Step |
Hyp |
Ref |
Expression |
1 |
|
subrgmpl.s |
⊢ 𝑆 = ( 𝐼 mPoly 𝑅 ) |
2 |
|
subrgmpl.h |
⊢ 𝐻 = ( 𝑅 ↾s 𝑇 ) |
3 |
|
subrgmpl.u |
⊢ 𝑈 = ( 𝐼 mPoly 𝐻 ) |
4 |
|
subrgmpl.b |
⊢ 𝐵 = ( Base ‘ 𝑈 ) |
5 |
|
simpl |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) → 𝐼 ∈ 𝑉 ) |
6 |
|
simpr |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) → 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) |
7 |
|
eqid |
⊢ ( 𝐼 mPwSer 𝐻 ) = ( 𝐼 mPwSer 𝐻 ) |
8 |
|
eqid |
⊢ ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) = ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
10 |
1 2 3 4 5 6 7 8 9
|
ressmplbas2 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) → 𝐵 = ( ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ∩ ( Base ‘ 𝑆 ) ) ) |
11 |
|
eqid |
⊢ ( 𝐼 mPwSer 𝑅 ) = ( 𝐼 mPwSer 𝑅 ) |
12 |
11 2 7 8
|
subrgpsr |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) → ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ∈ ( SubRing ‘ ( 𝐼 mPwSer 𝑅 ) ) ) |
13 |
|
subrgrcl |
⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → 𝑅 ∈ Ring ) |
14 |
13
|
adantl |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) → 𝑅 ∈ Ring ) |
15 |
11 1 9 5 14
|
mplsubrg |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) → ( Base ‘ 𝑆 ) ∈ ( SubRing ‘ ( 𝐼 mPwSer 𝑅 ) ) ) |
16 |
|
subrgin |
⊢ ( ( ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ∈ ( SubRing ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ ( Base ‘ 𝑆 ) ∈ ( SubRing ‘ ( 𝐼 mPwSer 𝑅 ) ) ) → ( ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ∩ ( Base ‘ 𝑆 ) ) ∈ ( SubRing ‘ ( 𝐼 mPwSer 𝑅 ) ) ) |
17 |
12 15 16
|
syl2anc |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) → ( ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ∩ ( Base ‘ 𝑆 ) ) ∈ ( SubRing ‘ ( 𝐼 mPwSer 𝑅 ) ) ) |
18 |
10 17
|
eqeltrd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) → 𝐵 ∈ ( SubRing ‘ ( 𝐼 mPwSer 𝑅 ) ) ) |
19 |
|
inss2 |
⊢ ( ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ∩ ( Base ‘ 𝑆 ) ) ⊆ ( Base ‘ 𝑆 ) |
20 |
10 19
|
eqsstrdi |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) → 𝐵 ⊆ ( Base ‘ 𝑆 ) ) |
21 |
1 11 9
|
mplval2 |
⊢ 𝑆 = ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ 𝑆 ) ) |
22 |
21
|
subsubrg |
⊢ ( ( Base ‘ 𝑆 ) ∈ ( SubRing ‘ ( 𝐼 mPwSer 𝑅 ) ) → ( 𝐵 ∈ ( SubRing ‘ 𝑆 ) ↔ ( 𝐵 ∈ ( SubRing ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝐵 ⊆ ( Base ‘ 𝑆 ) ) ) ) |
23 |
15 22
|
syl |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) → ( 𝐵 ∈ ( SubRing ‘ 𝑆 ) ↔ ( 𝐵 ∈ ( SubRing ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝐵 ⊆ ( Base ‘ 𝑆 ) ) ) ) |
24 |
18 20 23
|
mpbir2and |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) → 𝐵 ∈ ( SubRing ‘ 𝑆 ) ) |