Step | Hyp | Ref
| Expression |
1 | | nn0addcl 11205 |
. . . 4
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (𝑥 + 𝑦) ∈
ℕ0) |
2 | 1 | adantl 481 |
. . 3
⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ (𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0))
→ (𝑥 + 𝑦) ∈
ℕ0) |
3 | | simp2 1055 |
. . . . 5
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐹 ∈ 𝐷) |
4 | | psrbag.d |
. . . . . . 7
⊢ 𝐷 = {𝑓 ∈ (ℕ0
↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈
Fin} |
5 | 4 | psrbag 19185 |
. . . . . 6
⊢ (𝐼 ∈ 𝑉 → (𝐹 ∈ 𝐷 ↔ (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈
Fin))) |
6 | 5 | 3ad2ant1 1075 |
. . . . 5
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐹 ∈ 𝐷 ↔ (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈
Fin))) |
7 | 3, 6 | mpbid 221 |
. . . 4
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈
Fin)) |
8 | 7 | simpld 474 |
. . 3
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐹:𝐼⟶ℕ0) |
9 | | simp3 1056 |
. . . . 5
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐺 ∈ 𝐷) |
10 | 4 | psrbag 19185 |
. . . . . 6
⊢ (𝐼 ∈ 𝑉 → (𝐺 ∈ 𝐷 ↔ (𝐺:𝐼⟶ℕ0 ∧ (◡𝐺 “ ℕ) ∈
Fin))) |
11 | 10 | 3ad2ant1 1075 |
. . . . 5
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐺 ∈ 𝐷 ↔ (𝐺:𝐼⟶ℕ0 ∧ (◡𝐺 “ ℕ) ∈
Fin))) |
12 | 9, 11 | mpbid 221 |
. . . 4
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐺:𝐼⟶ℕ0 ∧ (◡𝐺 “ ℕ) ∈
Fin)) |
13 | 12 | simpld 474 |
. . 3
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐺:𝐼⟶ℕ0) |
14 | | simp1 1054 |
. . 3
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐼 ∈ 𝑉) |
15 | | inidm 3784 |
. . 3
⊢ (𝐼 ∩ 𝐼) = 𝐼 |
16 | 2, 8, 13, 14, 14, 15 | off 6810 |
. 2
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐹 ∘𝑓 + 𝐺):𝐼⟶ℕ0) |
17 | | frnnn0supp 11226 |
. . . . 5
⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∘𝑓 + 𝐺):𝐼⟶ℕ0) → ((𝐹 ∘𝑓 +
𝐺) supp 0) = (◡(𝐹 ∘𝑓 + 𝐺) “
ℕ)) |
18 | 14, 16, 17 | syl2anc 691 |
. . . 4
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → ((𝐹 ∘𝑓 + 𝐺) supp 0) = (◡(𝐹 ∘𝑓 + 𝐺) “
ℕ)) |
19 | | fvex 6113 |
. . . . . . 7
⊢ (𝐹‘𝑥) ∈ V |
20 | 19 | a1i 11 |
. . . . . 6
⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ V) |
21 | | fvex 6113 |
. . . . . . 7
⊢ (𝐺‘𝑥) ∈ V |
22 | 21 | a1i 11 |
. . . . . 6
⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ V) |
23 | 8 | feqmptd 6159 |
. . . . . 6
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥))) |
24 | 13 | feqmptd 6159 |
. . . . . 6
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐺 = (𝑥 ∈ 𝐼 ↦ (𝐺‘𝑥))) |
25 | 14, 20, 22, 23, 24 | offval2 6812 |
. . . . 5
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐹 ∘𝑓 + 𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥) + (𝐺‘𝑥)))) |
26 | 25 | oveq1d 6564 |
. . . 4
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → ((𝐹 ∘𝑓 + 𝐺) supp 0) = ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥) + (𝐺‘𝑥))) supp 0)) |
27 | 18, 26 | eqtr3d 2646 |
. . 3
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (◡(𝐹 ∘𝑓 + 𝐺) “ ℕ) = ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥) + (𝐺‘𝑥))) supp 0)) |
28 | | frnnn0supp 11226 |
. . . . . . 7
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ)) |
29 | 14, 8, 28 | syl2anc 691 |
. . . . . 6
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐹 supp 0) = (◡𝐹 “ ℕ)) |
30 | 7 | simprd 478 |
. . . . . 6
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (◡𝐹 “ ℕ) ∈
Fin) |
31 | 29, 30 | eqeltrd 2688 |
. . . . 5
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐹 supp 0) ∈ Fin) |
32 | | frnnn0supp 11226 |
. . . . . . 7
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺:𝐼⟶ℕ0) → (𝐺 supp 0) = (◡𝐺 “ ℕ)) |
33 | 14, 13, 32 | syl2anc 691 |
. . . . . 6
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐺 supp 0) = (◡𝐺 “ ℕ)) |
34 | 12 | simprd 478 |
. . . . . 6
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (◡𝐺 “ ℕ) ∈
Fin) |
35 | 33, 34 | eqeltrd 2688 |
. . . . 5
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐺 supp 0) ∈ Fin) |
36 | | unfi 8112 |
. . . . 5
⊢ (((𝐹 supp 0) ∈ Fin ∧ (𝐺 supp 0) ∈ Fin) →
((𝐹 supp 0) ∪ (𝐺 supp 0)) ∈
Fin) |
37 | 31, 35, 36 | syl2anc 691 |
. . . 4
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ∈ Fin) |
38 | | ssun1 3738 |
. . . . . . . . 9
⊢ (𝐹 supp 0) ⊆ ((𝐹 supp 0) ∪ (𝐺 supp 0)) |
39 | 38 | a1i 11 |
. . . . . . . 8
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐹 supp 0) ⊆ ((𝐹 supp 0) ∪ (𝐺 supp 0))) |
40 | | c0ex 9913 |
. . . . . . . . 9
⊢ 0 ∈
V |
41 | 40 | a1i 11 |
. . . . . . . 8
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 0 ∈ V) |
42 | 8, 39, 14, 41 | suppssr 7213 |
. . . . . . 7
⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → (𝐹‘𝑥) = 0) |
43 | | ssun2 3739 |
. . . . . . . . 9
⊢ (𝐺 supp 0) ⊆ ((𝐹 supp 0) ∪ (𝐺 supp 0)) |
44 | 43 | a1i 11 |
. . . . . . . 8
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐺 supp 0) ⊆ ((𝐹 supp 0) ∪ (𝐺 supp 0))) |
45 | 13, 44, 14, 41 | suppssr 7213 |
. . . . . . 7
⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → (𝐺‘𝑥) = 0) |
46 | 42, 45 | oveq12d 6567 |
. . . . . 6
⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → ((𝐹‘𝑥) + (𝐺‘𝑥)) = (0 + 0)) |
47 | | 00id 10090 |
. . . . . 6
⊢ (0 + 0) =
0 |
48 | 46, 47 | syl6eq 2660 |
. . . . 5
⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ 𝑥 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → ((𝐹‘𝑥) + (𝐺‘𝑥)) = 0) |
49 | 48, 14 | suppss2 7216 |
. . . 4
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥) + (𝐺‘𝑥))) supp 0) ⊆ ((𝐹 supp 0) ∪ (𝐺 supp 0))) |
50 | | ssfi 8065 |
. . . 4
⊢ ((((𝐹 supp 0) ∪ (𝐺 supp 0)) ∈ Fin ∧
((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥) + (𝐺‘𝑥))) supp 0) ⊆ ((𝐹 supp 0) ∪ (𝐺 supp 0))) → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥) + (𝐺‘𝑥))) supp 0) ∈ Fin) |
51 | 37, 49, 50 | syl2anc 691 |
. . 3
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥) + (𝐺‘𝑥))) supp 0) ∈ Fin) |
52 | 27, 51 | eqeltrd 2688 |
. 2
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (◡(𝐹 ∘𝑓 + 𝐺) “ ℕ) ∈
Fin) |
53 | 4 | psrbag 19185 |
. . 3
⊢ (𝐼 ∈ 𝑉 → ((𝐹 ∘𝑓 + 𝐺) ∈ 𝐷 ↔ ((𝐹 ∘𝑓 + 𝐺):𝐼⟶ℕ0 ∧ (◡(𝐹 ∘𝑓 + 𝐺) “ ℕ) ∈
Fin))) |
54 | 53 | 3ad2ant1 1075 |
. 2
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → ((𝐹 ∘𝑓 + 𝐺) ∈ 𝐷 ↔ ((𝐹 ∘𝑓 + 𝐺):𝐼⟶ℕ0 ∧ (◡(𝐹 ∘𝑓 + 𝐺) “ ℕ) ∈
Fin))) |
55 | 16, 52, 54 | mpbir2and 959 |
1
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐹 ∘𝑓 + 𝐺) ∈ 𝐷) |