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Metamath Proof Explorer

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 )