Database
BASIC LINEAR ALGEBRA
Abstract multivariate polynomials
Univariate polynomials
psr1mulr
Metamath Proof Explorer
Description: Value of multiplication in a univariate power series ring.
(Contributed by Stefan O'Rear , 21-Mar-2015) (Revised by Mario
Carneiro , 2-Oct-2015)
Ref
Expression
Hypotheses
psr1plusg.y
⊢ Y = PwSer 1 ⁡ R
psr1plusg.s
⊢ S = 1 𝑜 mPwSer R
psr1mulr.n
⊢ · ˙ = ⋅ Y
Assertion
psr1mulr
⊢ · ˙ = ⋅ S
Proof
Step
Hyp
Ref
Expression
1
psr1plusg.y
⊢ Y = PwSer 1 ⁡ R
2
psr1plusg.s
⊢ S = 1 𝑜 mPwSer R
3
psr1mulr.n
⊢ · ˙ = ⋅ Y
4
1
psr1val
⊢ Y = 1 𝑜 ordPwSer R ⁡ ∅
5
0ss
⊢ ∅ ⊆ 1 𝑜 × 1 𝑜
6
5
a1i
⊢ ⊤ → ∅ ⊆ 1 𝑜 × 1 𝑜
7
2 4 6
opsrmulr
⊢ ⊤ → ⋅ S = ⋅ Y
8
7
mptru
⊢ ⋅ S = ⋅ Y
9
3 8
eqtr4i
⊢ · ˙ = ⋅ S