Step |
Hyp |
Ref |
Expression |
1 |
|
chpmatfval.c |
⊢ 𝐶 = ( 𝑁 CharPlyMat 𝑅 ) |
2 |
|
chpmatfval.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
3 |
|
chpmatfval.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
4 |
|
chpmatfval.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
5 |
|
chpmatfval.y |
⊢ 𝑌 = ( 𝑁 Mat 𝑃 ) |
6 |
|
chpmatfval.d |
⊢ 𝐷 = ( 𝑁 maDet 𝑃 ) |
7 |
|
chpmatfval.s |
⊢ − = ( -g ‘ 𝑌 ) |
8 |
|
chpmatfval.x |
⊢ 𝑋 = ( var1 ‘ 𝑅 ) |
9 |
|
chpmatfval.m |
⊢ · = ( ·𝑠 ‘ 𝑌 ) |
10 |
|
chpmatfval.t |
⊢ 𝑇 = ( 𝑁 matToPolyMat 𝑅 ) |
11 |
|
chpmatfval.i |
⊢ 1 = ( 1r ‘ 𝑌 ) |
12 |
1 2 3 4 5 6 7 8 9 10 11
|
chpmatfval |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → 𝐶 = ( 𝑚 ∈ 𝐵 ↦ ( 𝐷 ‘ ( ( 𝑋 · 1 ) − ( 𝑇 ‘ 𝑚 ) ) ) ) ) |
13 |
12
|
3adant3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ) → 𝐶 = ( 𝑚 ∈ 𝐵 ↦ ( 𝐷 ‘ ( ( 𝑋 · 1 ) − ( 𝑇 ‘ 𝑚 ) ) ) ) ) |
14 |
|
fveq2 |
⊢ ( 𝑚 = 𝑀 → ( 𝑇 ‘ 𝑚 ) = ( 𝑇 ‘ 𝑀 ) ) |
15 |
14
|
oveq2d |
⊢ ( 𝑚 = 𝑀 → ( ( 𝑋 · 1 ) − ( 𝑇 ‘ 𝑚 ) ) = ( ( 𝑋 · 1 ) − ( 𝑇 ‘ 𝑀 ) ) ) |
16 |
15
|
fveq2d |
⊢ ( 𝑚 = 𝑀 → ( 𝐷 ‘ ( ( 𝑋 · 1 ) − ( 𝑇 ‘ 𝑚 ) ) ) = ( 𝐷 ‘ ( ( 𝑋 · 1 ) − ( 𝑇 ‘ 𝑀 ) ) ) ) |
17 |
16
|
adantl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑚 = 𝑀 ) → ( 𝐷 ‘ ( ( 𝑋 · 1 ) − ( 𝑇 ‘ 𝑚 ) ) ) = ( 𝐷 ‘ ( ( 𝑋 · 1 ) − ( 𝑇 ‘ 𝑀 ) ) ) ) |
18 |
|
simp3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ) → 𝑀 ∈ 𝐵 ) |
19 |
|
fvexd |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ) → ( 𝐷 ‘ ( ( 𝑋 · 1 ) − ( 𝑇 ‘ 𝑀 ) ) ) ∈ V ) |
20 |
13 17 18 19
|
fvmptd |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ) → ( 𝐶 ‘ 𝑀 ) = ( 𝐷 ‘ ( ( 𝑋 · 1 ) − ( 𝑇 ‘ 𝑀 ) ) ) ) |