Metamath Proof Explorer


Theorem psrmulcllem

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 · = ( .r𝑆 )
psrmulcl.r ( 𝜑𝑅 ∈ Ring )
psrmulcl.x ( 𝜑𝑋𝐵 )
psrmulcl.y ( 𝜑𝑌𝐵 )
psrmulcl.d 𝐷 = { 𝑓 ∈ ( ℕ0m 𝐼 ) ∣ ( 𝑓 “ ℕ ) ∈ Fin }
Assertion psrmulcllem ( 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 · = ( .r𝑆 )
4 psrmulcl.r ( 𝜑𝑅 ∈ Ring )
5 psrmulcl.x ( 𝜑𝑋𝐵 )
6 psrmulcl.y ( 𝜑𝑌𝐵 )
7 psrmulcl.d 𝐷 = { 𝑓 ∈ ( ℕ0m 𝐼 ) ∣ ( 𝑓 “ ℕ ) ∈ Fin }
8 eqid ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
9 1 8 7 2 5 psrelbas ( 𝜑𝑋 : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
10 1 8 7 2 6 psrelbas ( 𝜑𝑌 : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
11 7 4 9 10 rhmpsrlem2 ( ( 𝜑𝑘𝐷 ) → ( 𝑅 Σg ( 𝑥 ∈ { 𝑦𝐷𝑦r𝑘 } ↦ ( ( 𝑋𝑥 ) ( .r𝑅 ) ( 𝑌 ‘ ( 𝑘f𝑥 ) ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
12 11 fmpttd ( 𝜑 → ( 𝑘𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦𝐷𝑦r𝑘 } ↦ ( ( 𝑋𝑥 ) ( .r𝑅 ) ( 𝑌 ‘ ( 𝑘f𝑥 ) ) ) ) ) ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
13 fvex ( Base ‘ 𝑅 ) ∈ V
14 ovex ( ℕ0m 𝐼 ) ∈ V
15 7 14 rabex2 𝐷 ∈ V
16 13 15 elmap ( ( 𝑘𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦𝐷𝑦r𝑘 } ↦ ( ( 𝑋𝑥 ) ( .r𝑅 ) ( 𝑌 ‘ ( 𝑘f𝑥 ) ) ) ) ) ) ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐷 ) ↔ ( 𝑘𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦𝐷𝑦r𝑘 } ↦ ( ( 𝑋𝑥 ) ( .r𝑅 ) ( 𝑌 ‘ ( 𝑘f𝑥 ) ) ) ) ) ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
17 12 16 sylibr ( 𝜑 → ( 𝑘𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦𝐷𝑦r𝑘 } ↦ ( ( 𝑋𝑥 ) ( .r𝑅 ) ( 𝑌 ‘ ( 𝑘f𝑥 ) ) ) ) ) ) ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐷 ) )
18 eqid ( .r𝑅 ) = ( .r𝑅 )
19 1 2 18 3 7 5 6 psrmulfval ( 𝜑 → ( 𝑋 · 𝑌 ) = ( 𝑘𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦𝐷𝑦r𝑘 } ↦ ( ( 𝑋𝑥 ) ( .r𝑅 ) ( 𝑌 ‘ ( 𝑘f𝑥 ) ) ) ) ) ) )
20 reldmpsr Rel dom mPwSer
21 20 1 2 elbasov ( 𝑋𝐵 → ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) )
22 5 21 syl ( 𝜑 → ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) )
23 22 simpld ( 𝜑𝐼 ∈ V )
24 1 8 7 2 23 psrbas ( 𝜑𝐵 = ( ( Base ‘ 𝑅 ) ↑m 𝐷 ) )
25 17 19 24 3eltr4d ( 𝜑 → ( 𝑋 · 𝑌 ) ∈ 𝐵 )