Step |
Hyp |
Ref |
Expression |
1 |
|
lspsnel5.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lspsnel5.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑊 ) |
3 |
|
lspsnel5.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
4 |
|
lspsnel5.w |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
5 |
|
lspsnel5.a |
⊢ ( 𝜑 → 𝑈 ∈ 𝑆 ) |
6 |
1 2
|
lssel |
⊢ ( ( 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈 ) → 𝑋 ∈ 𝑉 ) |
7 |
5 6
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑈 ) → 𝑋 ∈ 𝑉 ) |
8 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑈 ) → 𝑊 ∈ LMod ) |
9 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑈 ) → 𝑈 ∈ 𝑆 ) |
10 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑈 ) → 𝑋 ∈ 𝑈 ) |
11 |
2 3
|
lspsnss |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈 ) → ( 𝑁 ‘ { 𝑋 } ) ⊆ 𝑈 ) |
12 |
8 9 10 11
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑈 ) → ( 𝑁 ‘ { 𝑋 } ) ⊆ 𝑈 ) |
13 |
7 12
|
jca |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑈 ) → ( 𝑋 ∈ 𝑉 ∧ ( 𝑁 ‘ { 𝑋 } ) ⊆ 𝑈 ) ) |
14 |
1 3
|
lspsnid |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
15 |
4 14
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
16 |
|
ssel |
⊢ ( ( 𝑁 ‘ { 𝑋 } ) ⊆ 𝑈 → ( 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) → 𝑋 ∈ 𝑈 ) ) |
17 |
15 16
|
syl5com |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑉 ) → ( ( 𝑁 ‘ { 𝑋 } ) ⊆ 𝑈 → 𝑋 ∈ 𝑈 ) ) |
18 |
17
|
impr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑉 ∧ ( 𝑁 ‘ { 𝑋 } ) ⊆ 𝑈 ) ) → 𝑋 ∈ 𝑈 ) |
19 |
13 18
|
impbida |
⊢ ( 𝜑 → ( 𝑋 ∈ 𝑈 ↔ ( 𝑋 ∈ 𝑉 ∧ ( 𝑁 ‘ { 𝑋 } ) ⊆ 𝑈 ) ) ) |