Theorem opsrvsca

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 ) )