Step |
Hyp |
Ref |
Expression |
1 |
|
lspun0.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lspun0.o |
⊢ 0 = ( 0g ‘ 𝑊 ) |
3 |
|
lspun0.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
4 |
|
lspun0.w |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
5 |
|
lspun0.x |
⊢ ( 𝜑 → 𝑋 ⊆ 𝑉 ) |
6 |
1 2
|
lmod0vcl |
⊢ ( 𝑊 ∈ LMod → 0 ∈ 𝑉 ) |
7 |
4 6
|
syl |
⊢ ( 𝜑 → 0 ∈ 𝑉 ) |
8 |
7
|
snssd |
⊢ ( 𝜑 → { 0 } ⊆ 𝑉 ) |
9 |
1 3
|
lspun |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ⊆ 𝑉 ∧ { 0 } ⊆ 𝑉 ) → ( 𝑁 ‘ ( 𝑋 ∪ { 0 } ) ) = ( 𝑁 ‘ ( ( 𝑁 ‘ 𝑋 ) ∪ ( 𝑁 ‘ { 0 } ) ) ) ) |
10 |
4 5 8 9
|
syl3anc |
⊢ ( 𝜑 → ( 𝑁 ‘ ( 𝑋 ∪ { 0 } ) ) = ( 𝑁 ‘ ( ( 𝑁 ‘ 𝑋 ) ∪ ( 𝑁 ‘ { 0 } ) ) ) ) |
11 |
2 3
|
lspsn0 |
⊢ ( 𝑊 ∈ LMod → ( 𝑁 ‘ { 0 } ) = { 0 } ) |
12 |
4 11
|
syl |
⊢ ( 𝜑 → ( 𝑁 ‘ { 0 } ) = { 0 } ) |
13 |
12
|
uneq2d |
⊢ ( 𝜑 → ( ( 𝑁 ‘ 𝑋 ) ∪ ( 𝑁 ‘ { 0 } ) ) = ( ( 𝑁 ‘ 𝑋 ) ∪ { 0 } ) ) |
14 |
|
eqid |
⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 ) |
15 |
1 14 3
|
lspcl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ⊆ 𝑉 ) → ( 𝑁 ‘ 𝑋 ) ∈ ( LSubSp ‘ 𝑊 ) ) |
16 |
4 5 15
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ 𝑋 ) ∈ ( LSubSp ‘ 𝑊 ) ) |
17 |
2 14
|
lss0ss |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑁 ‘ 𝑋 ) ∈ ( LSubSp ‘ 𝑊 ) ) → { 0 } ⊆ ( 𝑁 ‘ 𝑋 ) ) |
18 |
4 16 17
|
syl2anc |
⊢ ( 𝜑 → { 0 } ⊆ ( 𝑁 ‘ 𝑋 ) ) |
19 |
|
ssequn2 |
⊢ ( { 0 } ⊆ ( 𝑁 ‘ 𝑋 ) ↔ ( ( 𝑁 ‘ 𝑋 ) ∪ { 0 } ) = ( 𝑁 ‘ 𝑋 ) ) |
20 |
18 19
|
sylib |
⊢ ( 𝜑 → ( ( 𝑁 ‘ 𝑋 ) ∪ { 0 } ) = ( 𝑁 ‘ 𝑋 ) ) |
21 |
13 20
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝑁 ‘ 𝑋 ) ∪ ( 𝑁 ‘ { 0 } ) ) = ( 𝑁 ‘ 𝑋 ) ) |
22 |
21
|
fveq2d |
⊢ ( 𝜑 → ( 𝑁 ‘ ( ( 𝑁 ‘ 𝑋 ) ∪ ( 𝑁 ‘ { 0 } ) ) ) = ( 𝑁 ‘ ( 𝑁 ‘ 𝑋 ) ) ) |
23 |
1 3
|
lspidm |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ⊆ 𝑉 ) → ( 𝑁 ‘ ( 𝑁 ‘ 𝑋 ) ) = ( 𝑁 ‘ 𝑋 ) ) |
24 |
4 5 23
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ ( 𝑁 ‘ 𝑋 ) ) = ( 𝑁 ‘ 𝑋 ) ) |
25 |
22 24
|
eqtrd |
⊢ ( 𝜑 → ( 𝑁 ‘ ( ( 𝑁 ‘ 𝑋 ) ∪ ( 𝑁 ‘ { 0 } ) ) ) = ( 𝑁 ‘ 𝑋 ) ) |
26 |
10 25
|
eqtrd |
⊢ ( 𝜑 → ( 𝑁 ‘ ( 𝑋 ∪ { 0 } ) ) = ( 𝑁 ‘ 𝑋 ) ) |