Step |
Hyp |
Ref |
Expression |
1 |
|
subrgply1.s |
⊢ 𝑆 = ( Poly1 ‘ 𝑅 ) |
2 |
|
subrgply1.h |
⊢ 𝐻 = ( 𝑅 ↾s 𝑇 ) |
3 |
|
subrgply1.u |
⊢ 𝑈 = ( Poly1 ‘ 𝐻 ) |
4 |
|
subrgply1.b |
⊢ 𝐵 = ( Base ‘ 𝑈 ) |
5 |
|
1on |
⊢ 1o ∈ On |
6 |
|
eqid |
⊢ ( 1o mPoly 𝑅 ) = ( 1o mPoly 𝑅 ) |
7 |
|
eqid |
⊢ ( 1o mPoly 𝐻 ) = ( 1o mPoly 𝐻 ) |
8 |
|
eqid |
⊢ ( PwSer1 ‘ 𝐻 ) = ( PwSer1 ‘ 𝐻 ) |
9 |
3 8 4
|
ply1bas |
⊢ 𝐵 = ( Base ‘ ( 1o mPoly 𝐻 ) ) |
10 |
6 2 7 9
|
subrgmpl |
⊢ ( ( 1o ∈ On ∧ 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) → 𝐵 ∈ ( SubRing ‘ ( 1o mPoly 𝑅 ) ) ) |
11 |
5 10
|
mpan |
⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → 𝐵 ∈ ( SubRing ‘ ( 1o mPoly 𝑅 ) ) ) |
12 |
|
eqidd |
⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) ) |
13 |
|
eqid |
⊢ ( PwSer1 ‘ 𝑅 ) = ( PwSer1 ‘ 𝑅 ) |
14 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
15 |
1 13 14
|
ply1bas |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ ( 1o mPoly 𝑅 ) ) |
16 |
15
|
a1i |
⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → ( Base ‘ 𝑆 ) = ( Base ‘ ( 1o mPoly 𝑅 ) ) ) |
17 |
|
eqid |
⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 ) |
18 |
1 6 17
|
ply1plusg |
⊢ ( +g ‘ 𝑆 ) = ( +g ‘ ( 1o mPoly 𝑅 ) ) |
19 |
18
|
a1i |
⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → ( +g ‘ 𝑆 ) = ( +g ‘ ( 1o mPoly 𝑅 ) ) ) |
20 |
19
|
oveqdr |
⊢ ( ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) = ( 𝑥 ( +g ‘ ( 1o mPoly 𝑅 ) ) 𝑦 ) ) |
21 |
|
eqid |
⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 ) |
22 |
1 6 21
|
ply1mulr |
⊢ ( .r ‘ 𝑆 ) = ( .r ‘ ( 1o mPoly 𝑅 ) ) |
23 |
22
|
a1i |
⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → ( .r ‘ 𝑆 ) = ( .r ‘ ( 1o mPoly 𝑅 ) ) ) |
24 |
23
|
oveqdr |
⊢ ( ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( .r ‘ 𝑆 ) 𝑦 ) = ( 𝑥 ( .r ‘ ( 1o mPoly 𝑅 ) ) 𝑦 ) ) |
25 |
12 16 20 24
|
subrgpropd |
⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → ( SubRing ‘ 𝑆 ) = ( SubRing ‘ ( 1o mPoly 𝑅 ) ) ) |
26 |
11 25
|
eleqtrrd |
⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → 𝐵 ∈ ( SubRing ‘ 𝑆 ) ) |