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 |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) ∧ ( 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0 ) ) → ( 𝑥 + 𝑦 ) ∈ ℕ0 ) |
4 |
|
simp2 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → 𝐹 ∈ 𝐷 ) |
5 |
1
|
psrbag |
⊢ ( 𝐼 ∈ 𝑉 → ( 𝐹 ∈ 𝐷 ↔ ( 𝐹 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝐹 “ ℕ ) ∈ Fin ) ) ) |
6 |
5
|
3ad2ant1 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( 𝐹 ∈ 𝐷 ↔ ( 𝐹 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝐹 “ ℕ ) ∈ Fin ) ) ) |
7 |
4 6
|
mpbid |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( 𝐹 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝐹 “ ℕ ) ∈ Fin ) ) |
8 |
7
|
simpld |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → 𝐹 : 𝐼 ⟶ ℕ0 ) |
9 |
|
simp3 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → 𝐺 ∈ 𝐷 ) |
10 |
1
|
psrbag |
⊢ ( 𝐼 ∈ 𝑉 → ( 𝐺 ∈ 𝐷 ↔ ( 𝐺 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝐺 “ ℕ ) ∈ Fin ) ) ) |
11 |
10
|
3ad2ant1 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( 𝐺 ∈ 𝐷 ↔ ( 𝐺 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝐺 “ ℕ ) ∈ Fin ) ) ) |
12 |
9 11
|
mpbid |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( 𝐺 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝐺 “ ℕ ) ∈ Fin ) ) |
13 |
12
|
simpld |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → 𝐺 : 𝐼 ⟶ ℕ0 ) |
14 |
|
simp1 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → 𝐼 ∈ 𝑉 ) |
15 |
|
inidm |
⊢ ( 𝐼 ∩ 𝐼 ) = 𝐼 |
16 |
3 8 13 14 14 15
|
off |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( 𝐹 ∘f + 𝐺 ) : 𝐼 ⟶ ℕ0 ) |
17 |
|
frnnn0supp |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∘f + 𝐺 ) : 𝐼 ⟶ ℕ0 ) → ( ( 𝐹 ∘f + 𝐺 ) supp 0 ) = ( ◡ ( 𝐹 ∘f + 𝐺 ) “ ℕ ) ) |
18 |
14 16 17
|
syl2anc |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( ( 𝐹 ∘f + 𝐺 ) supp 0 ) = ( ◡ ( 𝐹 ∘f + 𝐺 ) “ ℕ ) ) |
19 |
|
fvexd |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑥 ) ∈ V ) |
20 |
|
fvexd |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑥 ) ∈ V ) |
21 |
8
|
feqmptd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → 𝐹 = ( 𝑥 ∈ 𝐼 ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
22 |
13
|
feqmptd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → 𝐺 = ( 𝑥 ∈ 𝐼 ↦ ( 𝐺 ‘ 𝑥 ) ) ) |
23 |
14 19 20 21 22
|
offval2 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( 𝐹 ∘f + 𝐺 ) = ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐺 ‘ 𝑥 ) ) ) ) |
24 |
23
|
oveq1d |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( ( 𝐹 ∘f + 𝐺 ) supp 0 ) = ( ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐺 ‘ 𝑥 ) ) ) supp 0 ) ) |
25 |
18 24
|
eqtr3d |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( ◡ ( 𝐹 ∘f + 𝐺 ) “ ℕ ) = ( ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐺 ‘ 𝑥 ) ) ) supp 0 ) ) |
26 |
|
