Step |
Hyp |
Ref |
Expression |
1 |
|
dia2dimlem12.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
dia2dimlem12.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
dia2dimlem12.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
dia2dimlem12.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
dia2dimlem12.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
dia2dimlem12.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
dia2dimlem12.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dia2dimlem12.y |
⊢ 𝑌 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
dia2dimlem12.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑌 ) |
10 |
|
dia2dimlem12.pl |
⊢ ⊕ = ( LSSum ‘ 𝑌 ) |
11 |
|
dia2dimlem12.n |
⊢ 𝑁 = ( LSpan ‘ 𝑌 ) |
12 |
|
dia2dimlem12.i |
⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) |
13 |
|
dia2dimlem12.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
14 |
|
dia2dimlem12.u |
⊢ ( 𝜑 → ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) |
15 |
|
dia2dimlem12.v |
⊢ ( 𝜑 → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) |
16 |
|
oveq2 |
⊢ ( 𝑈 = 𝑉 → ( 𝑈 ∨ 𝑈 ) = ( 𝑈 ∨ 𝑉 ) ) |
17 |
16
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑈 = 𝑉 ) → ( 𝑈 ∨ 𝑈 ) = ( 𝑈 ∨ 𝑉 ) ) |
18 |
13
|
simpld |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
19 |
14
|
simpld |
⊢ ( 𝜑 → 𝑈 ∈ 𝐴 ) |
20 |
2 4
|
hlatjidm |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ) → ( 𝑈 ∨ 𝑈 ) = 𝑈 ) |
21 |
18 19 20
|
syl2anc |
⊢ ( 𝜑 → ( 𝑈 ∨ 𝑈 ) = 𝑈 ) |
22 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑈 = 𝑉 ) → ( 𝑈 ∨ 𝑈 ) = 𝑈 ) |
23 |
17 22
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑈 = 𝑉 ) → ( 𝑈 ∨ 𝑉 ) = 𝑈 ) |
24 |
23
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑈 = 𝑉 ) → ( 𝐼 ‘ ( 𝑈 ∨ 𝑉 ) ) = ( 𝐼 ‘ 𝑈 ) ) |
25 |
|
ssid |
⊢ ( 𝐼 ‘ 𝑈 ) ⊆ ( 𝐼 ‘ 𝑈 ) |
26 |
24 25
|
eqsstrdi |
⊢ ( ( 𝜑 ∧ 𝑈 = 𝑉 ) → ( 𝐼 ‘ ( 𝑈 ∨ 𝑉 ) ) ⊆ ( 𝐼 ‘ 𝑈 ) ) |
27 |
5 8
|
dvalvec |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑌 ∈ LVec ) |
28 |
|
lveclmod |
⊢ ( 𝑌 ∈ LVec → 𝑌 ∈ LMod ) |
29 |
13 27 28
|
3syl |
⊢ ( 𝜑 → 𝑌 ∈ LMod ) |
30 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
31 |
30 4
|
atbase |
⊢ ( 𝑈 ∈ 𝐴 → 𝑈 ∈ ( Base ‘ 𝐾 ) ) |
32 |
19 31
|
syl |
⊢ ( 𝜑 → 𝑈 ∈ ( Base ‘ 𝐾 ) ) |
33 |
14
|
simprd |
⊢ ( 𝜑 → 𝑈 ≤ 𝑊 ) |
34 |
30 1 5 8 12 9
|
dialss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ ( Base ‘ 𝐾 ) ∧ 𝑈 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑈 ) ∈ 𝑆 ) |
35 |
13 32 33 34
|
syl12anc |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑈 ) ∈ 𝑆 ) |
36 |
9
|
lsssubg |
⊢ ( ( 𝑌 ∈ LMod ∧ ( 𝐼 ‘ 𝑈 ) ∈ 𝑆 ) → ( 𝐼 ‘ 𝑈 ) ∈ ( SubGrp ‘ 𝑌 ) ) |
37 |
29 35 36
|
syl2anc |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑈 ) ∈ ( SubGrp ‘ 𝑌 ) ) |
38 |
10
|
lsmidm |
⊢ ( ( 𝐼 ‘ 𝑈 ) ∈ ( SubGrp ‘ 𝑌 ) → ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑈 ) ) = ( 𝐼 ‘ 𝑈 ) ) |
39 |
37 38
|
syl |
⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑈 ) ) = ( 𝐼 ‘ 𝑈 ) ) |
40 |
|
fveq2 |
⊢ ( 𝑈 = 𝑉 → ( 𝐼 ‘ 𝑈 ) = ( 𝐼 ‘ 𝑉 ) ) |
41 |
40
|
oveq2d |
⊢ ( 𝑈 = 𝑉 → ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑈 ) ) = ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |
42 |
39 41
|
sylan9req |
⊢ ( ( 𝜑 ∧ 𝑈 = 𝑉 ) → ( 𝐼 ‘ 𝑈 ) = ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |
43 |
26 42
|
sseqtrd |
⊢ ( ( 𝜑 ∧ 𝑈 = 𝑉 ) → ( 𝐼 ‘ ( 𝑈 ∨ 𝑉 ) ) ⊆ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |
44 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑈 ≠ 𝑉 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
45 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑈 ≠ 𝑉 ) → ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) |
46 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑈 ≠ 𝑉 ) → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) |
47 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑈 ≠ 𝑉 ) → 𝑈 ≠ 𝑉 ) |
48 |
1 2 3 4 5 6 7 8 9 10 11 12 44 45 46 47
|
dia2dimlem12 |
⊢ ( ( 𝜑 ∧ 𝑈 ≠ 𝑉 ) → ( 𝐼 ‘ ( 𝑈 ∨ 𝑉 ) ) ⊆ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |
49 |
43 48
|
pm2.61dane |
⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑈 ∨ 𝑉 ) ) ⊆ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |