Step |
Hyp |
Ref |
Expression |
1 |
|
dia2dimlem5.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
dia2dimlem5.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
dia2dimlem5.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
dia2dimlem5.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
dia2dimlem5.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
dia2dimlem5.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
dia2dimlem5.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dia2dimlem5.y |
⊢ 𝑌 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
dia2dimlem5.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑌 ) |
10 |
|
dia2dimlem5.pl |
⊢ ⊕ = ( LSSum ‘ 𝑌 ) |
11 |
|
dia2dimlem5.n |
⊢ 𝑁 = ( LSpan ‘ 𝑌 ) |
12 |
|
dia2dimlem5.i |
⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) |
13 |
|
dia2dimlem5.q |
⊢ 𝑄 = ( ( 𝑃 ∨ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ 𝑉 ) ) |
14 |
|
dia2dimlem5.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
15 |
|
dia2dimlem5.u |
⊢ ( 𝜑 → ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) |
16 |
|
dia2dimlem5.v |
⊢ ( 𝜑 → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) |
17 |
|
dia2dimlem5.p |
⊢ ( 𝜑 → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
18 |
|
dia2dimlem5.f |
⊢ ( 𝜑 → ( 𝐹 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) |
19 |
|
dia2dimlem5.rf |
⊢ ( 𝜑 → ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑈 ∨ 𝑉 ) ) |
20 |
|
dia2dimlem5.uv |
⊢ ( 𝜑 → 𝑈 ≠ 𝑉 ) |
21 |
|
dia2dimlem5.ru |
⊢ ( 𝜑 → ( 𝑅 ‘ 𝐹 ) ≠ 𝑈 ) |
22 |
|
dia2dimlem5.rv |
⊢ ( 𝜑 → ( 𝑅 ‘ 𝐹 ) ≠ 𝑉 ) |
23 |
|
dia2dimlem5.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝑇 ) |
24 |
|
dia2dimlem5.gv |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑃 ) = 𝑄 ) |
25 |
|
dia2dimlem5.d |
⊢ ( 𝜑 → 𝐷 ∈ 𝑇 ) |
26 |
|
dia2dimlem5.dv |
⊢ ( 𝜑 → ( 𝐷 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) ) |
27 |
|
eqid |
⊢ ( +g ‘ 𝑌 ) = ( +g ‘ 𝑌 ) |
28 |
5 6 8 27
|
dvavadd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( 𝐷 ( +g ‘ 𝑌 ) 𝐺 ) = ( 𝐷 ∘ 𝐺 ) ) |
29 |
14 25 23 28
|
syl12anc |
⊢ ( 𝜑 → ( 𝐷 ( +g ‘ 𝑌 ) 𝐺 ) = ( 𝐷 ∘ 𝐺 ) ) |
30 |
18
|
simpld |
⊢ ( 𝜑 → 𝐹 ∈ 𝑇 ) |
31 |
1 4 5 6 14 17 30 23 24 25 26
|
dia2dimlem4 |
⊢ ( 𝜑 → ( 𝐷 ∘ 𝐺 ) = 𝐹 ) |
32 |
29 31
|
eqtr2d |
⊢ ( 𝜑 → 𝐹 = ( 𝐷 ( +g ‘ 𝑌 ) 𝐺 ) ) |
33 |
5 8
|
dvalvec |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑌 ∈ LVec ) |
34 |
|
lveclmod |
⊢ ( 𝑌 ∈ LVec → 𝑌 ∈ LMod ) |
35 |
14 33 34
|
3syl |
⊢ ( 𝜑 → 𝑌 ∈ LMod ) |
36 |
9
|
lsssssubg |
⊢ ( 𝑌 ∈ LMod → 𝑆 ⊆ ( SubGrp ‘ 𝑌 ) ) |
37 |
35 36
|
syl |
⊢ ( 𝜑 → 𝑆 ⊆ ( SubGrp ‘ 𝑌 ) ) |
38 |
16
|
simpld |
⊢ ( 𝜑 → 𝑉 ∈ 𝐴 ) |
39 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
40 |
39 4
|
atbase |
⊢ ( 𝑉 ∈ 𝐴 → 𝑉 ∈ ( Base ‘ 𝐾 ) ) |
41 |
38 40
|
syl |
⊢ ( 𝜑 → 𝑉 ∈ ( Base ‘ 𝐾 ) ) |
42 |
16
|
simprd |
⊢ ( 𝜑 → 𝑉 ≤ 𝑊 ) |
43 |
39 1 5 8 12 9
|
dialss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑉 ∈ ( Base ‘ 𝐾 ) ∧ 𝑉 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑉 ) ∈ 𝑆 ) |
44 |
14 41 42 43
|
syl12anc |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑉 ) ∈ 𝑆 ) |
45 |
37 44
|
sseldd |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑉 ) ∈ ( SubGrp ‘ 𝑌 ) ) |
46 |
15
|
simpld |
⊢ ( 𝜑 → 𝑈 ∈ 𝐴 ) |
47 |
39 4
|
atbase |
⊢ ( 𝑈 ∈ 𝐴 → 𝑈 ∈ ( Base ‘ 𝐾 ) ) |
48 |
46 47
|
syl |
⊢ ( 𝜑 → 𝑈 ∈ ( Base ‘ 𝐾 ) ) |
49 |
15
|
simprd |
⊢ ( 𝜑 → 𝑈 ≤ 𝑊 ) |
50 |
39 1 5 8 12 9
|
dialss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ ( Base ‘ 𝐾 ) ∧ 𝑈 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑈 ) ∈ 𝑆 ) |
51 |
14 48 49 50
|
syl12anc |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑈 ) ∈ 𝑆 ) |
52 |
37 51
|
sseldd |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑈 ) ∈ ( SubGrp ‘ 𝑌 ) ) |
53 |
5 6 7 8 12 11
|
dia1dim2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐷 ∈ 𝑇 ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐷 ) ) = ( 𝑁 ‘ { 𝐷 } ) ) |
54 |
14 25 53
|
syl2anc |
⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑅 ‘ 𝐷 ) ) = ( 𝑁 ‘ { 𝐷 } ) ) |
55 |
1 2 3 4 5 6 7 13 14 15 16 17 18 19 20 21 22 25 26
|
dia2dimlem3 |
⊢ ( 𝜑 → ( 𝑅 ‘ 𝐷 ) = 𝑉 ) |
56 |
55
|
fveq2d |
⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑅 ‘ 𝐷 ) ) = ( 𝐼 ‘ 𝑉 ) ) |
57 |
|
eqss |
⊢ ( ( 𝐼 ‘ ( 𝑅 ‘ 𝐷 ) ) = ( 𝐼 ‘ 𝑉 ) ↔ ( ( 𝐼 ‘ ( 𝑅 ‘ 𝐷 ) ) ⊆ ( 𝐼 ‘ 𝑉 ) ∧ ( 𝐼 ‘ 𝑉 ) ⊆ ( 𝐼 ‘ ( 𝑅 ‘ 𝐷 ) ) ) ) |
58 |
56 57
|
sylib |
⊢ ( 𝜑 → ( ( 𝐼 ‘ ( 𝑅 ‘ 𝐷 ) ) ⊆ ( 𝐼 ‘ 𝑉 ) ∧ ( 𝐼 ‘ 𝑉 ) ⊆ ( 𝐼 ‘ ( 𝑅 ‘ 𝐷 ) ) ) ) |
59 |
58
|
simpld |
⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑅 ‘ 𝐷 ) ) ⊆ ( 𝐼 ‘ 𝑉 ) ) |
60 |
54 59
|
eqsstrrd |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝐷 } ) ⊆ ( 𝐼 ‘ 𝑉 ) ) |
61 |
|
eqid |
⊢ ( Base ‘ 𝑌 ) = ( Base ‘ 𝑌 ) |
62 |
5 6 8 61
|
dvavbase |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( Base ‘ 𝑌 ) = 𝑇 ) |
63 |
14 62
|
syl |
⊢ ( 𝜑 → ( Base ‘ 𝑌 ) = 𝑇 ) |
64 |
25 63
|
eleqtrrd |
⊢ ( 𝜑 → 𝐷 ∈ ( Base ‘ 𝑌 ) ) |
65 |
61 9 11 35 44 64
|
lspsnel5 |
⊢ ( 𝜑 → ( 𝐷 ∈ ( 𝐼 ‘ 𝑉 ) ↔ ( 𝑁 ‘ { 𝐷 } ) ⊆ ( 𝐼 ‘ 𝑉 ) ) ) |
66 |
60 65
|
mpbird |
⊢ ( 𝜑 → 𝐷 ∈ ( 𝐼 ‘ 𝑉 ) ) |
67 |
5 6 7 8 12 11
|
dia1dim2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐺 ) ) = ( 𝑁 ‘ { 𝐺 } ) ) |
68 |
14 23 67
|
syl2anc |
⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑅 ‘ 𝐺 ) ) = ( 𝑁 ‘ { 𝐺 } ) ) |
69 |
1 2 3 4 5 6 7 13 14 15 16 17 18 19 22 23 24
|
dia2dimlem2 |
⊢ ( 𝜑 → ( 𝑅 ‘ 𝐺 ) = 𝑈 ) |
70 |
69
|
fveq2d |
⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑅 ‘ 𝐺 ) ) = ( 𝐼 ‘ 𝑈 ) ) |
71 |
|
eqss |
⊢ ( ( 𝐼 ‘ ( 𝑅 ‘ 𝐺 ) ) = ( 𝐼 ‘ 𝑈 ) ↔ ( ( 𝐼 ‘ ( 𝑅 ‘ 𝐺 ) ) ⊆ ( 𝐼 ‘ 𝑈 ) ∧ ( 𝐼 ‘ 𝑈 ) ⊆ ( 𝐼 ‘ ( 𝑅 ‘ 𝐺 ) ) ) ) |
72 |
70 71
|
sylib |
⊢ ( 𝜑 → ( ( 𝐼 ‘ ( 𝑅 ‘ 𝐺 ) ) ⊆ ( 𝐼 ‘ 𝑈 ) ∧ ( 𝐼 ‘ 𝑈 ) ⊆ ( 𝐼 ‘ ( 𝑅 ‘ 𝐺 ) ) ) ) |
73 |
72
|
simpld |
⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑅 ‘ 𝐺 ) ) ⊆ ( 𝐼 ‘ 𝑈 ) ) |
74 |
68 73
|
eqsstrrd |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝐺 } ) ⊆ ( 𝐼 ‘ 𝑈 ) ) |
75 |
23 63
|
eleqtrrd |
⊢ ( 𝜑 → 𝐺 ∈ ( Base ‘ 𝑌 ) ) |
76 |
61 9 11 35 51 75
|
lspsnel5 |
⊢ ( 𝜑 → ( 𝐺 ∈ ( 𝐼 ‘ 𝑈 ) ↔ ( 𝑁 ‘ { 𝐺 } ) ⊆ ( 𝐼 ‘ 𝑈 ) ) ) |
77 |
74 76
|
mpbird |
⊢ ( 𝜑 → 𝐺 ∈ ( 𝐼 ‘ 𝑈 ) ) |
78 |
27 10
|
lsmelvali |
⊢ ( ( ( ( 𝐼 ‘ 𝑉 ) ∈ ( SubGrp ‘ 𝑌 ) ∧ ( 𝐼 ‘ 𝑈 ) ∈ ( SubGrp ‘ 𝑌 ) ) ∧ ( 𝐷 ∈ ( 𝐼 ‘ 𝑉 ) ∧ 𝐺 ∈ ( 𝐼 ‘ 𝑈 ) ) ) → ( 𝐷 ( +g ‘ 𝑌 ) 𝐺 ) ∈ ( ( 𝐼 ‘ 𝑉 ) ⊕ ( 𝐼 ‘ 𝑈 ) ) ) |
79 |
45 52 66 77 78
|
syl22anc |
⊢ ( 𝜑 → ( 𝐷 ( +g ‘ 𝑌 ) 𝐺 ) ∈ ( ( 𝐼 ‘ 𝑉 ) ⊕ ( 𝐼 ‘ 𝑈 ) ) ) |
80 |
32 79
|
eqeltrd |
⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐼 ‘ 𝑉 ) ⊕ ( 𝐼 ‘ 𝑈 ) ) ) |
81 |
|
lmodabl |
⊢ ( 𝑌 ∈ LMod → 𝑌 ∈ Abel ) |
82 |
35 81
|
syl |
⊢ ( 𝜑 → 𝑌 ∈ Abel ) |
83 |
10
|
lsmcom |
⊢ ( ( 𝑌 ∈ Abel ∧ ( 𝐼 ‘ 𝑉 ) ∈ ( SubGrp ‘ 𝑌 ) ∧ ( 𝐼 ‘ 𝑈 ) ∈ ( SubGrp ‘ 𝑌 ) ) → ( ( 𝐼 ‘ 𝑉 ) ⊕ ( 𝐼 ‘ 𝑈 ) ) = ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |
84 |
82 45 52 83
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝑉 ) ⊕ ( 𝐼 ‘ 𝑈 ) ) = ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |
85 |
80 84
|
eleqtrd |
⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |