Step | Hyp | Ref
| Expression |
1 | | ulmdv.m |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℤ) |
2 | | uzid 11702 |
. . . . . 6
⊢ (𝑀 ∈ ℤ → 𝑀 ∈
(ℤ≥‘𝑀)) |
3 | 1, 2 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
4 | | ulmdv.z |
. . . . 5
⊢ 𝑍 =
(ℤ≥‘𝑀) |
5 | 3, 4 | syl6eleqr 2712 |
. . . 4
⊢ (𝜑 → 𝑀 ∈ 𝑍) |
6 | | ulmdv.s |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
7 | | ulmdv.f |
. . . . . . 7
⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑𝑚
𝑋)) |
8 | | ulmdv.g |
. . . . . . 7
⊢ (𝜑 → 𝐺:𝑋⟶ℂ) |
9 | | ulmdv.l |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧)) |
10 | | ulmdv.u |
. . . . . . 7
⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝𝑢‘𝑋)𝐻) |
11 | 4, 6, 1, 7, 8, 9, 10 | ulmdvlem2 24155 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom (𝑆 D (𝐹‘𝑘)) = 𝑋) |
12 | | recnprss 23668 |
. . . . . . . . 9
⊢ (𝑆 ∈ {ℝ, ℂ}
→ 𝑆 ⊆
ℂ) |
13 | 6, 12 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
14 | 13 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑆 ⊆ ℂ) |
15 | 7 | ffvelrnda 6359 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (ℂ ↑𝑚
𝑋)) |
16 | | elmapi 7879 |
. . . . . . . 8
⊢ ((𝐹‘𝑘) ∈ (ℂ ↑𝑚
𝑋) → (𝐹‘𝑘):𝑋⟶ℂ) |
17 | 15, 16 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘):𝑋⟶ℂ) |
18 | | dvbsss 23666 |
. . . . . . . 8
⊢ dom
(𝑆 D (𝐹‘𝑘)) ⊆ 𝑆 |
19 | 11, 18 | syl6eqssr 3656 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑋 ⊆ 𝑆) |
20 | | eqid 2622 |
. . . . . . 7
⊢
((TopOpen‘ℂfld) ↾t 𝑆) =
((TopOpen‘ℂfld) ↾t 𝑆) |
21 | | eqid 2622 |
. . . . . . 7
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
22 | 14, 17, 19, 20, 21 | dvbssntr 23664 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom (𝑆 D (𝐹‘𝑘)) ⊆
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋)) |
23 | 11, 22 | eqsstr3d 3640 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑋 ⊆
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋)) |
24 | 23 | ralrimiva 2966 |
. . . 4
⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝑋 ⊆
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋)) |
25 | | biidd 252 |
. . . . 5
⊢ (𝑘 = 𝑀 → (𝑋 ⊆
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋) ↔ 𝑋 ⊆
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋))) |
26 | 25 | rspcv 3305 |
. . . 4
⊢ (𝑀 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 𝑋 ⊆
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋) → 𝑋 ⊆
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋))) |
27 | 5, 24, 26 | sylc 65 |
. . 3
⊢ (𝜑 → 𝑋 ⊆
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋)) |
28 | 27 | sselda 3603 |
. 2
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋)) |
29 | | ulmcl 24135 |
. . . . 5
⊢ ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝𝑢‘𝑋)𝐻 → 𝐻:𝑋⟶ℂ) |
30 | 10, 29 | syl 17 |
. . . 4
⊢ (𝜑 → 𝐻:𝑋⟶ℂ) |
31 | 30 | ffvelrnda 6359 |
. . 3
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝐻‘𝑧) ∈ ℂ) |
32 | | rphalfcl 11858 |
. . . . . . . 8
⊢ (𝑟 ∈ ℝ+
→ (𝑟 / 2) ∈
ℝ+) |
33 | 32 | adantl 482 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑟 / 2) ∈
ℝ+) |
34 | | rphalfcl 11858 |
. . . . . . 7
⊢ ((𝑟 / 2) ∈ ℝ+
→ ((𝑟 / 2) / 2) ∈
ℝ+) |
35 | 33, 34 | syl 17 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) → ((𝑟 / 2) / 2) ∈
ℝ+) |
36 | | ulmrel 24132 |
. . . . . . . . . 10
⊢ Rel
(⇝𝑢‘𝑋) |
37 | | releldm 5358 |
. . . . . . . . . 10
⊢ ((Rel
(⇝𝑢‘𝑋) ∧ (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝𝑢‘𝑋)𝐻) → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) ∈ dom
(⇝𝑢‘𝑋)) |
38 | 36, 10, 37 | sylancr 695 |
. . . . . . . . 9
⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) ∈ dom
(⇝𝑢‘𝑋)) |
39 | | ulmscl 24133 |
. . . . . . . . . . 11
⊢ ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝𝑢‘𝑋)𝐻 → 𝑋 ∈ V) |
40 | 10, 39 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑋 ∈ V) |
41 | | ovex 6678 |
. . . . . . . . . . . . 13
⊢ (𝑆 D (𝐹‘𝑘)) ∈ V |
42 | 41 | rgenw 2924 |
. . . . . . . . . . . 12
⊢
∀𝑘 ∈
𝑍 (𝑆 D (𝐹‘𝑘)) ∈ V |
43 | | eqid 2622 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) = (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) |
44 | 43 | fnmpt 6020 |
. . . . . . . . . . . 12
⊢
(∀𝑘 ∈
𝑍 (𝑆 D (𝐹‘𝑘)) ∈ V → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) Fn 𝑍) |
