Metamath Proof Explorer


Theorem vr1cl2

Description: The variable X is a member of the power series algebra R [ [ X ] ] . (Contributed by Mario Carneiro, 8-Feb-2015)

Ref Expression
Hypotheses vr1val.1 𝑋 = ( var1𝑅 )
vr1cl2.2 𝑆 = ( PwSer1𝑅 )
vr1cl2.3 𝐵 = ( Base ‘ 𝑆 )
Assertion vr1cl2 ( 𝑅 ∈ Ring → 𝑋𝐵 )

Proof

Step Hyp Ref Expression
1 vr1val.1 𝑋 = ( var1𝑅 )
2 vr1cl2.2 𝑆 = ( PwSer1𝑅 )
3 vr1cl2.3 𝐵 = ( Base ‘ 𝑆 )
4 1 vr1val 𝑋 = ( ( 1o mVar 𝑅 ) ‘ ∅ )
5 eqid ( 1o mPwSer 𝑅 ) = ( 1o mPwSer 𝑅 )
6 eqid ( 1o mVar 𝑅 ) = ( 1o mVar 𝑅 )
7 eqid ( Base ‘ ( 1o mPwSer 𝑅 ) ) = ( Base ‘ ( 1o mPwSer 𝑅 ) )
8 1on 1o ∈ On
9 8 a1i ( 𝑅 ∈ Ring → 1o ∈ On )
10 id ( 𝑅 ∈ Ring → 𝑅 ∈ Ring )
11 0lt1o ∅ ∈ 1o
12 11 a1i ( 𝑅 ∈ Ring → ∅ ∈ 1o )
13 5 6 7 9 10 12 mvrcl2 ( 𝑅 ∈ Ring → ( ( 1o mVar 𝑅 ) ‘ ∅ ) ∈ ( Base ‘ ( 1o mPwSer 𝑅 ) ) )
14 2 psr1val 𝑆 = ( ( 1o ordPwSer 𝑅 ) ‘ ∅ )
15 0ss ∅ ⊆ ( 1o × 1o )
16 15 a1i ( 𝑅 ∈ Ring → ∅ ⊆ ( 1o × 1o ) )
17 5 14 16 opsrbas ( 𝑅 ∈ Ring → ( Base ‘ ( 1o mPwSer 𝑅 ) ) = ( Base ‘ 𝑆 ) )
18 17 3 eqtr4di ( 𝑅 ∈ Ring → ( Base ‘ ( 1o mPwSer 𝑅 ) ) = 𝐵 )
19 13 18 eleqtrd ( 𝑅 ∈ Ring → ( ( 1o mVar 𝑅 ) ‘ ∅ ) ∈ 𝐵 )
20 4 19 eqeltrid ( 𝑅 ∈ Ring → 𝑋𝐵 )