Step | Hyp | Ref
| Expression |
1 | | smflimsuplem7.d |
. . 3
⊢ 𝐷 = {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} |
2 | 1 | a1i 11 |
. 2
⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) |
3 | | simpl 473 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) → 𝜑) |
4 | | rabidim2 39284 |
. . . . . . . . . 10
⊢ (𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} → (lim
sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) |
5 | 4 | adantl 482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) → (lim
sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) |
6 | | rabidim1 3117 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} → 𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
7 | | eliun 4524 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
8 | 6, 7 | sylib 208 |
. . . . . . . . . 10
⊢ (𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
9 | 8 | adantl 482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
10 | | nfv 1843 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑛(𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) |
11 | | nfv 1843 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑚𝜑 |
12 | | nfcv 2764 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑚lim
sup |
13 | | nfmpt1 4747 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) |
14 | 12, 13 | nffv 6198 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑚(lim
sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) |
15 | | nfcv 2764 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑚ℝ |
16 | 14, 15 | nfel 2777 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑚(lim
sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ |
17 | 11, 16 | nfan 1828 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑚(𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) |
18 | | nfv 1843 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑚 𝑛 ∈ 𝑍 |
19 | | nfcv 2764 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑚𝑥 |
20 | | nfii1 4551 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑚∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) |
21 | 19, 20 | nfel 2777 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑚 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) |
22 | 17, 18, 21 | nf3an 1831 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑚((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
23 | | nfv 1843 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑚 𝑘 ∈
(ℤ≥‘𝑛) |
24 | 22, 23 | nfan 1828 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑚(((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) |
25 | | simpl1l 1112 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝜑) |
26 | | smflimsuplem7.m |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑀 ∈ ℤ) |
27 | 25, 26 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑀 ∈ ℤ) |
28 | | smflimsuplem7.z |
. . . . . . . . . . . . . . . 16
⊢ 𝑍 =
(ℤ≥‘𝑀) |
29 | | smflimsuplem7.s |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑆 ∈ SAlg) |
30 | 25, 29 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑆 ∈ SAlg) |
31 | | smflimsuplem7.f |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) |
32 | 25, 31 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝐹:𝑍⟶(SMblFn‘𝑆)) |
33 | | smflimsuplem7.e |
. . . . . . . . . . . . . . . 16
⊢ 𝐸 = (𝑘 ∈ 𝑍 ↦ {𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑘)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ}) |
34 | | smflimsuplem7.h |
. . . . . . . . . . . . . . . 16
⊢ 𝐻 = (𝑘 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑘) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, <
))) |
35 | 28 | uztrn2 11705 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑘 ∈ 𝑍) |
36 | 35 | 3ad2antl2 1224 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑘 ∈ 𝑍) |
37 | | simpl1r 1113 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) |
38 | | uzss 11708 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈
(ℤ≥‘𝑛) → (ℤ≥‘𝑘) ⊆
(ℤ≥‘𝑛)) |
39 | | iinss1 4533 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((ℤ≥‘𝑘) ⊆ (ℤ≥‘𝑛) → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ⊆ ∩
𝑚 ∈
(ℤ≥‘𝑘)dom (𝐹‘𝑚)) |
40 | 38, 39 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈
(ℤ≥‘𝑛) → ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ⊆ ∩
𝑚 ∈
(ℤ≥‘𝑘)dom (𝐹‘𝑚)) |
41 | 40 | adantl 482 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ⊆ ∩
𝑚 ∈
(ℤ≥‘𝑘)dom (𝐹‘𝑚)) |
42 | | simpl 473 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
43 | 41, 42 | sseldd 3604 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑘)dom (𝐹‘𝑚)) |
44 | 43 | 3ad2antl3 1225 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑘)dom (𝐹‘𝑚)) |
45 | 24, 27, 28, 30, 32, 33, 34, 36, 37, 44 | smflimsuplem2 41027 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑥 ∈ dom (𝐻‘𝑘)) |
46 | 45 | ralrimiva 2966 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) → ∀𝑘 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐻‘𝑘)) |
47 | | vex 3203 |
. . . . . . . . . . . . . . 15
⊢ 𝑥 ∈ V |
48 | | eliin 4525 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ V → (𝑥 ∈ ∩ 𝑘 ∈ (ℤ≥‘𝑛)dom (𝐻‘𝑘) ↔ ∀𝑘 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐻‘𝑘))) |
49 | 47, 48 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ∩ 𝑘 ∈ (ℤ≥‘𝑛)dom (𝐻‘𝑘) ↔ ∀𝑘 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐻‘𝑘)) |
50 | 46, 49 | sylibr 224 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) → 𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) |
51 | 50 | 3exp 1264 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → (𝑛 ∈ 𝑍 → (𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) → 𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)))) |
52 | 10, 51 | reximdai 3012 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → (∃𝑛 ∈ 𝑍 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘))) |
53 | 52 | imp 445 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) |
54 | | eliun 4524 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ↔ ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) |
