Step |
Hyp |
Ref |
Expression |
1 |
|
mat2pmatbas.t |
⊢ 𝑇 = ( 𝑁 matToPolyMat 𝑅 ) |
2 |
|
mat2pmatbas.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
3 |
|
mat2pmatbas.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
4 |
|
mat2pmatbas.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
5 |
|
mat2pmatbas.c |
⊢ 𝐶 = ( 𝑁 Mat 𝑃 ) |
6 |
|
mat2pmatbas0.h |
⊢ 𝐻 = ( Base ‘ 𝐶 ) |
7 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
8 |
2
|
matring |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝐴 ∈ Ring ) |
9 |
7 8
|
sylan2 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝐴 ∈ Ring ) |
10 |
4
|
ply1ring |
⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ Ring ) |
11 |
7 10
|
syl |
⊢ ( 𝑅 ∈ CRing → 𝑃 ∈ Ring ) |
12 |
5
|
matring |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ) → 𝐶 ∈ Ring ) |
13 |
11 12
|
sylan2 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝐶 ∈ Ring ) |
14 |
1 2 3 4 5 6
|
mat2pmatghm |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑇 ∈ ( 𝐴 GrpHom 𝐶 ) ) |
15 |
7 14
|
sylan2 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝑇 ∈ ( 𝐴 GrpHom 𝐶 ) ) |
16 |
1 2 3 4 5 6
|
mat2pmatmhm |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝑇 ∈ ( ( mulGrp ‘ 𝐴 ) MndHom ( mulGrp ‘ 𝐶 ) ) ) |
17 |
15 16
|
jca |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ( 𝑇 ∈ ( 𝐴 GrpHom 𝐶 ) ∧ 𝑇 ∈ ( ( mulGrp ‘ 𝐴 ) MndHom ( mulGrp ‘ 𝐶 ) ) ) ) |
18 |
|
eqid |
⊢ ( mulGrp ‘ 𝐴 ) = ( mulGrp ‘ 𝐴 ) |
19 |
|
eqid |
⊢ ( mulGrp ‘ 𝐶 ) = ( mulGrp ‘ 𝐶 ) |
20 |
18 19
|
isrhm |
⊢ ( 𝑇 ∈ ( 𝐴 RingHom 𝐶 ) ↔ ( ( 𝐴 ∈ Ring ∧ 𝐶 ∈ Ring ) ∧ ( 𝑇 ∈ ( 𝐴 GrpHom 𝐶 ) ∧ 𝑇 ∈ ( ( mulGrp ‘ 𝐴 ) MndHom ( mulGrp ‘ 𝐶 ) ) ) ) ) |
21 |
9 13 17 20
|
syl21anbrc |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝑇 ∈ ( 𝐴 RingHom 𝐶 ) ) |