This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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	$⊢ φ \to T \subseteq I \times I$
Assertion	opsrvsca	$⊢ φ \to \cdot_{S} = \cdot_{O}$

Proof

Step	Hyp	Ref	Expression
1		opsrbas.s	$⊢ S = I mPwSer R$
2		opsrbas.o	$⊢ O = (I ordPwSer R) (T)$
3		opsrbas.t	$⊢ φ \to T \subseteq I \times I$
4		vscaid	$⊢ \cdot_{𝑠} = Slot \cdot_{ndx}$
5		plendxnvscandx	$⊢ \leq_{ndx} \neq \cdot_{ndx}$
6	5	necomi	$⊢ \cdot_{ndx} \neq \leq_{ndx}$
7	1 2 3 4 6	opsrbaslem	$⊢ φ \to \cdot_{S} = \cdot_{O}$