45 | 42, 44 | mp1i 13 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) Fn 𝑍) |
46 | | ulmf2 24138 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) Fn 𝑍 ∧ (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝𝑢‘𝑋)𝐻) → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))):𝑍⟶(ℂ ↑𝑚
𝑋)) |
47 | 45, 10, 46 | syl2anc 693 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))):𝑍⟶(ℂ ↑𝑚
𝑋)) |
48 | 4, 1, 40, 47 | ulmcau2 24150 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) ∈ dom
(⇝𝑢‘𝑋) ↔ ∀𝑠 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘((((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑥) − (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚)‘𝑥))) < 𝑠)) |
49 | 38, 48 | mpbid 222 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑠 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘((((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑥) − (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚)‘𝑥))) < 𝑠) |
50 | 4 | uztrn2 11705 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗)) → 𝑛 ∈ 𝑍) |
51 | 50 | ad2ant2lr 784 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ≥‘𝑗) ∧ 𝑚 ∈ (ℤ≥‘𝑛))) → 𝑛 ∈ 𝑍) |
52 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛)) |
53 | 52 | oveq2d 6666 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑛 → (𝑆 D (𝐹‘𝑘)) = (𝑆 D (𝐹‘𝑛))) |
54 | | ovex 6678 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑆 D (𝐹‘𝑛)) ∈ V |
55 | 53, 43, 54 | fvmpt 6282 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛) = (𝑆 D (𝐹‘𝑛))) |
56 | 51, 55 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ≥‘𝑗) ∧ 𝑚 ∈ (ℤ≥‘𝑛))) → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛) = (𝑆 D (𝐹‘𝑛))) |
57 | 56 | fveq1d 6193 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ≥‘𝑗) ∧ 𝑚 ∈ (ℤ≥‘𝑛))) → (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑥) = ((𝑆 D (𝐹‘𝑛))‘𝑥)) |
58 | | simprr 796 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ≥‘𝑗) ∧ 𝑚 ∈ (ℤ≥‘𝑛))) → 𝑚 ∈ (ℤ≥‘𝑛)) |
59 | 4 | uztrn2 11705 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍) |
60 | 51, 58, 59 | syl2anc 693 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ≥‘𝑗) ∧ 𝑚 ∈ (ℤ≥‘𝑛))) → 𝑚 ∈ 𝑍) |
61 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚)) |
62 | 61 | oveq2d 6666 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑚 → (𝑆 D (𝐹‘𝑘)) = (𝑆 D (𝐹‘𝑚))) |
63 | | ovex 6678 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑆 D (𝐹‘𝑚)) ∈ V |
64 | 62, 43, 63 | fvmpt 6282 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚) = (𝑆 D (𝐹‘𝑚))) |
65 | 60, 64 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ≥‘𝑗) ∧ 𝑚 ∈ (ℤ≥‘𝑛))) → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚) = (𝑆 D (𝐹‘𝑚))) |
66 | 65 | fveq1d 6193 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ≥‘𝑗) ∧ 𝑚 ∈ (ℤ≥‘𝑛))) → (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚)‘𝑥) = ((𝑆 D (𝐹‘𝑚))‘𝑥)) |
67 | 57, 66 | oveq12d 6668 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ≥‘𝑗) ∧ 𝑚 ∈ (ℤ≥‘𝑛))) → ((((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑥) − (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚)‘𝑥)) = (((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) |
68 | 67 | fveq2d 6195 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ≥‘𝑗) ∧ 𝑚 ∈ (ℤ≥‘𝑛))) → (abs‘((((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑥) − (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚)‘𝑥))) = (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥)))) |
69 | 68 | breq1d 4663 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ≥‘𝑗) ∧ 𝑚 ∈ (ℤ≥‘𝑛))) → ((abs‘((((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑥) − (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚)‘𝑥))) < 𝑠 ↔ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < 𝑠)) |
70 | 69 | ralbidv 2986 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ≥‘𝑗) ∧ 𝑚 ∈ (ℤ≥‘𝑛))) → (∀𝑥 ∈ 𝑋 (abs‘((((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑥) − (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚)‘𝑥))) < 𝑠 ↔ ∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < 𝑠)) |
71 | 70 | 2ralbidva 2988 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑛 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘((((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑥) − (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚)‘𝑥))) < 𝑠 ↔ ∀𝑛 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < 𝑠)) |
72 | 71 | rexbidva 3049 |
. . . . . . . . 9
⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘((((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑥) − (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚)‘𝑥))) < 𝑠 ↔ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < 𝑠)) |
73 | 72 | ralbidv 2986 |
. . . . . . . 8
⊢ (𝜑 → (∀𝑠 ∈ ℝ+
∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘((((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑥) − (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚)‘𝑥))) < 𝑠 ↔ ∀𝑠 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < 𝑠)) |
74 | 49, 73 | mpbid 222 |
. . . . . . 7
⊢ (𝜑 → ∀𝑠 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < 𝑠) |
75 | 74 | ad2antrr 762 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) →
∀𝑠 ∈
ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < 𝑠) |
76 | | breq2 4657 |
. . . . . . . . 9
⊢ (𝑠 = ((𝑟 / 2) / 2) → ((abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < 𝑠 ↔ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2))) |
77 | 76 | 2ralbidv 2989 |
. . . . . . . 8
⊢ (𝑠 = ((𝑟 / 2) / 2) → (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < 𝑠 ↔ ∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2))) |
78 | 77 | rexralbidv 3058 |
. . . . . . 7
⊢ (𝑠 = ((𝑟 / 2) / 2) → (∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < 𝑠 ↔ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2))) |
79 | 78 | rspcv 3305 |
. . . . . 6
⊢ (((𝑟 / 2) / 2) ∈
ℝ+ → (∀𝑠 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < 𝑠 → ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2))) |
80 | 35, 75, 79 | sylc 65 |
. . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) →
∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2)) |
81 | 1 | ad2antrr 762 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) → 𝑀 ∈
ℤ) |
82 | 53 | fveq1d 6193 |
. . . . . . . 8
⊢ (𝑘 = 𝑛 → ((𝑆 D (𝐹‘𝑘))‘𝑧) = ((𝑆 D (𝐹‘𝑛))‘𝑧)) |
83 | | eqid 2622 |
. . . . . . . 8
⊢ (𝑘 ∈ 𝑍 ↦ ((𝑆 D (𝐹‘𝑘))‘𝑧)) = (𝑘 ∈ 𝑍 ↦ ((𝑆 D (𝐹‘𝑘))‘𝑧)) |
84 | | fvex 6201 |
. . . . . . . 8
⊢ ((𝑆 D (𝐹‘𝑛))‘𝑧) ∈ V |
85 | 82, 83, 84 | fvmpt 6282 |
. . . . . . 7
⊢ (𝑛 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ ((𝑆 D (𝐹‘𝑘))‘𝑧))‘𝑛) = ((𝑆 D (𝐹‘𝑛))‘𝑧)) |
86 | 85 | adantl 482 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ ((𝑆 D (𝐹‘𝑘))‘𝑧))‘𝑛) = ((𝑆 D (𝐹‘𝑛))‘𝑧)) |
87 | 47 | ad2antrr 762 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))):𝑍⟶(ℂ ↑𝑚
𝑋)) |
88 | | simplr 792 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) → 𝑧 ∈ 𝑋) |
89 | | fvex 6201 |
. . . . . . . . . 10
⊢
(ℤ≥‘𝑀) ∈ V |
90 | 4, 89 | eqeltri 2697 |
. . . . . . . . 9
⊢ 𝑍 ∈ V |
91 | 90 | mptex 6486 |
. . . . . . . 8
⊢ (𝑘 ∈ 𝑍 ↦ ((𝑆 D (𝐹‘𝑘))‘𝑧)) ∈ V |
92 | 91 | a1i 11 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑘 ∈ 𝑍 ↦ ((𝑆 D (𝐹‘𝑘))‘𝑧)) ∈ V) |
93 | 55 | adantl 482 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛) = (𝑆 D (𝐹‘𝑛))) |
94 | 93 | fveq1d 6193 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑧) = ((𝑆 D (𝐹‘𝑛))‘𝑧)) |
95 | 94, 86 | eqtr4d 2659 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑧) = ((𝑘 ∈ 𝑍 ↦ ((𝑆 D (𝐹‘𝑘))‘𝑧))‘𝑛)) |
96 | 10 | ad2antrr 762 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝𝑢‘𝑋)𝐻) |
97 | 4, 81, 87, 88, 92, 95, 96 | ulmclm 24141 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑘 ∈ 𝑍 ↦ ((𝑆 D (𝐹‘𝑘))‘𝑧)) ⇝ (𝐻‘𝑧)) |
98 | 4, 81, 33, 86, 97 | climi2 14242 |
. . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) →
∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) |
99 | 4 | rexanuz2 14089 |
. . . . . . 7
⊢
(∃𝑗 ∈
𝑍 ∀𝑛 ∈
(ℤ≥‘𝑗)(∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ↔ (∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2))) |
100 | 4 | r19.2uz 14091 |
. . . . . . 7
⊢
(∃𝑗 ∈
𝑍 ∀𝑛 ∈
(ℤ≥‘𝑗)(∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) → ∃𝑛 ∈ 𝑍 (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2))) |
101 | 99, 100 | sylbir 225 |
. . . . . 6
⊢
((∃𝑗 ∈
𝑍 ∀𝑛 ∈
(ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) → ∃𝑛 ∈ 𝑍 (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2))) |