frnnn0supp |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ ℕ0 ) → ( 𝐹 supp 0 ) = ( ◡ 𝐹 “ ℕ ) ) |
27 |
14 8 26
|
syl2anc |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( 𝐹 supp 0 ) = ( ◡ 𝐹 “ ℕ ) ) |
28 |
7
|
simprd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( ◡ 𝐹 “ ℕ ) ∈ Fin ) |
29 |
27 28
|
eqeltrd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( 𝐹 supp 0 ) ∈ Fin ) |
30 |
|
frnnn0supp |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ) → ( 𝐺 supp 0 ) = ( ◡ 𝐺 “ ℕ ) ) |
31 |
14 13 30
|
syl2anc |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( 𝐺 supp 0 ) = ( ◡ 𝐺 “ ℕ ) ) |
32 |
12
|
simprd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( ◡ 𝐺 “ ℕ ) ∈ Fin ) |
33 |
31 32
|
eqeltrd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( 𝐺 supp 0 ) ∈ Fin ) |
34 |
|
unfi |
⊢ ( ( ( 𝐹 supp 0 ) ∈ Fin ∧ ( 𝐺 supp 0 ) ∈ Fin ) → ( ( 𝐹 supp 0 ) ∪ ( 𝐺 supp 0 ) ) ∈ Fin ) |
35 |
29 33 34
|
syl2anc |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( ( 𝐹 supp 0 ) ∪ ( 𝐺 supp 0 ) ) ∈ Fin ) |
36 |
|
ssun1 |
⊢ ( 𝐹 supp 0 ) ⊆ ( ( 𝐹 supp 0 ) ∪ ( 𝐺 supp 0 ) ) |
37 |
36
|
a1i |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( 𝐹 supp 0 ) ⊆ ( ( 𝐹 supp 0 ) ∪ ( 𝐺 supp 0 ) ) ) |
38 |
|
c0ex |
⊢ 0 ∈ V |
39 |
38
|
a1i |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → 0 ∈ V ) |
40 |
8 37 14 39
|
suppssr |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) ∧ 𝑥 ∈ ( 𝐼 ∖ ( ( 𝐹 supp 0 ) ∪ ( 𝐺 supp 0 ) ) ) ) → ( 𝐹 ‘ 𝑥 ) = 0 ) |
41 |
|
ssun2 |
⊢ ( 𝐺 supp 0 ) ⊆ ( ( 𝐹 supp 0 ) ∪ ( 𝐺 supp 0 ) ) |
42 |
41
|
a1i |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( 𝐺 supp 0 ) ⊆ ( ( 𝐹 supp 0 ) ∪ ( 𝐺 supp 0 ) ) ) |
43 |
13 42 14 39
|
suppssr |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) ∧ 𝑥 ∈ ( 𝐼 ∖ ( ( 𝐹 supp 0 ) ∪ ( 𝐺 supp 0 ) ) ) ) → ( 𝐺 ‘ 𝑥 ) = 0 ) |
44 |
40 43
|
oveq12d |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) ∧ 𝑥 ∈ ( 𝐼 ∖ ( ( 𝐹 supp 0 ) ∪ ( 𝐺 supp 0 ) ) ) ) → ( ( 𝐹 ‘ 𝑥 ) + ( 𝐺 ‘ 𝑥 ) ) = ( 0 + 0 ) ) |
45 |
|
00id |
⊢ ( 0 + 0 ) = 0 |
46 |
44 45
|
eqtrdi |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) ∧ 𝑥 ∈ ( 𝐼 ∖ ( ( 𝐹 supp 0 ) ∪ ( 𝐺 supp 0 ) ) ) ) → ( ( 𝐹 ‘ 𝑥 ) + ( 𝐺 ‘ 𝑥 ) ) = 0 ) |
47 |
46 14
|
suppss2 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐺 ‘ 𝑥 ) ) ) supp 0 ) ⊆ ( ( 𝐹 supp 0 ) ∪ ( 𝐺 supp 0 ) ) ) |
48 |
35 47
|
ssfid |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) + ( 𝐺 ‘ 𝑥 ) ) ) supp 0 ) ∈ Fin ) |
49 |
25 48
|
eqeltrd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( ◡ ( 𝐹 ∘f + 𝐺 ) “ ℕ ) ∈ Fin ) |
50 |
1
|
psrbag |
⊢ ( 𝐼 ∈ 𝑉 → ( ( 𝐹 ∘f + 𝐺 ) ∈ 𝐷 ↔ ( ( 𝐹 ∘f + 𝐺 ) : 𝐼 ⟶ ℕ0 ∧ ( ◡ ( 𝐹 ∘f + 𝐺 ) “ ℕ ) ∈ Fin ) ) ) |
51 |
50
|
3ad2ant1 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( ( 𝐹 ∘f + 𝐺 ) ∈ 𝐷 ↔ ( ( 𝐹 ∘f + 𝐺 ) : 𝐼 ⟶ ℕ0 ∧ ( ◡ ( 𝐹 ∘f + 𝐺 ) “ ℕ ) ∈ Fin ) ) ) |
52 |
16 49 51
|
mpbir2and |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ) → ( 𝐹 ∘f + 𝐺 ) ∈ 𝐷 ) |