Database
BASIC LINEAR ALGEBRA
Abstract multivariate polynomials
Definition and basic properties
opsrmulr
Metamath Proof Explorer
Description: The multiplication operation of the ordered power series structure.
(Contributed by Mario Carneiro , 8-Feb-2015) (Revised by Mario
Carneiro , 30-Aug-2015) (Revised by AV , 1-Nov-2024)
Ref
Expression
Hypotheses
opsrbas.s
⊢ S = I mPwSer R
opsrbas.o
⊢ O = I ordPwSer R ⁡ T
opsrbas.t
⊢ φ → T ⊆ I × I
Assertion
opsrmulr
⊢ φ → ⋅ S = ⋅ 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
mulrid
⊢ ⋅ 𝑟 = Slot ⋅ ndx
5
plendxnmulrndx
⊢ ≤ ndx ≠ ⋅ ndx
6
5
necomi
⊢ ⋅ ndx ≠ ≤ ndx
7
1 2 3 4 6
opsrbaslem
⊢ φ → ⋅ S = ⋅ O