102 | 35 | adantr 481 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → ((𝑟 / 2) / 2) ∈
ℝ+) |
103 | | simpllr 799 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → 𝑧 ∈ 𝑋) |
104 | 87 | ffvelrnda 6359 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛) ∈ (ℂ ↑𝑚
𝑋)) |
105 | 93, 104 | eqeltrrd 2702 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → (𝑆 D (𝐹‘𝑛)) ∈ (ℂ ↑𝑚
𝑋)) |
106 | | elmapi 7879 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑆 D (𝐹‘𝑛)) ∈ (ℂ ↑𝑚
𝑋) → (𝑆 D (𝐹‘𝑛)):𝑋⟶ℂ) |
107 | | fdm 6051 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑆 D (𝐹‘𝑛)):𝑋⟶ℂ → dom (𝑆 D (𝐹‘𝑛)) = 𝑋) |
108 | 105, 106,
107 | 3syl 18 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → dom (𝑆 D (𝐹‘𝑛)) = 𝑋) |
109 | 103, 108 | eleqtrrd 2704 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → 𝑧 ∈ dom (𝑆 D (𝐹‘𝑛))) |
110 | 6 | ad3antrrr 766 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ {ℝ, ℂ}) |
111 | | dvfg 23670 |
. . . . . . . . . . . . . . . 16
⊢ (𝑆 ∈ {ℝ, ℂ}
→ (𝑆 D (𝐹‘𝑛)):dom (𝑆 D (𝐹‘𝑛))⟶ℂ) |
112 | | ffun 6048 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑆 D (𝐹‘𝑛)):dom (𝑆 D (𝐹‘𝑛))⟶ℂ → Fun (𝑆 D (𝐹‘𝑛))) |
113 | | funfvbrb 6330 |
. . . . . . . . . . . . . . . 16
⊢ (Fun
(𝑆 D (𝐹‘𝑛)) → (𝑧 ∈ dom (𝑆 D (𝐹‘𝑛)) ↔ 𝑧(𝑆 D (𝐹‘𝑛))((𝑆 D (𝐹‘𝑛))‘𝑧))) |
114 | 110, 111,
112, 113 | 4syl 19 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → (𝑧 ∈ dom (𝑆 D (𝐹‘𝑛)) ↔ 𝑧(𝑆 D (𝐹‘𝑛))((𝑆 D (𝐹‘𝑛))‘𝑧))) |
115 | 109, 114 | mpbid 222 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → 𝑧(𝑆 D (𝐹‘𝑛))((𝑆 D (𝐹‘𝑛))‘𝑧)) |
116 | | eqid 2622 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧))) = (𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧))) |
117 | 110, 12 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → 𝑆 ⊆ ℂ) |
118 | 7 | ad2antrr 762 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) → 𝐹:𝑍⟶(ℂ ↑𝑚
𝑋)) |
119 | 118 | ffvelrnda 6359 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (ℂ ↑𝑚
𝑋)) |
120 | | elmapi 7879 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐹‘𝑛) ∈ (ℂ ↑𝑚
𝑋) → (𝐹‘𝑛):𝑋⟶ℂ) |
121 | 119, 120 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):𝑋⟶ℂ) |
122 | 19 | ralrimiva 2966 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝑋 ⊆ 𝑆) |
123 | | biidd 252 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑀 → (𝑋 ⊆ 𝑆 ↔ 𝑋 ⊆ 𝑆)) |
124 | 123 | rspcv 3305 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 𝑋 ⊆ 𝑆 → 𝑋 ⊆ 𝑆)) |
125 | 5, 122, 124 | sylc 65 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
126 | 125 | ad3antrrr 766 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → 𝑋 ⊆ 𝑆) |
127 | 20, 21, 116, 117, 121, 126 | eldv 23662 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → (𝑧(𝑆 D (𝐹‘𝑛))((𝑆 D (𝐹‘𝑛))‘𝑧) ↔ (𝑧 ∈
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋) ∧ ((𝑆 D (𝐹‘𝑛))‘𝑧) ∈ ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧))) limℂ 𝑧)))) |
128 | 115, 127 | mpbid 222 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → (𝑧 ∈
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋) ∧ ((𝑆 D (𝐹‘𝑛))‘𝑧) ∈ ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧))) limℂ 𝑧))) |
129 | 128 | simprd 479 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → ((𝑆 D (𝐹‘𝑛))‘𝑧) ∈ ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧))) limℂ 𝑧)) |
130 | 125 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑋 ⊆ 𝑆) |
131 | 13 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑆 ⊆ ℂ) |
132 | 130, 131 | sstrd 3613 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑋 ⊆ ℂ) |
133 | 132 | ad2antrr 762 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → 𝑋 ⊆ ℂ) |
134 | 121, 133,
103 | dvlem 23660 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) ∧ 𝑦 ∈ (𝑋 ∖ {𝑧})) → ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)) ∈ ℂ) |
135 | 134, 116 | fmptd 6385 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → (𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧))):(𝑋 ∖ {𝑧})⟶ℂ) |
136 | 133 | ssdifssd 3748 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → (𝑋 ∖ {𝑧}) ⊆ ℂ) |
137 | 133, 103 | sseldd 3604 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → 𝑧 ∈ ℂ) |
138 | 135, 136,
