Step |
Hyp |
Ref |
Expression |
1 |
|
lbsacsbs.1 |
⊢ 𝐴 = ( LSubSp ‘ 𝑊 ) |
2 |
|
lbsacsbs.2 |
⊢ 𝑁 = ( mrCls ‘ 𝐴 ) |
3 |
|
lbsacsbs.3 |
⊢ 𝑋 = ( Base ‘ 𝑊 ) |
4 |
|
lbsacsbs.4 |
⊢ 𝐼 = ( mrInd ‘ 𝐴 ) |
5 |
|
lbsacsbs.5 |
⊢ 𝐽 = ( LBasis ‘ 𝑊 ) |
6 |
|
eqid |
⊢ ( LSpan ‘ 𝑊 ) = ( LSpan ‘ 𝑊 ) |
7 |
3 5 6
|
islbs2 |
⊢ ( 𝑊 ∈ LVec → ( 𝑆 ∈ 𝐽 ↔ ( 𝑆 ⊆ 𝑋 ∧ ( ( LSpan ‘ 𝑊 ) ‘ 𝑆 ) = 𝑋 ∧ ∀ 𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝑆 ∖ { 𝑥 } ) ) ) ) ) |
8 |
|
lveclmod |
⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod ) |
9 |
1 6 2
|
mrclsp |
⊢ ( 𝑊 ∈ LMod → ( LSpan ‘ 𝑊 ) = 𝑁 ) |
10 |
8 9
|
syl |
⊢ ( 𝑊 ∈ LVec → ( LSpan ‘ 𝑊 ) = 𝑁 ) |
11 |
10
|
fveq1d |
⊢ ( 𝑊 ∈ LVec → ( ( LSpan ‘ 𝑊 ) ‘ 𝑆 ) = ( 𝑁 ‘ 𝑆 ) ) |
12 |
11
|
eqeq1d |
⊢ ( 𝑊 ∈ LVec → ( ( ( LSpan ‘ 𝑊 ) ‘ 𝑆 ) = 𝑋 ↔ ( 𝑁 ‘ 𝑆 ) = 𝑋 ) ) |
13 |
10
|
fveq1d |
⊢ ( 𝑊 ∈ LVec → ( ( LSpan ‘ 𝑊 ) ‘ ( 𝑆 ∖ { 𝑥 } ) ) = ( 𝑁 ‘ ( 𝑆 ∖ { 𝑥 } ) ) ) |
14 |
13
|
eleq2d |
⊢ ( 𝑊 ∈ LVec → ( 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝑆 ∖ { 𝑥 } ) ) ↔ 𝑥 ∈ ( 𝑁 ‘ ( 𝑆 ∖ { 𝑥 } ) ) ) ) |
15 |
14
|
notbid |
⊢ ( 𝑊 ∈ LVec → ( ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝑆 ∖ { 𝑥 } ) ) ↔ ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝑆 ∖ { 𝑥 } ) ) ) ) |
16 |
15
|
ralbidv |
⊢ ( 𝑊 ∈ LVec → ( ∀ 𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝑆 ∖ { 𝑥 } ) ) ↔ ∀ 𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝑆 ∖ { 𝑥 } ) ) ) ) |
17 |
12 16
|
3anbi23d |
⊢ ( 𝑊 ∈ LVec → ( ( 𝑆 ⊆ 𝑋 ∧ ( ( LSpan ‘ 𝑊 ) ‘ 𝑆 ) = 𝑋 ∧ ∀ 𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝑆 ∖ { 𝑥 } ) ) ) ↔ ( 𝑆 ⊆ 𝑋 ∧ ( 𝑁 ‘ 𝑆 ) = 𝑋 ∧ ∀ 𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝑆 ∖ { 𝑥 } ) ) ) ) ) |
18 |
|
3anan32 |
⊢ ( ( 𝑆 ⊆ 𝑋 ∧ ( 𝑁 ‘ 𝑆 ) = 𝑋 ∧ ∀ 𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝑆 ∖ { 𝑥 } ) ) ) ↔ ( ( 𝑆 ⊆ 𝑋 ∧ ∀ 𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝑆 ∖ { 𝑥 } ) ) ) ∧ ( 𝑁 ‘ 𝑆 ) = 𝑋 ) ) |
19 |
3 1
|
lssmre |
⊢ ( 𝑊 ∈ LMod → 𝐴 ∈ ( Moore ‘ 𝑋 ) ) |
20 |
2 4
|
ismri |
⊢ ( 𝐴 ∈ ( Moore ‘ 𝑋 ) → ( 𝑆 ∈ 𝐼 ↔ ( 𝑆 ⊆ 𝑋 ∧ ∀ 𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝑆 ∖ { 𝑥 } ) ) ) ) ) |
21 |
8 19 20
|
3syl |
⊢ ( 𝑊 ∈ LVec → ( 𝑆 ∈ 𝐼 ↔ ( 𝑆 ⊆ 𝑋 ∧ ∀ 𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝑆 ∖ { 𝑥 } ) ) ) ) ) |
22 |
21
|
anbi1d |
⊢ ( 𝑊 ∈ LVec → ( ( 𝑆 ∈ 𝐼 ∧ ( 𝑁 ‘ 𝑆 ) = 𝑋 ) ↔ ( ( 𝑆 ⊆ 𝑋 ∧ ∀ 𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝑆 ∖ { 𝑥 } ) ) ) ∧ ( 𝑁 ‘ 𝑆 ) = 𝑋 ) ) ) |
23 |
18 22
|
bitr4id |
⊢ ( 𝑊 ∈ LVec → ( ( 𝑆 ⊆ 𝑋 ∧ ( 𝑁 ‘ 𝑆 ) = 𝑋 ∧ ∀ 𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝑆 ∖ { 𝑥 } ) ) ) ↔ ( 𝑆 ∈ 𝐼 ∧ ( 𝑁 ‘ 𝑆 ) = 𝑋 ) ) ) |
24 |
7 17 23
|
3bitrd |
⊢ ( 𝑊 ∈ LVec → ( 𝑆 ∈ 𝐽 ↔ ( 𝑆 ∈ 𝐼 ∧ ( 𝑁 ‘ 𝑆 ) = 𝑋 ) ) ) |