Step |
Hyp |
Ref |
Expression |
1 |
|
dia2dimlem10.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
dia2dimlem10.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
dia2dimlem10.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
dia2dimlem10.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
5 |
|
dia2dimlem10.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
dia2dimlem10.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
dia2dimlem10.y |
⊢ 𝑌 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dia2dimlem10.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑌 ) |
9 |
|
dia2dimlem10.n |
⊢ 𝑁 = ( LSpan ‘ 𝑌 ) |
10 |
|
dia2dimlem10.i |
⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) |
11 |
|
dia2dimlem10.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
12 |
|
dia2dimlem10.u |
⊢ ( 𝜑 → ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) |
13 |
|
dia2dimlem10.v |
⊢ ( 𝜑 → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) |
14 |
|
dia2dimlem10.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝑇 ) |
15 |
|
dia2dimlem10.fe |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝐼 ‘ ( 𝑈 ∨ 𝑉 ) ) ) |
16 |
4 5 6 7 10 9
|
dia1dim2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑁 ‘ { 𝐹 } ) ) |
17 |
11 14 16
|
syl2anc |
⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑁 ‘ { 𝐹 } ) ) |
18 |
4 7
|
dvalvec |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑌 ∈ LVec ) |
19 |
|
lveclmod |
⊢ ( 𝑌 ∈ LVec → 𝑌 ∈ LMod ) |
20 |
11 18 19
|
3syl |
⊢ ( 𝜑 → 𝑌 ∈ LMod ) |
21 |
11
|
simpld |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
22 |
12
|
simpld |
⊢ ( 𝜑 → 𝑈 ∈ 𝐴 ) |
23 |
13
|
simpld |
⊢ ( 𝜑 → 𝑉 ∈ 𝐴 ) |
24 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
25 |
24 2 3
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ) → ( 𝑈 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) ) |
26 |
21 22 23 25
|
syl3anc |
⊢ ( 𝜑 → ( 𝑈 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) ) |
27 |
12
|
simprd |
⊢ ( 𝜑 → 𝑈 ≤ 𝑊 ) |
28 |
13
|
simprd |
⊢ ( 𝜑 → 𝑉 ≤ 𝑊 ) |
29 |
21
|
hllatd |
⊢ ( 𝜑 → 𝐾 ∈ Lat ) |
30 |
24 3
|
atbase |
⊢ ( 𝑈 ∈ 𝐴 → 𝑈 ∈ ( Base ‘ 𝐾 ) ) |
31 |
22 30
|
syl |
⊢ ( 𝜑 → 𝑈 ∈ ( Base ‘ 𝐾 ) ) |
32 |
24 3
|
atbase |
⊢ ( 𝑉 ∈ 𝐴 → 𝑉 ∈ ( Base ‘ 𝐾 ) ) |
33 |
23 32
|
syl |
⊢ ( 𝜑 → 𝑉 ∈ ( Base ‘ 𝐾 ) ) |
34 |
11
|
simprd |
⊢ ( 𝜑 → 𝑊 ∈ 𝐻 ) |
35 |
24 4
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
36 |
34 35
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
37 |
24 1 2
|
latjle12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑈 ∈ ( Base ‘ 𝐾 ) ∧ 𝑉 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑈 ≤ 𝑊 ∧ 𝑉 ≤ 𝑊 ) ↔ ( 𝑈 ∨ 𝑉 ) ≤ 𝑊 ) ) |
38 |
29 31 33 36 37
|
syl13anc |
⊢ ( 𝜑 → ( ( 𝑈 ≤ 𝑊 ∧ 𝑉 ≤ 𝑊 ) ↔ ( 𝑈 ∨ 𝑉 ) ≤ 𝑊 ) ) |
39 |
27 28 38
|
mpbi2and |
⊢ ( 𝜑 → ( 𝑈 ∨ 𝑉 ) ≤ 𝑊 ) |
40 |
24 1 4 7 10 8
|
dialss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑈 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑈 ∨ 𝑉 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( 𝑈 ∨ 𝑉 ) ) ∈ 𝑆 ) |
41 |
11 26 39 40
|
syl12anc |
⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑈 ∨ 𝑉 ) ) ∈ 𝑆 ) |
42 |
8 9 20 41 15
|
lspsnel5a |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝐹 } ) ⊆ ( 𝐼 ‘ ( 𝑈 ∨ 𝑉 ) ) ) |
43 |
17 42
|
eqsstrd |
⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) ⊆ ( 𝐼 ‘ ( 𝑈 ∨ 𝑉 ) ) ) |
44 |
24 4 5 6
|
trlcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) ) |
45 |
11 14 44
|
syl2anc |
⊢ ( 𝜑 → ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) ) |
46 |
1 4 5 6
|
trlle |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ≤ 𝑊 ) |
47 |
11 14 46
|
syl2anc |
⊢ ( 𝜑 → ( 𝑅 ‘ 𝐹 ) ≤ 𝑊 ) |
48 |
24 1 4 10
|
diaord |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑅 ‘ 𝐹 ) ≤ 𝑊 ) ∧ ( ( 𝑈 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑈 ∨ 𝑉 ) ≤ 𝑊 ) ) → ( ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) ⊆ ( 𝐼 ‘ ( 𝑈 ∨ 𝑉 ) ) ↔ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑈 ∨ 𝑉 ) ) ) |
49 |
11 45 47 26 39 48
|
syl122anc |
⊢ ( 𝜑 → ( ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) ⊆ ( 𝐼 ‘ ( 𝑈 ∨ 𝑉 ) ) ↔ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑈 ∨ 𝑉 ) ) ) |
50 |
43 49
|
mpbid |
⊢ ( 𝜑 → ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑈 ∨ 𝑉 ) ) |