Step |
Hyp |
Ref |
Expression |
1 |
|
lspdisj2.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lspdisj2.o |
⊢ 0 = ( 0g ‘ 𝑊 ) |
3 |
|
lspdisj2.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
4 |
|
lspdisj2.w |
⊢ ( 𝜑 → 𝑊 ∈ LVec ) |
5 |
|
lspdisj2.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
6 |
|
lspdisj2.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) |
7 |
|
lspdisj2.q |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) |
8 |
|
sneq |
⊢ ( 𝑋 = 0 → { 𝑋 } = { 0 } ) |
9 |
8
|
fveq2d |
⊢ ( 𝑋 = 0 → ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 0 } ) ) |
10 |
|
lveclmod |
⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod ) |
11 |
4 10
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
12 |
2 3
|
lspsn0 |
⊢ ( 𝑊 ∈ LMod → ( 𝑁 ‘ { 0 } ) = { 0 } ) |
13 |
11 12
|
syl |
⊢ ( 𝜑 → ( 𝑁 ‘ { 0 } ) = { 0 } ) |
14 |
9 13
|
sylan9eqr |
⊢ ( ( 𝜑 ∧ 𝑋 = 0 ) → ( 𝑁 ‘ { 𝑋 } ) = { 0 } ) |
15 |
14
|
ineq1d |
⊢ ( ( 𝜑 ∧ 𝑋 = 0 ) → ( ( 𝑁 ‘ { 𝑋 } ) ∩ ( 𝑁 ‘ { 𝑌 } ) ) = ( { 0 } ∩ ( 𝑁 ‘ { 𝑌 } ) ) ) |
16 |
|
eqid |
⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 ) |
17 |
1 16 3
|
lspsncl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
18 |
11 6 17
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
19 |
2 16
|
lss0ss |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑊 ) ) → { 0 } ⊆ ( 𝑁 ‘ { 𝑌 } ) ) |
20 |
11 18 19
|
syl2anc |
⊢ ( 𝜑 → { 0 } ⊆ ( 𝑁 ‘ { 𝑌 } ) ) |
21 |
|
df-ss |
⊢ ( { 0 } ⊆ ( 𝑁 ‘ { 𝑌 } ) ↔ ( { 0 } ∩ ( 𝑁 ‘ { 𝑌 } ) ) = { 0 } ) |
22 |
20 21
|
sylib |
⊢ ( 𝜑 → ( { 0 } ∩ ( 𝑁 ‘ { 𝑌 } ) ) = { 0 } ) |
23 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = 0 ) → ( { 0 } ∩ ( 𝑁 ‘ { 𝑌 } ) ) = { 0 } ) |
24 |
15 23
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑋 = 0 ) → ( ( 𝑁 ‘ { 𝑋 } ) ∩ ( 𝑁 ‘ { 𝑌 } ) ) = { 0 } ) |
25 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 0 ) → 𝑊 ∈ LVec ) |
26 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 0 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
27 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝑉 ) |
28 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 0 ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) |
29 |
25
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ≠ 0 ) ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) ) → 𝑊 ∈ LVec ) |
30 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 0 ) → 𝑌 ∈ 𝑉 ) |
31 |
30
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ≠ 0 ) ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) ) → 𝑌 ∈ 𝑉 ) |
32 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ≠ 0 ) ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) ) → 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) ) |
33 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ≠ 0 ) ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) ) → 𝑋 ≠ 0 ) |
34 |
1 2 3 29 31 32 33
|
lspsneleq |
⊢ ( ( ( 𝜑 ∧ 𝑋 ≠ 0 ) ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) ) → ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑌 } ) ) |
35 |
34
|
ex |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 0 ) → ( 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) → ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑌 } ) ) ) |
36 |
35
|
necon3ad |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 0 ) → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) ) ) |
37 |
28 36
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 0 ) → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) ) |
38 |
1 2 3 16 25 26 27 37
|
lspdisj |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 0 ) → ( ( 𝑁 ‘ { 𝑋 } ) ∩ ( 𝑁 ‘ { 𝑌 } ) ) = { 0 } ) |
39 |
24 38
|
pm2.61dane |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ∩ ( 𝑁 ‘ { 𝑌 } ) ) = { 0 } ) |