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Metamath Proof Explorer

Theorem psr1vsca

Description: Value of scalar multiplication in a univariate power series ring. (Contributed by Stefan O'Rear, 21-Mar-2015) (Revised by Mario Carneiro, 2-Oct-2015)

		Ref	Expression
	Hypotheses	psr1plusg.y	$⊢ Y = {PwSer}_{1} (R)$
		psr1plusg.s	$⊢ S = 1_{𝑜} mPwSer R$
		psr1vscafval.n	$⊢ \underset{˙}{\cdot} = \cdot_{Y}$
	Assertion	psr1vsca	$⊢ \underset{˙}{\cdot} = \cdot_{S}$

Proof

Step	Hyp	Ref	Expression
1		psr1plusg.y	$⊢ Y = {PwSer}_{1} (R)$
2		psr1plusg.s	$⊢ S = 1_{𝑜} mPwSer R$
3		psr1vscafval.n	$⊢ \underset{˙}{\cdot} = \cdot_{Y}$
4	1	psr1val	$⊢ Y = (1_{𝑜} ordPwSer R) (\emptyset)$
5		0ss	$⊢ \emptyset \subseteq 1_{𝑜} \times 1_{𝑜}$
6	5	a1i	$⊢ ⊤ \to \emptyset \subseteq 1_{𝑜} \times 1_{𝑜}$
7	2 4 6	opsrvsca	$⊢ ⊤ \to \cdot_{S} = \cdot_{Y}$
8	7	mptru	$⊢ \cdot_{S} = \cdot_{Y}$
9	3 8	eqtr4i	$⊢ \underset{˙}{\cdot} = \cdot_{S}$