Metamath Proof Explorer


Theorem vr1cl

Description: The generator of a univariate polynomial algebra is contained in the base set. (Contributed by Stefan O'Rear, 19-Mar-2015)

Ref Expression
Hypotheses vr1cl.x 𝑋 = ( var1𝑅 )
vr1cl.p 𝑃 = ( Poly1𝑅 )
vr1cl.b 𝐵 = ( Base ‘ 𝑃 )
Assertion vr1cl ( 𝑅 ∈ Ring → 𝑋𝐵 )

Proof

Step Hyp Ref Expression
1 vr1cl.x 𝑋 = ( var1𝑅 )
2 vr1cl.p 𝑃 = ( Poly1𝑅 )
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 → 𝑋𝐵 )