Theorem psr1ring

Description: Univariate power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015)

	Ref	Expression
Hypothesis	psr1ring.s	$$⊢S={\mathrm{PwSer}}_1\left(R\right)$$
Assertion	psr1ring	$$⊢R\in \mathrm{Ring}\to S\in \mathrm{Ring}$$

Proof

Step	Hyp	Ref	Expression
1		psr1ring.s	$$⊢S={\mathrm{PwSer}}_1\left(R\right)$$
2	1	psr1val	$$⊢S=\left(1_{𝑜}\mathrm{ordPwSer}R\right)\left(\varnothing \right)$$
3		1on	$$⊢1_{𝑜}\in \mathrm{On}$$
4	3	a1i	$$⊢R\in \mathrm{Ring}\to 1_{𝑜}\in \mathrm{On}$$
5		id	$$⊢R\in \mathrm{Ring}\to R\in \mathrm{Ring}$$
6		0ss	$$⊢\varnothing \subseteq 1_{𝑜}\times 1_{𝑜}$$
7	6	a1i	$$⊢R\in \mathrm{Ring}\to \varnothing \subseteq 1_{𝑜}\times 1_{𝑜}$$
8	2 4 5 7	opsrring	$$⊢R\in \mathrm{Ring}\to S\in \mathrm{Ring}$$