Theorem psrmulcl

Description: Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014)

	Ref	Expression
Hypotheses	psrmulcl.s	⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
	psrmulcl.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
	psrmulcl.t	⊢ · = ( .r ‘ 𝑆 )
	psrmulcl.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
	psrmulcl.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	psrmulcl.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	psrmulcl	⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		psrmulcl.s	⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
2		psrmulcl.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
3		psrmulcl.t	⊢ · = ( .r ‘ 𝑆 )
4		psrmulcl.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
5		psrmulcl.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		psrmulcl.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7		eqid	⊢ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
8	1 2 3 4 5 6 7	psrmulcllem	⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) ∈ 𝐵 )