Step |
Hyp |
Ref |
Expression |
1 |
|
psrvscacl.s |
⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 ) |
2 |
|
psrvscacl.n |
⊢ · = ( ·𝑠 ‘ 𝑆 ) |
3 |
|
psrvscacl.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
4 |
|
psrvscacl.b |
⊢ 𝐵 = ( Base ‘ 𝑆 ) |
5 |
|
psrvscacl.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
6 |
|
psrvscacl.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐾 ) |
7 |
|
psrvscacl.y |
⊢ ( 𝜑 → 𝐹 ∈ 𝐵 ) |
8 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
9 |
3 8
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ) → ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ 𝐾 ) |
10 |
9
|
3expb |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ) ) → ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ 𝐾 ) |
11 |
5 10
|
sylan |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ) ) → ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ 𝐾 ) |
12 |
|
fconst6g |
⊢ ( 𝑋 ∈ 𝐾 → ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑋 } ) : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ 𝐾 ) |
13 |
6 12
|
syl |
⊢ ( 𝜑 → ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑋 } ) : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ 𝐾 ) |
14 |
|
eqid |
⊢ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
15 |
1 3 14 4 7
|
psrelbas |
⊢ ( 𝜑 → 𝐹 : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ 𝐾 ) |
16 |
|
ovex |
⊢ ( ℕ0 ↑m 𝐼 ) ∈ V |
17 |
16
|
rabex |
⊢ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∈ V |
18 |
17
|
a1i |
⊢ ( 𝜑 → { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∈ V ) |
19 |
|
inidm |
⊢ ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∩ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
20 |
11 13 15 18 18 19
|
off |
⊢ ( 𝜑 → ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑋 } ) ∘f ( .r ‘ 𝑅 ) 𝐹 ) : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ 𝐾 ) |
21 |
3
|
fvexi |
⊢ 𝐾 ∈ V |
22 |
21 17
|
elmap |
⊢ ( ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑋 } ) ∘f ( .r ‘ 𝑅 ) 𝐹 ) ∈ ( 𝐾 ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ↔ ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑋 } ) ∘f ( .r ‘ 𝑅 ) 𝐹 ) : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ 𝐾 ) |
23 |
20 22
|
sylibr |
⊢ ( 𝜑 → ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑋 } ) ∘f ( .r ‘ 𝑅 ) 𝐹 ) ∈ ( 𝐾 ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ) |
24 |
1 2 3 4 8 14 6 7
|
psrvsca |
⊢ ( 𝜑 → ( 𝑋 · 𝐹 ) = ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { 𝑋 } ) ∘f ( .r ‘ 𝑅 ) 𝐹 ) ) |
25 |
|
reldmpsr |
⊢ Rel dom mPwSer |
26 |
25 1 4
|
elbasov |
⊢ ( 𝐹 ∈ 𝐵 → ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) |
27 |
7 26
|
syl |
⊢ ( 𝜑 → ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) |
28 |
27
|
simpld |
⊢ ( 𝜑 → 𝐼 ∈ V ) |
29 |
1 3 14 4 28
|
psrbas |
⊢ ( 𝜑 → 𝐵 = ( 𝐾 ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ) |
30 |
23 24 29
|
3eltr4d |
⊢ ( 𝜑 → ( 𝑋 · 𝐹 ) ∈ 𝐵 ) |