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 |
|
cpmatel2.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
6 |
|
cpmatel2.a |
⊢ 𝐴 = ( algSc ‘ 𝑃 ) |
7 |
1 2 3 4
|
cpmatpmat |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆 ) → 𝑀 ∈ 𝐵 ) |
8 |
7
|
3expa |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑀 ∈ 𝑆 ) → 𝑀 ∈ 𝐵 ) |
9 |
1 2 3 4 5 6
|
cpmatel2 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → ( 𝑀 ∈ 𝑆 ↔ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∃ 𝑘 ∈ 𝐾 ( 𝑖 𝑀 𝑗 ) = ( 𝐴 ‘ 𝑘 ) ) ) |
10 |
9
|
3expa |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑀 ∈ 𝐵 ) → ( 𝑀 ∈ 𝑆 ↔ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∃ 𝑘 ∈ 𝐾 ( 𝑖 𝑀 𝑗 ) = ( 𝐴 ‘ 𝑘 ) ) ) |
11 |
10
|
biimpd |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑀 ∈ 𝐵 ) → ( 𝑀 ∈ 𝑆 → ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∃ 𝑘 ∈ 𝐾 ( 𝑖 𝑀 𝑗 ) = ( 𝐴 ‘ 𝑘 ) ) ) |
12 |
11
|
impancom |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑀 ∈ 𝑆 ) → ( 𝑀 ∈ 𝐵 → ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∃ 𝑘 ∈ 𝐾 ( 𝑖 𝑀 𝑗 ) = ( 𝐴 ‘ 𝑘 ) ) ) |
13 |
8 12
|
jcai |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑀 ∈ 𝑆 ) → ( 𝑀 ∈ 𝐵 ∧ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∃ 𝑘 ∈ 𝐾 ( 𝑖 𝑀 𝑗 ) = ( 𝐴 ‘ 𝑘 ) ) ) |
14 |
13
|
ex |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 𝑀 ∈ 𝑆 → ( 𝑀 ∈ 𝐵 ∧ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∃ 𝑘 ∈ 𝐾 ( 𝑖 𝑀 𝑗 ) = ( 𝐴 ‘ 𝑘 ) ) ) ) |