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
|
mat2pmatmhm |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝑇 ∈ ( ( mulGrp ‘ 𝐴 ) MndHom ( mulGrp ‘ 𝐶 ) ) ) |
10 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
11 |
10
|
anim2i |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ) |
12 |
1 5 6
|
cpmatsrgpmat |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑆 ∈ ( SubRing ‘ 𝐶 ) ) |
13 |
|
eqid |
⊢ ( mulGrp ‘ 𝐶 ) = ( mulGrp ‘ 𝐶 ) |
14 |
13
|
subrgsubm |
⊢ ( 𝑆 ∈ ( SubRing ‘ 𝐶 ) → 𝑆 ∈ ( SubMnd ‘ ( mulGrp ‘ 𝐶 ) ) ) |
15 |
11 12 14
|
3syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝑆 ∈ ( SubMnd ‘ ( mulGrp ‘ 𝐶 ) ) ) |
16 |
1 2 3 4
|
m2cpmf |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑇 : 𝐵 ⟶ 𝑆 ) |
17 |
|
frn |
⊢ ( 𝑇 : 𝐵 ⟶ 𝑆 → ran 𝑇 ⊆ 𝑆 ) |
18 |
11 16 17
|
3syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ran 𝑇 ⊆ 𝑆 ) |
19 |
6
|
ovexi |
⊢ 𝐶 ∈ V |
20 |
1
|
ovexi |
⊢ 𝑆 ∈ V |
21 |
7 13
|
mgpress |
⊢ ( ( 𝐶 ∈ V ∧ 𝑆 ∈ V ) → ( ( mulGrp ‘ 𝐶 ) ↾s 𝑆 ) = ( mulGrp ‘ 𝑈 ) ) |
22 |
19 20 21
|
mp2an |
⊢ ( ( mulGrp ‘ 𝐶 ) ↾s 𝑆 ) = ( mulGrp ‘ 𝑈 ) |
23 |
22
|
eqcomi |
⊢ ( mulGrp ‘ 𝑈 ) = ( ( mulGrp ‘ 𝐶 ) ↾s 𝑆 ) |
24 |
23
|
resmhm2b |
⊢ ( ( 𝑆 ∈ ( SubMnd ‘ ( mulGrp ‘ 𝐶 ) ) ∧ ran 𝑇 ⊆ 𝑆 ) → ( 𝑇 ∈ ( ( mulGrp ‘ 𝐴 ) MndHom ( mulGrp ‘ 𝐶 ) ) ↔ 𝑇 ∈ ( ( mulGrp ‘ 𝐴 ) MndHom ( mulGrp ‘ 𝑈 ) ) ) ) |
25 |
15 18 24
|
syl2anc |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ( 𝑇 ∈ ( ( mulGrp ‘ 𝐴 ) MndHom ( mulGrp ‘ 𝐶 ) ) ↔ 𝑇 ∈ ( ( mulGrp ‘ 𝐴 ) MndHom ( mulGrp ‘ 𝑈 ) ) ) ) |
26 |
9 25
|
mpbid |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝑇 ∈ ( ( mulGrp ‘ 𝐴 ) MndHom ( mulGrp ‘ 𝑈 ) ) ) |