Step |
Hyp |
Ref |
Expression |
1 |
|
lspprvacl.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lspprvacl.p |
⊢ + = ( +g ‘ 𝑊 ) |
3 |
|
lspprvacl.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
4 |
|
lspprvacl.w |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
5 |
|
lspprvacl.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
6 |
|
lspprvacl.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) |
7 |
|
eqid |
⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 ) |
8 |
1 7 3 4 5 6
|
lspprcl |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
9 |
1 3 4 5 6
|
lspprid1 |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) |
10 |
1 3 4 5 6
|
lspprid2 |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) |
11 |
2 7
|
lssvacl |
⊢ ( ( ( 𝑊 ∈ LMod ∧ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∈ ( LSubSp ‘ 𝑊 ) ) ∧ ( 𝑋 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ) → ( 𝑋 + 𝑌 ) ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) |
12 |
4 8 9 10 11
|
syl22anc |
⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) |