137 | ellimc3 23643 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → (((𝑆 D (𝐹‘𝑛))‘𝑧) ∈ ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧))) limℂ 𝑧) ↔ (((𝑆 D (𝐹‘𝑛))‘𝑧) ∈ ℂ ∧ ∀𝑠 ∈ ℝ+
∃𝑤 ∈
ℝ+ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < 𝑠)))) |
139 | 129, 138 | mpbid 222 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → (((𝑆 D (𝐹‘𝑛))‘𝑧) ∈ ℂ ∧ ∀𝑠 ∈ ℝ+
∃𝑤 ∈
ℝ+ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < 𝑠))) |
140 | 139 | simprd 479 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → ∀𝑠 ∈ ℝ+ ∃𝑤 ∈ ℝ+
∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < 𝑠)) |
141 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 𝑣 → ((𝐹‘𝑛)‘𝑦) = ((𝐹‘𝑛)‘𝑣)) |
142 | 141 | oveq1d 6665 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = 𝑣 → (((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) = (((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧))) |
143 | | oveq1 6657 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = 𝑣 → (𝑦 − 𝑧) = (𝑣 − 𝑧)) |
144 | 142, 143 | oveq12d 6668 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 = 𝑣 → ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)) = ((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧))) |
145 | | ovex 6678 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) ∈ V |
146 | 144, 116,
145 | fvmpt 6282 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑣 ∈ (𝑋 ∖ {𝑧}) → ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) = ((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧))) |
147 | 146 | oveq1d 6665 |
. . . . . . . . . . . . . . . 16
⊢ (𝑣 ∈ (𝑋 ∖ {𝑧}) → (((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − ((𝑆 D (𝐹‘𝑛))‘𝑧)) = (((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) |
148 | 147 | fveq2d 6195 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 ∈ (𝑋 ∖ {𝑧}) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) = (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧)))) |
149 | | id 22 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 = ((𝑟 / 2) / 2) → 𝑠 = ((𝑟 / 2) / 2)) |
150 | 148, 149 | breqan12rd 4670 |
. . . . . . . . . . . . . 14
⊢ ((𝑠 = ((𝑟 / 2) / 2) ∧ 𝑣 ∈ (𝑋 ∖ {𝑧})) → ((abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < 𝑠 ↔ (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2))) |
151 | 150 | imbi2d 330 |
. . . . . . . . . . . . 13
⊢ ((𝑠 = ((𝑟 / 2) / 2) ∧ 𝑣 ∈ (𝑋 ∖ {𝑧})) → (((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < 𝑠) ↔ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) |
152 | 151 | ralbidva 2985 |
. . . . . . . . . . . 12
⊢ (𝑠 = ((𝑟 / 2) / 2) → (∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < 𝑠) ↔ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) |
153 | 152 | rexbidv 3052 |
. . . . . . . . . . 11
⊢ (𝑠 = ((𝑟 / 2) / 2) → (∃𝑤 ∈ ℝ+ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < 𝑠) ↔ ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) |
154 | 153 | rspcv 3305 |
. . . . . . . . . 10
⊢ (((𝑟 / 2) / 2) ∈
ℝ+ → (∀𝑠 ∈ ℝ+ ∃𝑤 ∈ ℝ+
∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < 𝑠) → ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) |
155 | 102, 140,
154 | sylc 65 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ 𝑍) → ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2))) |
156 | 155 | adantrr 753 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) → ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2))) |
157 | | anass 681 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ 𝑤 ∈ ℝ+) ↔ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ ((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈
ℝ+))) |
158 | | df-3an 1039 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))) ↔ ((𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2))) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))))) |
159 | | anass 681 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ↔ (𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈
ℝ+))) |
160 | 9 | ralrimiva 2966 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ∀𝑧 ∈ 𝑋 (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧)) |
161 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑧 = 𝑠 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑠)) |
