Step |
Hyp |
Ref |
Expression |
1 |
|
subrgvr1.x |
⊢ 𝑋 = ( var1 ‘ 𝑅 ) |
2 |
|
subrgvr1.r |
⊢ ( 𝜑 → 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) |
3 |
|
subrgvr1.h |
⊢ 𝐻 = ( 𝑅 ↾s 𝑇 ) |
4 |
|
subrgvr1cl.u |
⊢ 𝑈 = ( Poly1 ‘ 𝐻 ) |
5 |
|
subrgvr1cl.b |
⊢ 𝐵 = ( Base ‘ 𝑈 ) |
6 |
1
|
vr1val |
⊢ 𝑋 = ( ( 1o mVar 𝑅 ) ‘ ∅ ) |
7 |
|
eqid |
⊢ ( 1o mVar 𝑅 ) = ( 1o mVar 𝑅 ) |
8 |
|
1on |
⊢ 1o ∈ On |
9 |
8
|
a1i |
⊢ ( 𝜑 → 1o ∈ On ) |
10 |
|
eqid |
⊢ ( 1o mPoly 𝐻 ) = ( 1o mPoly 𝐻 ) |
11 |
|
eqid |
⊢ ( PwSer1 ‘ 𝐻 ) = ( PwSer1 ‘ 𝐻 ) |
12 |
4 11 5
|
ply1bas |
⊢ 𝐵 = ( Base ‘ ( 1o mPoly 𝐻 ) ) |
13 |
7 9 2 3 10 12
|
subrgmvrf |
⊢ ( 𝜑 → ( 1o mVar 𝑅 ) : 1o ⟶ 𝐵 ) |
14 |
|
0lt1o |
⊢ ∅ ∈ 1o |
15 |
|
ffvelrn |
⊢ ( ( ( 1o mVar 𝑅 ) : 1o ⟶ 𝐵 ∧ ∅ ∈ 1o ) → ( ( 1o mVar 𝑅 ) ‘ ∅ ) ∈ 𝐵 ) |
16 |
13 14 15
|
sylancl |
⊢ ( 𝜑 → ( ( 1o mVar 𝑅 ) ‘ ∅ ) ∈ 𝐵 ) |
17 |
6 16
|
eqeltrid |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |