Database
BASIC LINEAR ALGEBRA
Abstract multivariate polynomials
Definition and basic properties
opsrplusg
Metamath Proof Explorer
Description: The addition operation 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 ) )
Assertion
opsrplusg
⊢ ( 𝜑 → ( +_{g} ‘ 𝑆 ) = ( +_{g} ‘ 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
df-plusg
⊢ +_{g} = Slot 2
5
2nn
⊢ 2 ∈ ℕ
6
2lt10
⊢ 2 < ; 1 0
7
1 2 3 4 5 6
opsrbaslem
⊢ ( 𝜑 → ( +_{g} ‘ 𝑆 ) = ( +_{g} ‘ 𝑂 ) )