55 | 53, 54 | sylibr 224 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) → 𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) |
56 | 3, 5, 9, 55 | syl21anc 1325 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) |
57 | 7 | biimpi 206 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
58 | 6, 57 | syl 17 |
. . . . . . . . . 10
⊢ (𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
59 | 58 | adantl 482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
60 | | nfv 1843 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛𝜑 |
61 | | nfcv 2764 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑛𝑥 |
62 | | nfv 1843 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑛(lim
sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ |
63 | | nfiu1 4550 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑛∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) |
64 | 62, 63 | nfrab 3123 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑛{𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} |
65 | 61, 64 | nfel 2777 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} |
66 | 60, 65 | nfan 1828 |
. . . . . . . . . 10
⊢
Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) |
67 | | nfv 1843 |
. . . . . . . . . 10
⊢
Ⅎ𝑛(𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ |
68 | | nfv 1843 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
69 | | simp1l 1085 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) → 𝜑) |
70 | 69, 26 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) → 𝑀 ∈ ℤ) |
71 | 69, 29 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) → 𝑆 ∈ SAlg) |
72 | 69, 31 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) → 𝐹:𝑍⟶(SMblFn‘𝑆)) |
73 | | simp1r 1086 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) |
74 | | simp2 1062 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) → 𝑛 ∈ 𝑍) |
75 | | simp3 1063 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) → 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
76 | 68, 22, 70, 28, 71, 72, 33, 34, 73, 74, 75 | smflimsuplem6 41031 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) → (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) |
77 | 76 | 3exp 1264 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → (𝑛 ∈ 𝑍 → (𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) → (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ))) |
78 | 5, 77 | syldan 487 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) → (𝑛 ∈ 𝑍 → (𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) → (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ))) |
79 | 66, 67, 78 | rexlimd 3026 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) → (∃𝑛 ∈ 𝑍 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) → (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ )) |
80 | 59, 79 | mpd 15 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) → (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) |
81 | 56, 80 | jca 554 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) → (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ )) |
82 | | rabid 3116 |
. . . . . . 7
⊢ (𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } ↔ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ )) |
83 | 81, 82 | sylibr 224 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) → 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }) |
84 | 83 | ex 450 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} → 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ })) |
85 | | ssrab2 3687 |
. . . . . . . . . 10
⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) |
86 | 85 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) |
87 | 28 | eluzelz2 39627 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ) |
88 | 87 | uzidd 39631 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑛)) |
89 | 88 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ (ℤ≥‘𝑛)) |
90 | | nfv 1843 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑥(𝜑 ∧ 𝑛 ∈ 𝑍) |
91 | | xrltso 11974 |
. . . . . . . . . . . . . . . . . . 19
⊢ < Or
ℝ* |
92 | 91 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (𝐸‘𝑛)) → < Or
ℝ*) |
93 | 92 | supexd 8359 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (𝐸‘𝑛)) → sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
V) |
94 | | eqid 2622 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, <
)) |
95 | 90, 93, 94 | fnmptd 39434 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )) Fn (𝐸‘𝑛)) |
96 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑛 → (𝐸‘𝑘) = (𝐸‘𝑛)) |
97 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 = 𝑛 → (ℤ≥‘𝑘) =
(ℤ≥‘𝑛)) |
98 | 97 | mpteq1d 4738 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = 𝑛 → (𝑚 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥))) |
99 | 98 | rneqd 5353 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑛 → ran (𝑚 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥))) |
100 | 99 | supeq1d 8352 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑛 → sup(ran (𝑚 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) = sup(ran
(𝑚 ∈
(ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, <
)) |
101 | 96, 100 | mpteq12dv 4733 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑛 → (𝑥 ∈ (𝐸‘𝑘) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, <
))) |
102 | | fvex 6201 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐸‘𝑛) ∈ V |
103 | 102 | mptex 6486 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )) ∈
V |
104 | 101, 34, 103 | fvmpt 6282 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ 𝑍 → (𝐻‘𝑛) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, <
))) |
105 | 104 | adantl 482 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, <
))) |
106 | 105 | fneq1d 5981 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐻‘𝑛) Fn (𝐸‘𝑛) ↔ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )) Fn (𝐸‘𝑛))) |
107 | 95, 106 | mpbird 247 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) Fn (𝐸‘𝑛)) |
