Database
BASIC LINEAR ALGEBRA
Abstract multivariate polynomials
Definition and basic properties
opsrvsca
Metamath Proof Explorer
Description: The scalar product 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
opsrvsca
⊢ ( 𝜑 → ( · 𝑠 ‘ 𝑆 ) = ( · 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
vscaid
⊢ · 𝑠 = Slot ( · 𝑠 ‘ ndx )
5
plendxnvscandx
⊢ ( le ‘ ndx ) ≠ ( · 𝑠 ‘ ndx )
6
5
necomi
⊢ ( · 𝑠 ‘ ndx ) ≠ ( le ‘ ndx )
7
1 2 3 4 6
opsrbaslem
⊢ ( 𝜑 → ( · 𝑠 ‘ 𝑆 ) = ( · 𝑠 ‘ 𝑂 ) )