Step |
Hyp |
Ref |
Expression |
1 |
|
lkrlsp3.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lkrlsp3.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
3 |
|
lkrlsp3.f |
⊢ 𝐹 = ( LFnl ‘ 𝑊 ) |
4 |
|
lkrlsp3.k |
⊢ 𝐾 = ( LKer ‘ 𝑊 ) |
5 |
|
lveclmod |
⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod ) |
6 |
5
|
3ad2ant1 |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ¬ 𝑋 ∈ ( 𝐾 ‘ 𝐺 ) ) → 𝑊 ∈ LMod ) |
7 |
|
simp2r |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ¬ 𝑋 ∈ ( 𝐾 ‘ 𝐺 ) ) → 𝐺 ∈ 𝐹 ) |
8 |
|
eqid |
⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 ) |
9 |
3 4 8
|
lkrlss |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ) → ( 𝐾 ‘ 𝐺 ) ∈ ( LSubSp ‘ 𝑊 ) ) |
10 |
6 7 9
|
syl2anc |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ¬ 𝑋 ∈ ( 𝐾 ‘ 𝐺 ) ) → ( 𝐾 ‘ 𝐺 ) ∈ ( LSubSp ‘ 𝑊 ) ) |
11 |
8 2
|
lspid |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝐾 ‘ 𝐺 ) ∈ ( LSubSp ‘ 𝑊 ) ) → ( 𝑁 ‘ ( 𝐾 ‘ 𝐺 ) ) = ( 𝐾 ‘ 𝐺 ) ) |
12 |
6 10 11
|
syl2anc |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ¬ 𝑋 ∈ ( 𝐾 ‘ 𝐺 ) ) → ( 𝑁 ‘ ( 𝐾 ‘ 𝐺 ) ) = ( 𝐾 ‘ 𝐺 ) ) |
13 |
12
|
uneq1d |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ¬ 𝑋 ∈ ( 𝐾 ‘ 𝐺 ) ) → ( ( 𝑁 ‘ ( 𝐾 ‘ 𝐺 ) ) ∪ ( 𝑁 ‘ { 𝑋 } ) ) = ( ( 𝐾 ‘ 𝐺 ) ∪ ( 𝑁 ‘ { 𝑋 } ) ) ) |
14 |
13
|
fveq2d |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ¬ 𝑋 ∈ ( 𝐾 ‘ 𝐺 ) ) → ( 𝑁 ‘ ( ( 𝑁 ‘ ( 𝐾 ‘ 𝐺 ) ) ∪ ( 𝑁 ‘ { 𝑋 } ) ) ) = ( 𝑁 ‘ ( ( 𝐾 ‘ 𝐺 ) ∪ ( 𝑁 ‘ { 𝑋 } ) ) ) ) |
15 |
1 3 4 6 7
|
lkrssv |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ¬ 𝑋 ∈ ( 𝐾 ‘ 𝐺 ) ) → ( 𝐾 ‘ 𝐺 ) ⊆ 𝑉 ) |
16 |
|
simp2l |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ¬ 𝑋 ∈ ( 𝐾 ‘ 𝐺 ) ) → 𝑋 ∈ 𝑉 ) |
17 |
16
|
snssd |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ¬ 𝑋 ∈ ( 𝐾 ‘ 𝐺 ) ) → { 𝑋 } ⊆ 𝑉 ) |
18 |
1 2
|
lspun |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝐾 ‘ 𝐺 ) ⊆ 𝑉 ∧ { 𝑋 } ⊆ 𝑉 ) → ( 𝑁 ‘ ( ( 𝐾 ‘ 𝐺 ) ∪ { 𝑋 } ) ) = ( 𝑁 ‘ ( ( 𝑁 ‘ ( 𝐾 ‘ 𝐺 ) ) ∪ ( 𝑁 ‘ { 𝑋 } ) ) ) ) |
19 |
6 15 17 18
|
syl3anc |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ¬ 𝑋 ∈ ( 𝐾 ‘ 𝐺 ) ) → ( 𝑁 ‘ ( ( 𝐾 ‘ 𝐺 ) ∪ { 𝑋 } ) ) = ( 𝑁 ‘ ( ( 𝑁 ‘ ( 𝐾 ‘ 𝐺 ) ) ∪ ( 𝑁 ‘ { 𝑋 } ) ) ) ) |
20 |
1 8 2
|
lspsncl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
21 |
6 16 20
|
syl2anc |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ¬ 𝑋 ∈ ( 𝐾 ‘ 𝐺 ) ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
22 |
|
eqid |
⊢ ( LSSum ‘ 𝑊 ) = ( LSSum ‘ 𝑊 ) |
23 |
8 2 22
|
lsmsp |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝐾 ‘ 𝐺 ) ∈ ( LSubSp ‘ 𝑊 ) ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) ) → ( ( 𝐾 ‘ 𝐺 ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝑁 ‘ ( ( 𝐾 ‘ 𝐺 ) ∪ ( 𝑁 ‘ { 𝑋 } ) ) ) ) |
24 |
6 10 21 23
|
syl3anc |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ¬ 𝑋 ∈ ( 𝐾 ‘ 𝐺 ) ) → ( ( 𝐾 ‘ 𝐺 ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝑁 ‘ ( ( 𝐾 ‘ 𝐺 ) ∪ ( 𝑁 ‘ { 𝑋 } ) ) ) ) |
25 |
14 19 24
|
3eqtr4d |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ¬ 𝑋 ∈ ( 𝐾 ‘ 𝐺 ) ) → ( 𝑁 ‘ ( ( 𝐾 ‘ 𝐺 ) ∪ { 𝑋 } ) ) = ( ( 𝐾 ‘ 𝐺 ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑋 } ) ) ) |
26 |
1 2 22 3 4
|
lkrlsp2 |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ¬ 𝑋 ∈ ( 𝐾 ‘ 𝐺 ) ) → ( ( 𝐾 ‘ 𝐺 ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) |
27 |
25 26
|
eqtrd |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ¬ 𝑋 ∈ ( 𝐾 ‘ 𝐺 ) ) → ( 𝑁 ‘ ( ( 𝐾 ‘ 𝐺 ) ∪ { 𝑋 } ) ) = 𝑉 ) |