108 | 107 | fndmd 39441 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐻‘𝑛) = (𝐸‘𝑛)) |
109 | 97 | iineq1d 39267 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑛 → ∩
𝑚 ∈
(ℤ≥‘𝑘)dom (𝐹‘𝑚) = ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
110 | 109 | eleq2d 2687 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑛 → (𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑘)dom (𝐹‘𝑚) ↔ 𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚))) |
111 | 100 | eleq1d 2686 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑛 → (sup(ran (𝑚 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ ↔ sup(ran (𝑚
∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ)) |
112 | 110, 111 | anbi12d 747 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑛 → ((𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑘)dom (𝐹‘𝑚) ∧ sup(ran (𝑚 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ) ↔ (𝑥 ∈
∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∧ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ))) |
113 | 112 | rabbidva2 3186 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑛 → {𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑘)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ} = {𝑥 ∈
∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ}) |
114 | | id 22 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ 𝑍) |
115 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 𝑦 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑦)) |
116 | 115 | mpteq2dv 4745 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑦 → (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦))) |
117 | 116 | rneqd 5353 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 𝑦 → ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦))) |
118 | 117 | supeq1d 8352 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 𝑦 → sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) = sup(ran
(𝑚 ∈
(ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ*, <
)) |
119 | 118 | eleq1d 2686 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑦 → (sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ ↔ sup(ran (𝑚
∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ*, < ) ∈
ℝ)) |
120 | 119 | cbvrabv 3199 |
. . . . . . . . . . . . . . . . . 18
⊢ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ} = {𝑦 ∈
∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ*, < ) ∈
ℝ} |
121 | | ne0i 3921 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈
(ℤ≥‘𝑛) → (ℤ≥‘𝑛) ≠ ∅) |
122 | 88, 121 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ≠ ∅) |
123 | | fvex 6201 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐹‘𝑚) ∈ V |
124 | 123 | dmex 7099 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ dom
(𝐹‘𝑚) ∈ V |
125 | 124 | rgenw 2924 |
. . . . . . . . . . . . . . . . . . . 20
⊢
∀𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V |
126 | 125 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ 𝑍 → ∀𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V) |
127 | 122, 126 | iinexd 39318 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ 𝑍 → ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V) |
128 | 120, 127 | rabexd 4814 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ 𝑍 → {𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ} ∈ V) |
129 | 33, 113, 114, 128 | fvmptd3 39447 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ 𝑍 → (𝐸‘𝑛) = {𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ}) |
130 | 129 | adantl 482 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) = {𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ}) |
131 | | ssrab2 3687 |
. . . . . . . . . . . . . . . 16
⊢ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ} ⊆ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) |
132 | 131 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → {𝑥 ∈ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈
ℝ} ⊆ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
133 | 130, 132 | eqsstrd 3639 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
134 | 108, 133 | eqsstrd 3639 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐻‘𝑛) ⊆ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
135 | | fveq2 6191 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑛 → (𝐻‘𝑘) = (𝐻‘𝑛)) |
136 | 135 | dmeqd 5326 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑛 → dom (𝐻‘𝑘) = dom (𝐻‘𝑛)) |
137 | 136 | sseq1d 3632 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑛 → (dom (𝐻‘𝑘) ⊆ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ dom (𝐻‘𝑛) ⊆ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚))) |
138 | 137 | rspcev 3309 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈
(ℤ≥‘𝑛) ∧ dom (𝐻‘𝑛) ⊆ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) → ∃𝑘 ∈ (ℤ≥‘𝑛)dom (𝐻‘𝑘) ⊆ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
139 | 89, 134, 138 | syl2anc 693 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑘 ∈ (ℤ≥‘𝑛)dom (𝐻‘𝑘) ⊆ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
140 | | iinss 4571 |
. . . . . . . . . . . 12
⊢
(∃𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ⊆ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) → ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ⊆ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
141 | 139, 140 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ⊆ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
142 | 141 | ralrimiva 2966 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ⊆ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
143 | | ss2iun 4536 |
. . . . . . . . . 10
⊢
(∀𝑛 ∈
𝑍 ∩ 𝑘 ∈ (ℤ≥‘𝑛)dom (𝐻‘𝑘) ⊆ ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) → ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ⊆ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
144 | 142, 143 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ⊆ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
145 | 86, 144 | sstrd 3613 |
. . . . . . . 8
⊢ (𝜑 → {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
