Step |
Hyp |
Ref |
Expression |
1 |
|
ressmpl.s |
⊢ 𝑆 = ( 𝐼 mPoly 𝑅 ) |
2 |
|
ressmpl.h |
⊢ 𝐻 = ( 𝑅 ↾s 𝑇 ) |
3 |
|
ressmpl.u |
⊢ 𝑈 = ( 𝐼 mPoly 𝐻 ) |
4 |
|
ressmpl.b |
⊢ 𝐵 = ( Base ‘ 𝑈 ) |
5 |
|
ressmpl.1 |
⊢ ( 𝜑 → 𝐼 ∈ 𝑉 ) |
6 |
|
ressmpl.2 |
⊢ ( 𝜑 → 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) |
7 |
|
ressmpl.p |
⊢ 𝑃 = ( 𝑆 ↾s 𝐵 ) |
8 |
|
eqid |
⊢ ( 𝐼 mPwSer 𝐻 ) = ( 𝐼 mPwSer 𝐻 ) |
9 |
|
eqid |
⊢ ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) = ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
11 |
1 2 3 4 5 6 8 9 10
|
ressmplbas2 |
⊢ ( 𝜑 → 𝐵 = ( ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ∩ ( Base ‘ 𝑆 ) ) ) |
12 |
|
inss2 |
⊢ ( ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ∩ ( Base ‘ 𝑆 ) ) ⊆ ( Base ‘ 𝑆 ) |
13 |
11 12
|
eqsstrdi |
⊢ ( 𝜑 → 𝐵 ⊆ ( Base ‘ 𝑆 ) ) |
14 |
7 10
|
ressbas2 |
⊢ ( 𝐵 ⊆ ( Base ‘ 𝑆 ) → 𝐵 = ( Base ‘ 𝑃 ) ) |
15 |
13 14
|
syl |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑃 ) ) |