Step |
Hyp |
Ref |
Expression |
1 |
|
vr1cl.x |
⊢ 𝑋 = ( var1 ‘ 𝑅 ) |
2 |
|
vr1cl.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
3 |
|
vr1cl.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
4 |
1
|
vr1val |
⊢ 𝑋 = ( ( 1o mVar 𝑅 ) ‘ ∅ ) |
5 |
|
eqid |
⊢ ( 1o mPoly 𝑅 ) = ( 1o mPoly 𝑅 ) |
6 |
|
eqid |
⊢ ( 1o mVar 𝑅 ) = ( 1o mVar 𝑅 ) |
7 |
2 3
|
ply1bas |
⊢ 𝐵 = ( Base ‘ ( 1o mPoly 𝑅 ) ) |
8 |
|
1onn |
⊢ 1o ∈ ω |
9 |
8
|
a1i |
⊢ ( 𝑅 ∈ Ring → 1o ∈ ω ) |
10 |
|
id |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Ring ) |
11 |
|
0lt1o |
⊢ ∅ ∈ 1o |
12 |
11
|
a1i |
⊢ ( 𝑅 ∈ Ring → ∅ ∈ 1o ) |
13 |
5 6 7 9 10 12
|
mvrcl |
⊢ ( 𝑅 ∈ Ring → ( ( 1o mVar 𝑅 ) ‘ ∅ ) ∈ 𝐵 ) |
14 |
4 13
|
eqeltrid |
⊢ ( 𝑅 ∈ Ring → 𝑋 ∈ 𝐵 ) |