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