Step |
Hyp |
Ref |
Expression |
1 |
|
cpmat.s |
⊢ 𝑆 = ( 𝑁 ConstPolyMat 𝑅 ) |
2 |
|
cpmat.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
3 |
|
cpmat.c |
⊢ 𝐶 = ( 𝑁 Mat 𝑃 ) |
4 |
|
cpmat.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
5 |
1 2 3 4
|
cpmatpmat |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆 ) → 𝑀 ∈ 𝐵 ) |
6 |
5
|
3expa |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑀 ∈ 𝑆 ) → 𝑀 ∈ 𝐵 ) |
7 |
1 2 3 4
|
cpmatel |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → ( 𝑀 ∈ 𝑆 ↔ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑀 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑅 ) ) ) |
8 |
7
|
3expa |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑀 ∈ 𝐵 ) → ( 𝑀 ∈ 𝑆 ↔ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑀 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑅 ) ) ) |
9 |
8
|
biimpd |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑀 ∈ 𝐵 ) → ( 𝑀 ∈ 𝑆 → ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑀 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑅 ) ) ) |
10 |
9
|
impancom |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑀 ∈ 𝑆 ) → ( 𝑀 ∈ 𝐵 → ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑀 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑅 ) ) ) |
11 |
6 10
|
jcai |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑀 ∈ 𝑆 ) → ( 𝑀 ∈ 𝐵 ∧ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑀 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑅 ) ) ) |
12 |
11
|
ex |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 𝑀 ∈ 𝑆 → ( 𝑀 ∈ 𝐵 ∧ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑀 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑅 ) ) ) ) |