Theorem opsrring

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

	Ref	Expression
Hypotheses	opsrring.o	⊢ 𝑂 = ( ( 𝐼 ordPwSer 𝑅 ) ‘ 𝑇 )
	opsrring.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
	opsrring.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
	opsrring.t	⊢ ( 𝜑 → 𝑇 ⊆ ( 𝐼 × 𝐼 ) )
Assertion	opsrring	⊢ ( 𝜑 → 𝑂 ∈ Ring )

Proof

Step	Hyp	Ref	Expression
1		opsrring.o	⊢ 𝑂 = ( ( 𝐼 ordPwSer 𝑅 ) ‘ 𝑇 )
2		opsrring.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
3		opsrring.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
4		opsrring.t	⊢ ( 𝜑 → 𝑇 ⊆ ( 𝐼 × 𝐼 ) )
5		eqid	⊢ ( 𝐼 mPwSer 𝑅 ) = ( 𝐼 mPwSer 𝑅 )
6	5 2 3	psrring	⊢ ( 𝜑 → ( 𝐼 mPwSer 𝑅 ) ∈ Ring )
7		eqidd	⊢ ( 𝜑 → ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) )
8	5 1 4	opsrbas	⊢ ( 𝜑 → ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( Base ‘ 𝑂 ) )
9	5 1 4	opsrplusg	⊢ ( 𝜑 → ( +g ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( +g ‘ 𝑂 ) )
10	9	oveqdr	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑦 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) ) → ( 𝑥 ( +g ‘ ( 𝐼 mPwSer 𝑅 ) ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑂 ) 𝑦 ) )
11	5 1 4	opsrmulr	⊢ ( 𝜑 → ( .r ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( .r ‘ 𝑂 ) )
12	11	oveqdr	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑦 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) ) → ( 𝑥 ( .r ‘ ( 𝐼 mPwSer 𝑅 ) ) 𝑦 ) = ( 𝑥 ( .r ‘ 𝑂 ) 𝑦 ) )
13	7 8 10 12	ringpropd	⊢ ( 𝜑 → ( ( 𝐼 mPwSer 𝑅 ) ∈ Ring ↔ 𝑂 ∈ Ring ) )
14	6 13	mpbid	⊢ ( 𝜑 → 𝑂 ∈ Ring )