Step |
Hyp |
Ref |
Expression |
1 |
|
lsatel.o |
⊢ 0 = ( 0g ‘ 𝑊 ) |
2 |
|
lsatel.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
3 |
|
lsatel.a |
⊢ 𝐴 = ( LSAtoms ‘ 𝑊 ) |
4 |
|
lsatel.w |
⊢ ( 𝜑 → 𝑊 ∈ LVec ) |
5 |
|
lsatel.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝐴 ) |
6 |
|
lsatel.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑈 ) |
7 |
|
lsatel.e |
⊢ ( 𝜑 → 𝑋 ≠ 0 ) |
8 |
|
eqid |
⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 ) |
9 |
|
lveclmod |
⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod ) |
10 |
4 9
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
11 |
8 3 10 5
|
lsatlssel |
⊢ ( 𝜑 → 𝑈 ∈ ( LSubSp ‘ 𝑊 ) ) |
12 |
8 2 10 11 6
|
lspsnel5a |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ⊆ 𝑈 ) |
13 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
14 |
13 8
|
lssel |
⊢ ( ( 𝑈 ∈ ( LSubSp ‘ 𝑊 ) ∧ 𝑋 ∈ 𝑈 ) → 𝑋 ∈ ( Base ‘ 𝑊 ) ) |
15 |
11 6 14
|
syl2anc |
⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝑊 ) ) |
16 |
13 2 1 3
|
lsatlspsn2 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ∧ 𝑋 ≠ 0 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ 𝐴 ) |
17 |
10 15 7 16
|
syl3anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ∈ 𝐴 ) |
18 |
3 4 17 5
|
lsatcmp |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ⊆ 𝑈 ↔ ( 𝑁 ‘ { 𝑋 } ) = 𝑈 ) ) |
19 |
12 18
|
mpbid |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) = 𝑈 ) |
20 |
19
|
eqcomd |
⊢ ( 𝜑 → 𝑈 = ( 𝑁 ‘ { 𝑋 } ) ) |