Step |
Hyp |
Ref |
Expression |
1 |
|
lspsneq0.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lspsneq0.z |
⊢ 0 = ( 0g ‘ 𝑊 ) |
3 |
|
lspsneq0.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
4 |
1 3
|
lspsnid |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
5 |
|
eleq2 |
⊢ ( ( 𝑁 ‘ { 𝑋 } ) = { 0 } → ( 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) ↔ 𝑋 ∈ { 0 } ) ) |
6 |
4 5
|
syl5ibcom |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( ( 𝑁 ‘ { 𝑋 } ) = { 0 } → 𝑋 ∈ { 0 } ) ) |
7 |
|
elsni |
⊢ ( 𝑋 ∈ { 0 } → 𝑋 = 0 ) |
8 |
6 7
|
syl6 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( ( 𝑁 ‘ { 𝑋 } ) = { 0 } → 𝑋 = 0 ) ) |
9 |
2 3
|
lspsn0 |
⊢ ( 𝑊 ∈ LMod → ( 𝑁 ‘ { 0 } ) = { 0 } ) |
10 |
9
|
adantr |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 0 } ) = { 0 } ) |
11 |
|
sneq |
⊢ ( 𝑋 = 0 → { 𝑋 } = { 0 } ) |
12 |
11
|
fveqeq2d |
⊢ ( 𝑋 = 0 → ( ( 𝑁 ‘ { 𝑋 } ) = { 0 } ↔ ( 𝑁 ‘ { 0 } ) = { 0 } ) ) |
13 |
10 12
|
syl5ibrcom |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑋 = 0 → ( 𝑁 ‘ { 𝑋 } ) = { 0 } ) ) |
14 |
8 13
|
impbid |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( ( 𝑁 ‘ { 𝑋 } ) = { 0 } ↔ 𝑋 = 0 ) ) |