Step |
Hyp |
Ref |
Expression |
1 |
|
dia2dimlem9.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
dia2dimlem9.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
dia2dimlem9.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
dia2dimlem9.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
dia2dimlem9.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
dia2dimlem9.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
dia2dimlem9.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dia2dimlem9.y |
⊢ 𝑌 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
dia2dimlem9.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑌 ) |
10 |
|
dia2dimlem9.pl |
⊢ ⊕ = ( LSSum ‘ 𝑌 ) |
11 |
|
dia2dimlem9.n |
⊢ 𝑁 = ( LSpan ‘ 𝑌 ) |
12 |
|
dia2dimlem9.i |
⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) |
13 |
|
dia2dimlem9.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
14 |
|
dia2dimlem9.u |
⊢ ( 𝜑 → ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) |
15 |
|
dia2dimlem9.v |
⊢ ( 𝜑 → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) |
16 |
|
dia2dimlem9.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝑇 ) |
17 |
|
dia2dimlem9.rf |
⊢ ( 𝜑 → ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑈 ∨ 𝑉 ) ) |
18 |
|
dia2dimlem9.uv |
⊢ ( 𝜑 → 𝑈 ≠ 𝑉 ) |
19 |
5 8
|
dvalvec |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑌 ∈ LVec ) |
20 |
|
lveclmod |
⊢ ( 𝑌 ∈ LVec → 𝑌 ∈ LMod ) |
21 |
9
|
lsssssubg |
⊢ ( 𝑌 ∈ LMod → 𝑆 ⊆ ( SubGrp ‘ 𝑌 ) ) |
22 |
13 19 20 21
|
4syl |
⊢ ( 𝜑 → 𝑆 ⊆ ( SubGrp ‘ 𝑌 ) ) |
23 |
14
|
simpld |
⊢ ( 𝜑 → 𝑈 ∈ 𝐴 ) |
24 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
25 |
24 4
|
atbase |
⊢ ( 𝑈 ∈ 𝐴 → 𝑈 ∈ ( Base ‘ 𝐾 ) ) |
26 |
23 25
|
syl |
⊢ ( 𝜑 → 𝑈 ∈ ( Base ‘ 𝐾 ) ) |
27 |
14
|
simprd |
⊢ ( 𝜑 → 𝑈 ≤ 𝑊 ) |
28 |
24 1 5 8 12 9
|
dialss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ ( Base ‘ 𝐾 ) ∧ 𝑈 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑈 ) ∈ 𝑆 ) |
29 |
13 26 27 28
|
syl12anc |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑈 ) ∈ 𝑆 ) |
30 |
22 29
|
sseldd |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑈 ) ∈ ( SubGrp ‘ 𝑌 ) ) |
31 |
15
|
simpld |
⊢ ( 𝜑 → 𝑉 ∈ 𝐴 ) |
32 |
24 4
|
atbase |
⊢ ( 𝑉 ∈ 𝐴 → 𝑉 ∈ ( Base ‘ 𝐾 ) ) |
33 |
31 32
|
syl |
⊢ ( 𝜑 → 𝑉 ∈ ( Base ‘ 𝐾 ) ) |
34 |
15
|
simprd |
⊢ ( 𝜑 → 𝑉 ≤ 𝑊 ) |
35 |
24 1 5 8 12 9
|
dialss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑉 ∈ ( Base ‘ 𝐾 ) ∧ 𝑉 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑉 ) ∈ 𝑆 ) |
36 |
13 33 34 35
|
syl12anc |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑉 ) ∈ 𝑆 ) |
37 |
22 36
|
sseldd |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑉 ) ∈ ( SubGrp ‘ 𝑌 ) ) |
38 |
10
|
lsmub1 |
⊢ ( ( ( 𝐼 ‘ 𝑈 ) ∈ ( SubGrp ‘ 𝑌 ) ∧ ( 𝐼 ‘ 𝑉 ) ∈ ( SubGrp ‘ 𝑌 ) ) → ( 𝐼 ‘ 𝑈 ) ⊆ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |
39 |
30 37 38
|
syl2anc |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑈 ) ⊆ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |
40 |
39
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑅 ‘ 𝐹 ) = 𝑈 ) → ( 𝐼 ‘ 𝑈 ) ⊆ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |
41 |
5 6 7 12
|
