ply1subrg

Description: Univariate polynomials form a subring of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015)

Step	Hyp	Ref	Expression
1		ply1val.1	$⊢ P = {Poly}_{1} (R)$
2		ply1lss.2	$⊢ S = {PwSer}_{1} (R)$
3		ply1lss.u	$⊢ U = {Base}_{P}$
4		eqid	$⊢ 1_{𝑜} mPwSer R = 1_{𝑜} mPwSer R$
5		eqid	$⊢ 1_{𝑜} mPoly R = 1_{𝑜} mPoly R$
6	1 3	ply1bas	$⊢ U = {Base}_{(1_{𝑜} mPoly R)}$
7		1on	$⊢ 1_{𝑜} \in On$
8	7	a1i	$⊢ R \in Ring \to 1_{𝑜} \in On$
9		id	$⊢ R \in Ring \to R \in Ring$
10	4 5 6 8 9	mplsubrg	$⊢ R \in Ring \to U \in SubRing (1_{𝑜} mPwSer R)$
11		eqidd	$⊢ R \in Ring \to {Base}_{(1_{𝑜} mPwSer R)} = {Base}_{(1_{𝑜} mPwSer R)}$
12	2	psr1val	$⊢ S = (1_{𝑜} ordPwSer R) (\emptyset)$
13		0ss	$⊢ \emptyset \subseteq 1_{𝑜} \times 1_{𝑜}$
14	13	a1i	$⊢ R \in Ring \to \emptyset \subseteq 1_{𝑜} \times 1_{𝑜}$
15	4 12 14	opsrbas	$⊢ R \in Ring \to {Base}_{(1_{𝑜} mPwSer R)} = {Base}_{S}$
16	4 12 14	opsrplusg	$⊢ R \in Ring \to +_{(1_{𝑜} mPwSer R)} = +_{S}$
17	16	oveqdr	$⊢ (R \in Ring \land (x \in {Base}_{(1_{𝑜} mPwSer R)} \land y \in {Base}_{(1_{𝑜} mPwSer R)})) \to x +_{(1_{𝑜} mPwSer R)} y = x +_{S} y$
18	4 12 14	opsrmulr	$⊢ R \in Ring \to \cdot_{(1_{𝑜} mPwSer R)} = \cdot_{S}$
19	18	oveqdr	$⊢ (R \in Ring \land (x \in {Base}_{(1_{𝑜} mPwSer R)} \land y \in {Base}_{(1_{𝑜} mPwSer R)})) \to x \cdot_{(1_{𝑜} mPwSer R)} y = x \cdot_{S} y$
20	11 15 17 19	subrgpropd	$⊢ R \in Ring \to SubRing (1_{𝑜} mPwSer R) = SubRing (S)$
21	10 20	eleqtrd	$⊢ R \in Ring \to U \in SubRing (S)$

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