Step |
Hyp |
Ref |
Expression |
1 |
|
psrbag.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
2 |
|
nn0addcl |
⊢ ( ( 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0 ) → ( 𝑥 + 𝑦 ) ∈ ℕ0 ) |
3 |
2
|
adantl |
⊢ ( ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) ∧ ( 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0 ) ) → ( 𝑥 + 𝑦 ) ∈ ℕ0 ) |
4 |
1
|
psrbagf |
⊢ ( 𝐹 ∈ 𝐷 → 𝐹 : 𝐼 ⟶ ℕ0 ) |
5 |
4
|
adantr |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → 𝐹 : 𝐼 ⟶ ℕ0 ) |
6 |
1
|
psrbagf |
⊢ ( 𝐺 ∈ 𝐷 → 𝐺 : 𝐼 ⟶ ℕ0 ) |
7 |
6
|
adantl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → 𝐺 : 𝐼 ⟶ ℕ0 ) |
8 |
|
simpl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → 𝐹 ∈ 𝐷 ) |
9 |
5
|
ffnd |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → 𝐹 Fn 𝐼 ) |
10 |
8 9
|
fndmexd |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → 𝐼 ∈ V ) |
11 |
|
inidm |
⊢ ( 𝐼 ∩ 𝐼 ) = 𝐼 |
12 |
3 5 7 10 10 11
|
off |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( 𝐹 ∘f + 𝐺 ) : 𝐼 ⟶ ℕ0 ) |
13 |
|
ovex |
⊢ ( 𝐹 ∘f + 𝐺 ) ∈ V |
14 |
|
frnnn0suppg |
⊢ ( ( ( 𝐹 ∘f + 𝐺 ) ∈ V ∧ ( 𝐹 ∘f + 𝐺 ) : 𝐼 ⟶ ℕ0 ) → ( ( 𝐹 ∘f + 𝐺 ) supp 0 ) = ( ◡ ( 𝐹 ∘f + 𝐺 ) “ ℕ ) ) |
15 |
13 12 14
|
sylancr |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( ( 𝐹 ∘f + 𝐺 ) supp 0 ) = ( ◡ ( 𝐹 ∘f + 𝐺 ) “ ℕ ) ) |
16 |
1
|
psrbagfsupp |
⊢ ( 𝐹 ∈ 𝐷 → 𝐹 finSupp 0 ) |
17 |
16
|
adantr |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → 𝐹 finSupp 0 ) |
18 |
1
|
psrbagfsupp |
⊢ ( 𝐺 ∈ 𝐷 → 𝐺 finSupp 0 ) |
19 |
18
|
adantl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → 𝐺 finSupp 0 ) |
20 |
17 19
|
fsuppunfi |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( ( 𝐹 supp 0 ) ∪ ( 𝐺 supp 0 ) ) ∈ Fin ) |
21 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
22 |
21
|
a1i |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → 0 ∈ ℕ0 ) |
23 |
|
00id |
⊢ ( 0 + 0 ) = 0 |
24 |
23
|
a1i |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( 0 + 0 ) = 0 ) |
25 |
10 22 5 7 24
|
suppofssd |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( ( 𝐹 ∘f + 𝐺 ) supp 0 ) ⊆ ( ( 𝐹 supp 0 ) ∪ ( 𝐺 supp 0 ) ) ) |
26 |
20 25
|
ssfid |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( ( 𝐹 ∘f + 𝐺 ) supp 0 ) ∈ Fin ) |
27 |
15 26
|
eqeltrrd |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( ◡ ( 𝐹 ∘f + 𝐺 ) “ ℕ ) ∈ Fin ) |
28 |
1
|
psrbag |
⊢ ( 𝐼 ∈ V → ( ( 𝐹 ∘f + 𝐺 ) ∈ 𝐷 ↔ ( ( 𝐹 ∘f + 𝐺 ) : 𝐼 ⟶ ℕ0 ∧ ( ◡ ( 𝐹 ∘f + 𝐺 ) “ ℕ ) ∈ Fin ) ) ) |
29 |
10 28
|
syl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( ( 𝐹 ∘f + 𝐺 ) ∈ 𝐷 ↔ ( ( 𝐹 ∘f + 𝐺 ) : 𝐼 ⟶ ℕ0 ∧ ( ◡ ( 𝐹 ∘f + 𝐺 ) “ ℕ ) ∈ Fin ) ) ) |
30 |
12 27 29
|
mpbir2and |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( 𝐹 ∘f + 𝐺 ) ∈ 𝐷 ) |