Metamath Proof Explorer


Theorem opsrsca

Description: The scalar ring 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 ( 𝜑𝑇 ⊆ ( 𝐼 × 𝐼 ) )
opsrsca.i ( 𝜑𝐼𝑉 )
opsrsca.r ( 𝜑𝑅𝑊 )
Assertion opsrsca ( 𝜑𝑅 = ( 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 ( 𝜑𝑇 ⊆ ( 𝐼 × 𝐼 ) )
4 opsrsca.i ( 𝜑𝐼𝑉 )
5 opsrsca.r ( 𝜑𝑅𝑊 )
6 1 4 5 psrsca ( 𝜑𝑅 = ( Scalar ‘ 𝑆 ) )
7 scaid Scalar = Slot ( Scalar ‘ ndx )
8 plendxnscandx ( le ‘ ndx ) ≠ ( Scalar ‘ ndx )
9 8 necomi ( Scalar ‘ ndx ) ≠ ( le ‘ ndx )
10 1 2 3 7 9 opsrbaslem ( 𝜑 → ( Scalar ‘ 𝑆 ) = ( Scalar ‘ 𝑂 ) )
11 6 10 eqtrd ( 𝜑𝑅 = ( Scalar ‘ 𝑂 ) )