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 |
|
psrmulcl.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
8 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
9 |
1 8 7 2 5
|
psrelbas |
⊢ ( 𝜑 → 𝑋 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
10 |
1 8 7 2 6
|
psrelbas |
⊢ ( 𝜑 → 𝑌 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
11 |
7 4 9 10
|
rhmpsrlem2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ∈ ( Base ‘ 𝑅 ) ) |
12 |
11
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
13 |
|
fvex |
⊢ ( Base ‘ 𝑅 ) ∈ V |
14 |
|
ovex |
⊢ ( ℕ0 ↑m 𝐼 ) ∈ V |
15 |
7 14
|
rabex2 |
⊢ 𝐷 ∈ V |
16 |
13 15
|
elmap |
⊢ ( ( 𝑘 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐷 ) ↔ ( 𝑘 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
17 |
12 16
|
sylibr |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐷 ) ) |
18 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
19 |
1 2 18 3 7 5 6
|
psrmulfval |
⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) = ( 𝑘 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) |
20 |
|
reldmpsr |
⊢ Rel dom mPwSer |
21 |
20 1 2
|
elbasov |
⊢ ( 𝑋 ∈ 𝐵 → ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) |
22 |
5 21
|
syl |
⊢ ( 𝜑 → ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) |
23 |
22
|
simpld |
⊢ ( 𝜑 → 𝐼 ∈ V ) |
24 |
1 8 7 2 23
|
psrbas |
⊢ ( 𝜑 → 𝐵 = ( ( Base ‘ 𝑅 ) ↑m 𝐷 ) ) |
25 |
17 19 24
|
3eltr4d |
⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) ∈ 𝐵 ) |