162 | 161 | mpteq2dv 4745 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑧 = 𝑠 → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) = (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑠))) |
163 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑧 = 𝑠 → (𝐺‘𝑧) = (𝐺‘𝑠)) |
164 | 162, 163 | breq12d 4666 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑧 = 𝑠 → ((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧) ↔ (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑠)) ⇝ (𝐺‘𝑠))) |
165 | 164 | rspccva 3308 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((∀𝑧 ∈
𝑋 (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧) ∧ 𝑠 ∈ 𝑋) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑠)) ⇝ (𝐺‘𝑠)) |
166 | 160, 165 | sylan 488 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑠)) ⇝ (𝐺‘𝑠)) |
167 | | simprll 802 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → 𝑧 ∈ 𝑋) |
168 | | simprlr 803 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → 𝑟 ∈ ℝ+) |
169 | | simprr3 1111 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))) |
170 | | simplll 798 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑢 ∈ ℝ+
∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))) → 𝑢 ∈ ℝ+) |
171 | 169, 170 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → 𝑢 ∈ ℝ+) |
172 | | simplr 792 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑢 ∈ ℝ+
∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))) → 𝑤 ∈ ℝ+) |
173 | 169, 172 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → 𝑤 ∈ ℝ+) |
174 | | simpllr 799 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑢 ∈ ℝ+
∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))) → (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) |
175 | 169, 174 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) |
176 | 175 | simpld 475 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → 𝑢 < 𝑤) |
177 | 175 | simprd 479 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋) |
178 | | simpr3 1069 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑢 ∈ ℝ+
∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))) → (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)) |
179 | 169, 178 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)) |
180 | 179 | simprd 479 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → (abs‘(𝑣 − 𝑧)) < 𝑢) |
181 | | simprr1 1109 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → 𝑛 ∈ 𝑍) |
182 | | simprr2 1110 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2))) |
183 | 182 | simpld 475 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → ∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2)) |
184 | 182 | simprd 479 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) |
185 | | simpr1 1067 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑢 ∈ ℝ+
∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))) → 𝑣 ∈ (𝑋 ∖ {𝑧})) |
186 | 169, 185 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → 𝑣 ∈ (𝑋 ∖ {𝑧})) |
187 | 186 | eldifad 3586 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → 𝑣 ∈ 𝑋) |
188 | 179 | simpld 475 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → 𝑣 ≠ 𝑧) |
189 | | simpr2 1068 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑢 ∈ ℝ+
∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))) → ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2))) |
190 | 169, 189 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2))) |
191 | 188, 190 | mpand 711 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → ((abs‘(𝑣 − 𝑧)) < 𝑤 → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2))) |
192 | 4, 6, 1, 7, 8, 166, 10, 167, 168, 171, 173, 176, 177, 180, 181, 183, 184, 187, 188, 191 | ulmdvlem1 24154 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟) |
193 | 192 | anassrs 680 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+)) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))))) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟) |
194 | 159, 193 | sylanb 489 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))))) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟) |
