 ) |
23 |
11 21 12 22
|
syl3anc |
⊢ ( 𝜑 → 𝐺 ∈ 𝑇 ) |
24 |
2 6 3 15 19
|
ltrniotacl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐷 ∈ 𝑇 ) |
25 |
11 21 13 24
|
syl3anc |
⊢ ( 𝜑 → 𝐷 ∈ 𝑇 ) |
26 |
3 15
|
ltrncnv |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐷 ∈ 𝑇 ) → ◡ 𝐷 ∈ 𝑇 ) |
27 |
11 25 26
|
syl2anc |
⊢ ( 𝜑 → ◡ 𝐷 ∈ 𝑇 ) |
28 |
3 15
|
ltrnco |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ◡ 𝐷 ∈ 𝑇 ) → ( 𝐺 ∘ ◡ 𝐷 ) ∈ 𝑇 ) |
29 |
11 23 27 28
|
syl3anc |
⊢ ( 𝜑 → ( 𝐺 ∘ ◡ 𝐷 ) ∈ 𝑇 ) |
30 |
2 4 5 6 3 15 16
|
trlval2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐺 ∘ ◡ 𝐷 ) ∈ 𝑇 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐷 ) ) = ( ( 𝑄 ∨ ( ( 𝐺 ∘ ◡ 𝐷 ) ‘ 𝑄 ) ) ∧ 𝑊 ) ) |
31 |
11 29 13 30
|
syl3anc |
⊢ ( 𝜑 → ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐷 ) ) = ( ( 𝑄 ∨ ( ( 𝐺 ∘ ◡ 𝐷 ) ‘ 𝑄 ) ) ∧ 𝑊 ) ) |
32 |
13
|
simpld |
⊢ ( 𝜑 → 𝑄 ∈ 𝐴 ) |
33 |
2 6 3 15
|
ltrncoval |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ◡ 𝐷 ∈ 𝑇 ) ∧ 𝑄 ∈ 𝐴 ) → ( ( 𝐺 ∘ ◡ 𝐷 ) ‘ 𝑄 ) = ( 𝐺 ‘ ( ◡ 𝐷 ‘ 𝑄 ) ) ) |
34 |
11 23 27 32 33
|
syl121anc |
⊢ ( 𝜑 → ( ( 𝐺 ∘ ◡ 𝐷 ) ‘ 𝑄 ) = ( 𝐺 ‘ ( ◡ 𝐷 ‘ 𝑄 ) ) ) |
35 |
2 6 3 15 19
|
ltrniotacnvval |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ◡ 𝐷 ‘ 𝑄 ) = 𝐶 ) |
36 |
11 21 13 35
|
syl3anc |
⊢ ( 𝜑 → ( ◡ 𝐷 ‘ 𝑄 ) = 𝐶 ) |
37 |
36
|
fveq2d |
⊢ ( 𝜑 → ( 𝐺 ‘ ( ◡ 𝐷 ‘ 𝑄 ) ) = ( 𝐺 ‘ 𝐶 ) ) |
38 |
2 6 3 15 18
|
ltrniotaval |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐺 ‘ 𝐶 ) = 𝑃 ) |
39 |
11 21 12 38
|
syl3anc |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝐶 ) = 𝑃 ) |
40 |
37 39
|
eqtrd |
⊢ ( 𝜑 → ( 𝐺 ‘ ( ◡ 𝐷 ‘ 𝑄 ) ) = 𝑃 ) |
41 |
34 40
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝐺 ∘ ◡ 𝐷 ) ‘ 𝑄 ) = 𝑃 ) |
42 |
41
|
oveq2d |
⊢ ( 𝜑 → ( 𝑄 ∨ ( ( 𝐺 ∘ ◡ 𝐷 ) ‘ 𝑄 ) ) = ( 𝑄 ∨ 𝑃 ) ) |
43 |
11
|
simpld |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
44 |
12
|
simpld |
⊢ ( 𝜑 → 𝑃 ∈ 𝐴 ) |
45 |
4 6
|
hlatjcom |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑃 ) ) |
46 |
43 44 32 45
|
syl3anc |
⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑃 ) ) |
47 |
42 46
|
eqtr4d |
⊢ ( 𝜑 → ( 𝑄 ∨ ( ( 𝐺 ∘ ◡ 𝐷 ) ‘ 𝑄 ) ) = ( 𝑃 ∨ 𝑄 ) ) |
48 |
47
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝑄 ∨ ( ( 𝐺 ∘ ◡ 𝐷 ) ‘ 𝑄 ) ) ∧ 𝑊 ) = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) |
49 |
48 10
|
eqtr4di |
⊢ ( 𝜑 → ( ( 𝑄 ∨ ( ( 𝐺 ∘ ◡ 𝐷 ) ‘ 𝑄 ) ) ∧ 𝑊 ) = 𝑉 ) |
50 |
31 49
|
eqtrd |
⊢ ( 𝜑 → ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐷 ) ) = 𝑉 ) |