Theorem psr1crng

Description: The ring of univariate power series is a commutative ring. (Contributed by Mario Carneiro, 8-Feb-2015)

	Ref	Expression
Hypothesis	psr1val.1	⊢ 𝑆 = ( PwSer1 ‘ 𝑅 )
Assertion	psr1crng	⊢ ( 𝑅 ∈ CRing → 𝑆 ∈ CRing )

Proof

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