Step |
Hyp |
Ref |
Expression |
1 |
|
dihprrn.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
dihprrn.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
dihprrn.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
4 |
|
dihprrn.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
5 |
|
dihprrn.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
dihprrn.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
7 |
|
dihprrn.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
8 |
|
dihprrn.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) |
9 |
|
dihprrnlem1.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
10 |
|
dihprrnlem1.o |
⊢ 0 = ( 0g ‘ 𝑈 ) |
11 |
|
dihprrnlem1.nz |
⊢ ( 𝜑 → 𝑌 ≠ 0 ) |
12 |
|
dihprrnlem1.x |
⊢ ( 𝜑 → ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ≤ 𝑊 ) |
13 |
|
dihprrnlem1.y |
⊢ ( 𝜑 → ¬ ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ≤ 𝑊 ) |
14 |
|
df-pr |
⊢ { 𝑋 , 𝑌 } = ( { 𝑋 } ∪ { 𝑌 } ) |
15 |
14
|
fveq2i |
⊢ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) = ( 𝑁 ‘ ( { 𝑋 } ∪ { 𝑌 } ) ) |
16 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
17 |
|
eqid |
⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 ) |
18 |
|
eqid |
⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 ) |
19 |
|
eqid |
⊢ ( LSSum ‘ 𝑈 ) = ( LSSum ‘ 𝑈 ) |
20 |
1 2 3 4 5
|
dihlsprn |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ran 𝐼 ) |
21 |
6 7 20
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ∈ ran 𝐼 ) |
22 |
16 1 5
|
dihcnvcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ran 𝐼 ) → ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ∈ ( Base ‘ 𝐾 ) ) |
23 |
6 21 22
|
syl2anc |
⊢ ( 𝜑 → ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ∈ ( Base ‘ 𝐾 ) ) |
24 |
23 12
|
jca |
⊢ ( 𝜑 → ( ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ≤ 𝑊 ) ) |
25 |
18 1 2 3 10 4 5
|
dihlspsnat |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ) → ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( Atoms ‘ 𝐾 ) ) |
26 |
6 8 11 25
|
syl3anc |
⊢ ( 𝜑 → ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( Atoms ‘ 𝐾 ) ) |
27 |
26 13
|
jca |
⊢ ( 𝜑 → ( ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ≤ 𝑊 ) ) |
28 |
16 9 1 17 18 2 19 5 6 24 27
|
dihjatc |
⊢ ( 𝜑 → ( 𝐼 ‘ ( ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( join ‘ 𝐾 ) ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ) = ( ( 𝐼 ‘ ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ) ( LSSum ‘ 𝑈 ) ( 𝐼 ‘ ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ) ) |
29 |
1 5
|
dihcnvid2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ran 𝐼 ) → ( 𝐼 ‘ ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ) = ( 𝑁 ‘ { 𝑋 } ) ) |
30 |
6 21 29
|
syl2anc |
⊢ ( 𝜑 → ( 𝐼 ‘ ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ) = ( 𝑁 ‘ { 𝑋 } ) ) |
31 |
1 2 3 4 5
|
dihlsprn |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ran 𝐼 ) |
32 |
6 8 31
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ran 𝐼 ) |
33 |
1 5
|
dihcnvid2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑁 ‘ { 𝑌 } ) ∈ ran 𝐼 ) → ( 𝐼 ‘ ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) = ( 𝑁 ‘ { 𝑌 } ) ) |
34 |
6 32 33
|
syl2anc |
⊢ ( 𝜑 → ( 𝐼 ‘ ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) = ( 𝑁 ‘ { 𝑌 } ) ) |
35 |
30 34
|
oveq12d |
⊢ ( 𝜑 → ( ( 𝐼 ‘ ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ) ( LSSum ‘ 𝑈 ) ( 𝐼 ‘ ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ) = ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑌 } ) ) ) |
36 |
1 2 6
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
37 |
7
|
snssd |
⊢ ( 𝜑 → { 𝑋 } ⊆ 𝑉 ) |
38 |
8
|
snssd |
⊢ ( 𝜑 → { 𝑌 } ⊆ 𝑉 ) |
39 |
3 4 19
|
lsmsp2 |
⊢ ( ( 𝑈 ∈ LMod ∧ { 𝑋 } ⊆ 𝑉 ∧ { 𝑌 } ⊆ 𝑉 ) → ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝑁 ‘ ( { 𝑋 } ∪ { 𝑌 } ) ) ) |
40 |
36 37 38 39
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝑁 ‘ ( { 𝑋 } ∪ { 𝑌 } ) ) ) |
41 |
28 35 40
|
3eqtrrd |
⊢ ( 𝜑 → ( 𝑁 ‘ ( { 𝑋 } ∪ { 𝑌 } ) ) = ( 𝐼 ‘ ( ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( join ‘ 𝐾 ) ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ) ) |
42 |
15 41
|
syl5eq |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 , 𝑌 } ) = ( 𝐼 ‘ ( ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( join ‘ 𝐾 ) ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ) ) |
43 |
6
|
simpld |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
44 |
43
|
hllatd |
⊢ ( 𝜑 → 𝐾 ∈ Lat ) |
45 |
16 1 5
|
dihcnvcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑁 ‘ { 𝑌 } ) ∈ ran 𝐼 ) → ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( Base ‘ 𝐾 ) ) |
46 |
6 32 45
|
syl2anc |
⊢ ( 𝜑 → ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( Base ‘ 𝐾 ) ) |
47 |
16 17
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( Base ‘ 𝐾 ) ) → ( ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( join ‘ 𝐾 ) ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∈ ( Base ‘ 𝐾 ) ) |
48 |
44 23 46 47
|
syl3anc |
⊢ ( 𝜑 → ( ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( join ‘ 𝐾 ) ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∈ ( Base ‘ 𝐾 ) ) |
49 |
16 1 5
|
dihcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( join ‘ 𝐾 ) ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝐼 ‘ ( ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( join ‘ 𝐾 ) ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ) ∈ ran 𝐼 ) |
50 |
6 48 49
|
syl2anc |
⊢ ( 𝜑 → ( 𝐼 ‘ ( ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( join ‘ 𝐾 ) ( ◡ 𝐼 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ) ∈ ran 𝐼 ) |
51 |
42 50
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∈ ran 𝐼 ) |