dia1dimid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) ) |
42 |
13 16 41
|
syl2anc |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) ) |
43 |
42
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑅 ‘ 𝐹 ) = 𝑈 ) → 𝐹 ∈ ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) ) |
44 |
|
fveq2 |
⊢ ( ( 𝑅 ‘ 𝐹 ) = 𝑈 → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = ( 𝐼 ‘ 𝑈 ) ) |
45 |
44
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑅 ‘ 𝐹 ) = 𝑈 ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = ( 𝐼 ‘ 𝑈 ) ) |
46 |
43 45
|
eleqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑅 ‘ 𝐹 ) = 𝑈 ) → 𝐹 ∈ ( 𝐼 ‘ 𝑈 ) ) |
47 |
40 46
|
sseldd |
⊢ ( ( 𝜑 ∧ ( 𝑅 ‘ 𝐹 ) = 𝑈 ) → 𝐹 ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |
48 |
30
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑅 ‘ 𝐹 ) = 𝑉 ) → ( 𝐼 ‘ 𝑈 ) ∈ ( SubGrp ‘ 𝑌 ) ) |
49 |
37
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑅 ‘ 𝐹 ) = 𝑉 ) → ( 𝐼 ‘ 𝑉 ) ∈ ( SubGrp ‘ 𝑌 ) ) |
50 |
10
|
lsmub2 |
⊢ ( ( ( 𝐼 ‘ 𝑈 ) ∈ ( SubGrp ‘ 𝑌 ) ∧ ( 𝐼 ‘ 𝑉 ) ∈ ( SubGrp ‘ 𝑌 ) ) → ( 𝐼 ‘ 𝑉 ) ⊆ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |
51 |
48 49 50
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑅 ‘ 𝐹 ) = 𝑉 ) → ( 𝐼 ‘ 𝑉 ) ⊆ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |
52 |
42
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑅 ‘ 𝐹 ) = 𝑉 ) → 𝐹 ∈ ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) ) |
53 |
|
fveq2 |
⊢ ( ( 𝑅 ‘ 𝐹 ) = 𝑉 → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = ( 𝐼 ‘ 𝑉 ) ) |
54 |
53
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑅 ‘ 𝐹 ) = 𝑉 ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = ( 𝐼 ‘ 𝑉 ) ) |
55 |
52 54
|
eleqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑅 ‘ 𝐹 ) = 𝑉 ) → 𝐹 ∈ ( 𝐼 ‘ 𝑉 ) ) |
56 |
51 55
|
sseldd |
⊢ ( ( 𝜑 ∧ ( 𝑅 ‘ 𝐹 ) = 𝑉 ) → 𝐹 ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |
57 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑅 ‘ 𝐹 ) ≠ 𝑈 ∧ ( 𝑅 ‘ 𝐹 ) ≠ 𝑉 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
58 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑅 ‘ 𝐹 ) ≠ 𝑈 ∧ ( 𝑅 ‘ 𝐹 ) ≠ 𝑉 ) ) → ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) |
59 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑅 ‘ 𝐹 ) ≠ 𝑈 ∧ ( 𝑅 ‘ 𝐹 ) ≠ 𝑉 ) ) → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) |
60 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑅 ‘ 𝐹 ) ≠ 𝑈 ∧ ( 𝑅 ‘ 𝐹 ) ≠ 𝑉 ) ) → 𝐹 ∈ 𝑇 ) |
61 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑅 ‘ 𝐹 ) ≠ 𝑈 ∧ ( 𝑅 ‘ 𝐹 ) ≠ 𝑉 ) ) → ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑈 ∨ 𝑉 ) ) |
62 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑅 ‘ 𝐹 ) ≠ 𝑈 ∧ ( 𝑅 ‘ 𝐹 ) ≠ 𝑉 ) ) → 𝑈 ≠ 𝑉 ) |
63 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( ( 𝑅 ‘ 𝐹 ) ≠ 𝑈 ∧ ( 𝑅 ‘ 𝐹 ) ≠ 𝑉 ) ) → ( 𝑅 ‘ 𝐹 ) ≠ 𝑈 ) |
64 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( ( 𝑅 ‘ 𝐹 ) ≠ 𝑈 ∧ ( 𝑅 ‘ 𝐹 ) ≠ 𝑉 ) ) → ( 𝑅 ‘ 𝐹 ) ≠ 𝑉 ) |
65 |
1 2 3 4 5 6 7 8 9 10 11 12 57 58 59 60 61 62 63 64
|
dia2dimlem8 |
⊢ ( ( 𝜑 ∧ ( ( 𝑅 ‘ 𝐹 ) ≠ 𝑈 ∧ ( 𝑅 ‘ 𝐹 ) ≠ 𝑉 ) ) → 𝐹 ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |
66 |
47 56 65
|
pm2.61da2ne |
⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |