Theorem psr1ring

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

	Ref	Expression
Hypothesis	psr1ring.s	⊢ 𝑆 = ( PwSer1 ‘ 𝑅 )
Assertion	psr1ring	⊢ ( 𝑅 ∈ Ring → 𝑆 ∈ Ring )

Proof

Step	Hyp	Ref	Expression
1		psr1ring.s	⊢ 𝑆 = ( PwSer1 ‘ 𝑅 )
2	1	psr1val	⊢ 𝑆 = ( ( 1o ordPwSer 𝑅 ) ‘ ∅ )
3		1on	⊢ 1o ∈ On
4	3	a1i	⊢ ( 𝑅 ∈ Ring → 1o ∈ On )
5		id	⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Ring )
6		0ss	⊢ ∅ ⊆ ( 1o × 1o )
7	6	a1i	⊢ ( 𝑅 ∈ Ring → ∅ ⊆ ( 1o × 1o ) )
8	2 4 5 7	opsrring	⊢ ( 𝑅 ∈ Ring → 𝑆 ∈ Ring )