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
|
cpmatel |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → ( 𝑀 ∈ 𝑆 ↔ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∀ 𝑙 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑀 𝑗 ) ) ‘ 𝑙 ) = ( 0g ‘ 𝑅 ) ) ) |
8 |
|
simpl2 |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → 𝑅 ∈ Ring ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 ) |
10 |
|
simprl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → 𝑖 ∈ 𝑁 ) |
11 |
|
simprr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → 𝑗 ∈ 𝑁 ) |
12 |
|
simpl3 |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → 𝑀 ∈ 𝐵 ) |
13 |
3 9 4 10 11 12
|
matecld |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( 𝑖 𝑀 𝑗 ) ∈ ( Base ‘ 𝑃 ) ) |
14 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
15 |
5 14 2 9 6
|
cply1coe0bi |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑖 𝑀 𝑗 ) ∈ ( Base ‘ 𝑃 ) ) → ( ∃ 𝑘 ∈ 𝐾 ( 𝑖 𝑀 𝑗 ) = ( 𝐴 ‘ 𝑘 ) ↔ ∀ 𝑙 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑀 𝑗 ) ) ‘ 𝑙 ) = ( 0g ‘ 𝑅 ) ) ) |
16 |
8 13 15
|
syl2anc |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( ∃ 𝑘 ∈ 𝐾 ( 𝑖 𝑀 𝑗 ) = ( 𝐴 ‘ 𝑘 ) ↔ ∀ 𝑙 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑀 𝑗 ) ) ‘ 𝑙 ) = ( 0g ‘ 𝑅 ) ) ) |
17 |
16
|
bicomd |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( ∀ 𝑙 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑀 𝑗 ) ) ‘ 𝑙 ) = ( 0g ‘ 𝑅 ) ↔ ∃ 𝑘 ∈ 𝐾 ( 𝑖 𝑀 𝑗 ) = ( 𝐴 ‘ 𝑘 ) ) ) |
18 |
17
|
2ralbidva |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → ( ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∀ 𝑙 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑀 𝑗 ) ) ‘ 𝑙 ) = ( 0g ‘ 𝑅 ) ↔ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∃ 𝑘 ∈ 𝐾 ( 𝑖 𝑀 𝑗 ) = ( 𝐴 ‘ 𝑘 ) ) ) |
19 |
7 18
|
bitrd |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → ( 𝑀 ∈ 𝑆 ↔ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∃ 𝑘 ∈ 𝐾 ( 𝑖 𝑀 𝑗 ) = ( 𝐴 ‘ 𝑘 ) ) ) |