146 | 82 | simplbi 476 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) |
147 | 54 | biimpi 206 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) |
148 | 146, 147 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) |
149 | 148 | adantl 482 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) |
150 | | nfiu1 4550 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑛∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) |
151 | 67, 150 | nfrab 3123 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑛{𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } |
152 | 61, 151 | nfel 2777 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑛 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } |
153 | 60, 152 | nfan 1828 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }) |
154 | 82 | simprbi 480 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } → (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) |
155 | | nfv 1843 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑘𝜑 |
156 | | nfmpt1 4747 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘(𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) |
157 | | nfcv 2764 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘dom
⇝ |
158 | 156, 157 | nfel 2777 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑘(𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ |
159 | 155, 158 | nfan 1828 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑘(𝜑 ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) |
160 | | nfv 1843 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑘 𝑛 ∈ 𝑍 |
161 | | nfcv 2764 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑘𝑥 |
162 | | nfii1 4551 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑘∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) |
163 | 161, 162 | nfel 2777 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑘 𝑥 ∈ ∩ 𝑘 ∈ (ℤ≥‘𝑛)dom (𝐻‘𝑘) |
164 | 159, 160,
163 | nf3an 1831 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘((𝜑 ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) |
165 | 26 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑀 ∈ ℤ) |
166 | 165 | 3adant3 1081 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) → 𝑀 ∈ ℤ) |
167 | 166 | 3adant1r 1319 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) → 𝑀 ∈ ℤ) |
168 | 29 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg) |
169 | 168 | 3adant3 1081 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) → 𝑆 ∈ SAlg) |
170 | 169 | 3adant1r 1319 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) → 𝑆 ∈ SAlg) |
171 | 31 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐹:𝑍⟶(SMblFn‘𝑆)) |
172 | 171 | 3adant3 1081 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) → 𝐹:𝑍⟶(SMblFn‘𝑆)) |
173 | 172 | 3adant1r 1319 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) → 𝐹:𝑍⟶(SMblFn‘𝑆)) |
174 | | simp2 1062 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) → 𝑛 ∈ 𝑍) |
175 | | simp3 1063 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) → 𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) |
176 | | simp1r 1086 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) → (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) |
177 | 164, 167,
28, 170, 173, 33, 34, 174, 175, 176 | smflimsuplem4 41029 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘)) → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) |
178 | 177 | 3exp 1264 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) → (𝑛 ∈ 𝑍 → (𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ))) |
179 | 154, 178 | sylan2 491 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }) → (𝑛 ∈ 𝑍 → (𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ))) |
180 | 153, 62, 179 | rexlimd 3026 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }) → (∃𝑛 ∈ 𝑍 𝑥 ∈ ∩
𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)) |
181 | 149, 180 | mpd 15 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }) → (lim
sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) |
182 | 181 | ralrimiva 2966 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) |
183 | 145, 182 | jca 554 |
. . . . . . 7
⊢ (𝜑 → ({𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∧ ∀𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)) |
184 | | nfrab1 3122 |
. . . . . . . 8
⊢
Ⅎ𝑥{𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } |
185 | | nfcv 2764 |
. . . . . . . 8
⊢
Ⅎ𝑥∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) |
186 | 184, 185 | ssrabf 39298 |
. . . . . . 7
⊢ ({𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } ⊆ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} ↔ ({𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∧ ∀𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)) |
187 | 183, 186 | sylibr 224 |
. . . . . 6
⊢ (𝜑 → {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } ⊆ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) |
188 | 187 | sseld 3602 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } → 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ})) |
189 | 84, 188 | impbid 202 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} ↔ 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ })) |
190 | 189 | alrimiv 1855 |
. . 3
⊢ (𝜑 → ∀𝑥(𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} ↔ 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ })) |
191 | | nfrab1 3122 |
. . . 4
⊢
Ⅎ𝑥{𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} |
192 | 191, 184 | dfcleqf 39255 |
. . 3
⊢ ({𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} = {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } ↔ ∀𝑥(𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} ↔ 𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ })) |
193 | 190, 192 | sylibr 224 |
. 2
⊢ (𝜑 → {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} = {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }) |
194 | 2, 193 | eqtrd 2656 |
1
⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑘 ∈
(ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }) |