Step |
Hyp |
Ref |
Expression |
1 |
|
lcdlkreq.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
lcdlkreq.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
lcdlkreq.l |
⊢ 𝐿 = ( LKer ‘ 𝑈 ) |
4 |
|
lcdlkreq.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
lcdlkreq.o |
⊢ 0 = ( 0g ‘ 𝐶 ) |
6 |
|
lcdlkreq.n |
⊢ 𝑁 = ( LSpan ‘ 𝐶 ) |
7 |
|
lcdlkreq.v |
⊢ 𝑉 = ( Base ‘ 𝐶 ) |
8 |
|
lcdlkreq.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
9 |
|
lcdlkreq.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑉 ) |
10 |
|
lcdlkreq.g |
⊢ ( 𝜑 → 𝐺 ∈ ( 𝑁 ‘ { 𝐼 } ) ) |
11 |
|
lcdlkreq.z |
⊢ ( 𝜑 → 𝐺 ≠ 0 ) |
12 |
|
eqid |
⊢ ( LFnl ‘ 𝑈 ) = ( LFnl ‘ 𝑈 ) |
13 |
|
eqid |
⊢ ( LDual ‘ 𝑈 ) = ( LDual ‘ 𝑈 ) |
14 |
|
eqid |
⊢ ( 0g ‘ ( LDual ‘ 𝑈 ) ) = ( 0g ‘ ( LDual ‘ 𝑈 ) ) |
15 |
|
eqid |
⊢ ( LSpan ‘ ( LDual ‘ 𝑈 ) ) = ( LSpan ‘ ( LDual ‘ 𝑈 ) ) |
16 |
1 2 8
|
dvhlvec |
⊢ ( 𝜑 → 𝑈 ∈ LVec ) |
17 |
1 4 7 2 12 8 9
|
lcdvbaselfl |
⊢ ( 𝜑 → 𝐼 ∈ ( LFnl ‘ 𝑈 ) ) |
18 |
9
|
snssd |
⊢ ( 𝜑 → { 𝐼 } ⊆ 𝑉 ) |
19 |
1 2 13 15 4 7 6 8 18
|
lcdlsp |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝐼 } ) = ( ( LSpan ‘ ( LDual ‘ 𝑈 ) ) ‘ { 𝐼 } ) ) |
20 |
10 19
|
eleqtrd |
⊢ ( 𝜑 → 𝐺 ∈ ( ( LSpan ‘ ( LDual ‘ 𝑈 ) ) ‘ { 𝐼 } ) ) |
21 |
1 2 13 14 4 5 8
|
lcd0v2 |
⊢ ( 𝜑 → 0 = ( 0g ‘ ( LDual ‘ 𝑈 ) ) ) |
22 |
11 21
|
neeqtrd |
⊢ ( 𝜑 → 𝐺 ≠ ( 0g ‘ ( LDual ‘ 𝑈 ) ) ) |
23 |
|
eldifsn |
⊢ ( 𝐺 ∈ ( ( ( LSpan ‘ ( LDual ‘ 𝑈 ) ) ‘ { 𝐼 } ) ∖ { ( 0g ‘ ( LDual ‘ 𝑈 ) ) } ) ↔ ( 𝐺 ∈ ( ( LSpan ‘ ( LDual ‘ 𝑈 ) ) ‘ { 𝐼 } ) ∧ 𝐺 ≠ ( 0g ‘ ( LDual ‘ 𝑈 ) ) ) ) |
24 |
20 22 23
|
sylanbrc |
⊢ ( 𝜑 → 𝐺 ∈ ( ( ( LSpan ‘ ( LDual ‘ 𝑈 ) ) ‘ { 𝐼 } ) ∖ { ( 0g ‘ ( LDual ‘ 𝑈 ) ) } ) ) |
25 |
12 3 13 14 15 16 17 24
|
lkrlspeqN |
⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) = ( 𝐿 ‘ 𝐼 ) ) |