Step |
Hyp |
Ref |
Expression |
1 |
|
lspexchn1.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lspexchn1.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
3 |
|
lspexchn1.w |
⊢ ( 𝜑 → 𝑊 ∈ LVec ) |
4 |
|
lspexchn1.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
5 |
|
lspexchn1.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) |
6 |
|
lspexchn1.z |
⊢ ( 𝜑 → 𝑍 ∈ 𝑉 ) |
7 |
|
lspexchn1.q |
⊢ ( 𝜑 → ¬ 𝑌 ∈ ( 𝑁 ‘ { 𝑍 } ) ) |
8 |
|
lspexchn1.e |
⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) |
9 |
|
eqid |
⊢ ( 0g ‘ 𝑊 ) = ( 0g ‘ 𝑊 ) |
10 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) ) → 𝑊 ∈ LVec ) |
11 |
|
eqid |
⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 ) |
12 |
|
lveclmod |
⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod ) |
13 |
3 12
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
14 |
1 11 2
|
lspsncl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑍 } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
15 |
13 6 14
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑍 } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
16 |
9 11 13 15 5 7
|
lssneln0 |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { ( 0g ‘ 𝑊 ) } ) ) |
17 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) ) → 𝑌 ∈ ( 𝑉 ∖ { ( 0g ‘ 𝑊 ) } ) ) |
18 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) ) → 𝑋 ∈ 𝑉 ) |
19 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) ) → 𝑍 ∈ 𝑉 ) |
20 |
1 2 13 5 6 7
|
lspsnne2 |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) |
21 |
20
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) ) → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) |
22 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) ) → 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) ) |
23 |
1 9 2 10 17 18 19 21 22
|
lspexch |
⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) ) → 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) |
24 |
8 23
|
mtand |
⊢ ( 𝜑 → ¬ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) ) |