Step |
Hyp |
Ref |
Expression |
1 |
|
cpmatsrngpmat.s |
⊢ 𝑆 = ( 𝑁 ConstPolyMat 𝑅 ) |
2 |
|
cpmatsrngpmat.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
3 |
|
cpmatsrngpmat.c |
⊢ 𝐶 = ( 𝑁 Mat 𝑃 ) |
4 |
1 2 3
|
cpmatsubgpmat |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑆 ∈ ( SubGrp ‘ 𝐶 ) ) |
5 |
1 2 3
|
1elcpmat |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 1r ‘ 𝐶 ) ∈ 𝑆 ) |
6 |
1 2 3
|
cpmatmcl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( .r ‘ 𝐶 ) 𝑦 ) ∈ 𝑆 ) |
7 |
2 3
|
pmatring |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝐶 ∈ Ring ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
9 |
|
eqid |
⊢ ( 1r ‘ 𝐶 ) = ( 1r ‘ 𝐶 ) |
10 |
|
eqid |
⊢ ( .r ‘ 𝐶 ) = ( .r ‘ 𝐶 ) |
11 |
8 9 10
|
issubrg2 |
⊢ ( 𝐶 ∈ Ring → ( 𝑆 ∈ ( SubRing ‘ 𝐶 ) ↔ ( 𝑆 ∈ ( SubGrp ‘ 𝐶 ) ∧ ( 1r ‘ 𝐶 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( .r ‘ 𝐶 ) 𝑦 ) ∈ 𝑆 ) ) ) |
12 |
7 11
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 𝑆 ∈ ( SubRing ‘ 𝐶 ) ↔ ( 𝑆 ∈ ( SubGrp ‘ 𝐶 ) ∧ ( 1r ‘ 𝐶 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( .r ‘ 𝐶 ) 𝑦 ) ∈ 𝑆 ) ) ) |
13 |
4 5 6 12
|
mpbir3and |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑆 ∈ ( SubRing ‘ 𝐶 ) ) |