Database
BASIC LINEAR ALGEBRA
Abstract multivariate polynomials
Definition and basic properties
opsrsca
Metamath Proof Explorer
Description: The scalar ring of the ordered power series structure. (Contributed by Mario Carneiro , 8-Feb-2015) (Revised by Mario Carneiro , 30-Aug-2015)
Ref
Expression
Hypotheses
opsrbas.s
⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
opsrbas.o
⊢ 𝑂 = ( ( 𝐼 ordPwSer 𝑅 ) ‘ 𝑇 )
opsrbas.t
⊢ ( 𝜑 → 𝑇 ⊆ ( 𝐼 × 𝐼 ) )
opsrsca.i
⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
opsrsca.r
⊢ ( 𝜑 → 𝑅 ∈ 𝑊 )
Assertion
opsrsca
⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ 𝑂 ) )
Proof
Step
Hyp
Ref
Expression
1
opsrbas.s
⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
2
opsrbas.o
⊢ 𝑂 = ( ( 𝐼 ordPwSer 𝑅 ) ‘ 𝑇 )
3
opsrbas.t
⊢ ( 𝜑 → 𝑇 ⊆ ( 𝐼 × 𝐼 ) )
4
opsrsca.i
⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
5
opsrsca.r
⊢ ( 𝜑 → 𝑅 ∈ 𝑊 )
6
1 4 5
psrsca
⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ 𝑆 ) )
7
df-sca
⊢ Scalar = Slot 5
8
5nn
⊢ 5 ∈ ℕ
9
5lt10
⊢ 5 < ; 1 0
10
1 2 3 7 8 9
opsrbaslem
⊢ ( 𝜑 → ( Scalar ‘ 𝑆 ) = ( Scalar ‘ 𝑂 ) )
11
6 10
eqtrd
⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ 𝑂 ) )