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 |
|
m2cpmghm.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
6 |
|
m2cpmghm.c |
⊢ 𝐶 = ( 𝑁 Mat 𝑃 ) |
7 |
|
m2cpmghm.u |
⊢ 𝑈 = ( 𝐶 ↾s 𝑆 ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
9 |
2 3 4 5 6 8
|
mat2pmatghm |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑇 ∈ ( 𝐴 GrpHom 𝐶 ) ) |
10 |
1 5 6
|
cpmatsubgpmat |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑆 ∈ ( SubGrp ‘ 𝐶 ) ) |
11 |
1 2 3 4
|
m2cpmf |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑇 : 𝐵 ⟶ 𝑆 ) |
12 |
11
|
frnd |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ran 𝑇 ⊆ 𝑆 ) |
13 |
7
|
resghm2b |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐶 ) ∧ ran 𝑇 ⊆ 𝑆 ) → ( 𝑇 ∈ ( 𝐴 GrpHom 𝐶 ) ↔ 𝑇 ∈ ( 𝐴 GrpHom 𝑈 ) ) ) |
14 |
13
|
bicomd |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐶 ) ∧ ran 𝑇 ⊆ 𝑆 ) → ( 𝑇 ∈ ( 𝐴 GrpHom 𝑈 ) ↔ 𝑇 ∈ ( 𝐴 GrpHom 𝐶 ) ) ) |
15 |
10 12 14
|
syl2anc |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 𝑇 ∈ ( 𝐴 GrpHom 𝑈 ) ↔ 𝑇 ∈ ( 𝐴 GrpHom 𝐶 ) ) ) |
16 |
9 15
|
mpbird |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑇 ∈ ( 𝐴 GrpHom 𝑈 ) ) |