Step |
Hyp |
Ref |
Expression |
1 |
|
m2cpm.s |
⊢ 𝑆 = ( 𝑁 ConstPolyMat 𝑅 ) |
2 |
|
m2cpm.t |
⊢ 𝑇 = ( 𝑁 matToPolyMat 𝑅 ) |
3 |
|
m2cpm.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
4 |
|
m2cpm.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
5 |
|
eqid |
⊢ ( Poly1 ‘ 𝑅 ) = ( Poly1 ‘ 𝑅 ) |
6 |
|
eqid |
⊢ ( 𝑁 Mat ( Poly1 ‘ 𝑅 ) ) = ( 𝑁 Mat ( Poly1 ‘ 𝑅 ) ) |
7 |
|
eqid |
⊢ ( Base ‘ ( 𝑁 Mat ( Poly1 ‘ 𝑅 ) ) ) = ( Base ‘ ( 𝑁 Mat ( Poly1 ‘ 𝑅 ) ) ) |
8 |
2 3 4 5 6 7
|
mat2pmatf1 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑇 : 𝐵 –1-1→ ( Base ‘ ( 𝑁 Mat ( Poly1 ‘ 𝑅 ) ) ) ) |
9 |
1 2 3 4
|
m2cpmf |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑇 : 𝐵 ⟶ 𝑆 ) |
10 |
9
|
frnd |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ran 𝑇 ⊆ 𝑆 ) |
11 |
|
f1ssr |
⊢ ( ( 𝑇 : 𝐵 –1-1→ ( Base ‘ ( 𝑁 Mat ( Poly1 ‘ 𝑅 ) ) ) ∧ ran 𝑇 ⊆ 𝑆 ) → 𝑇 : 𝐵 –1-1→ 𝑆 ) |
12 |
8 10 11
|
syl2anc |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑇 : 𝐵 –1-1→ 𝑆 ) |