Step |
Hyp |
Ref |
Expression |
1 |
|
mapdindp1.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
mapdindp1.p |
⊢ + = ( +g ‘ 𝑊 ) |
3 |
|
mapdindp1.o |
⊢ 0 = ( 0g ‘ 𝑊 ) |
4 |
|
mapdindp1.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
5 |
|
mapdindp1.w |
⊢ ( 𝜑 → 𝑊 ∈ LVec ) |
6 |
|
mapdindp1.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) |
7 |
|
mapdindp1.y |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) ) |
8 |
|
mapdindp1.z |
⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) ) |
9 |
|
mapdindp1.W |
⊢ ( 𝜑 → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) ) |
10 |
|
mapdindp1.e |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) ) |
11 |
|
mapdindp1.ne |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) |
12 |
|
mapdindp1.f |
⊢ ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) |
13 |
|
lveclmod |
⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod ) |
14 |
5 13
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
15 |
9
|
eldifad |
⊢ ( 𝜑 → 𝑤 ∈ 𝑉 ) |
16 |
7
|
eldifad |
⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) |
17 |
1 2 4
|
lspvadd |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑤 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) } ) ⊆ ( 𝑁 ‘ { 𝑤 , 𝑌 } ) ) |
18 |
14 15 16 17
|
syl3anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) } ) ⊆ ( 𝑁 ‘ { 𝑤 , 𝑌 } ) ) |
19 |
1 3 4 5 6 16 15 11 12
|
lspindp1 |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑤 , 𝑌 } ) ) ) |
20 |
19
|
simprd |
⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑤 , 𝑌 } ) ) |
21 |
18 20
|
ssneldd |
⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) } ) ) |
22 |
6
|
eldifad |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
23 |
1 4
|
lspsnid |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
24 |
14 22 23
|
syl2anc |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
25 |
|
eleq2 |
⊢ ( ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) } ) → ( 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) ↔ 𝑋 ∈ ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) } ) ) ) |
26 |
24 25
|
syl5ibcom |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) } ) → 𝑋 ∈ ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) } ) ) ) |
27 |
26
|
necon3bd |
⊢ ( 𝜑 → ( ¬ 𝑋 ∈ ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) } ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) } ) ) ) |
28 |
21 27
|
mpd |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) } ) ) |