195 | 158, 194 | sylan2br 493 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ ((𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2))) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))))) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟) |
196 | 195 | anassrs 680 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟) |
197 | 196 | anassrs 680 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ ((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+)) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟) |
198 | 157, 197 | sylanb 489 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟) |
199 | 198 | 3exp2 1285 |
. . . . . . . . . . . . . 14
⊢
((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ 𝑤 ∈ ℝ+) → (𝑣 ∈ (𝑋 ∖ {𝑧}) → (((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) → ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟)))) |
200 | 199 | imp 445 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ 𝑤 ∈ ℝ+) ∧ 𝑣 ∈ (𝑋 ∖ {𝑧})) → (((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) → ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟))) |
201 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = 𝑣 → (𝐺‘𝑦) = (𝐺‘𝑣)) |
202 | 201 | oveq1d 6665 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 𝑣 → ((𝐺‘𝑦) − (𝐺‘𝑧)) = ((𝐺‘𝑣) − (𝐺‘𝑧))) |
203 | 202, 143 | oveq12d 6668 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = 𝑣 → (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)) = (((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧))) |
204 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧))) = (𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧))) |
205 | | ovex 6678 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) ∈ V |
206 | 203, 204,
205 | fvmpt 6282 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑣 ∈ (𝑋 ∖ {𝑧}) → ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) = (((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧))) |
207 | 206 | oveq1d 6665 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑣 ∈ (𝑋 ∖ {𝑧}) → (((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧)) = ((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) |
208 | 207 | fveq2d 6195 |
. . . . . . . . . . . . . . . 16
⊢ (𝑣 ∈ (𝑋 ∖ {𝑧}) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) = (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧)))) |
209 | 208 | breq1d 4663 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 ∈ (𝑋 ∖ {𝑧}) → ((abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟 ↔ (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟)) |
210 | 209 | imbi2d 330 |
. . . . . . . . . . . . . 14
⊢ (𝑣 ∈ (𝑋 ∖ {𝑧}) → (((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟) ↔ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟))) |
211 | 210 | adantl 482 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ 𝑤 ∈ ℝ+) ∧ 𝑣 ∈ (𝑋 ∖ {𝑧})) → (((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟) ↔ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟))) |
212 | 200, 211 | sylibrd 249 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ 𝑤 ∈ ℝ+) ∧ 𝑣 ∈ (𝑋 ∖ {𝑧})) → (((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) → ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟))) |
213 | 212 | ralimdva 2962 |
. . . . . . . . . . 11
⊢
((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ 𝑤 ∈ ℝ+) →
(∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) → ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟))) |
214 | 213 | impr 649 |
. . . . . . . . . 10
⊢
((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ (𝑤 ∈ ℝ+ ∧
∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) → ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟)) |
215 | 214 | an32s 846 |
. . . . . . . . 9
⊢
((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑤 ∈ ℝ+ ∧
∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) ∧ (𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) → ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟)) |
216 | | cnxmet 22576 |
. . . . . . . . . . . 12
⊢ (abs
∘ − ) ∈ (∞Met‘ℂ) |
217 | | xmetres2 22166 |
. . . . . . . . . . . 12
⊢ (((abs
∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((abs ∘
− ) ↾ (𝑆
× 𝑆)) ∈
(∞Met‘𝑆)) |
218 | 216, 131,
217 | sylancr 695 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ((abs ∘ − ) ↾
(𝑆 × 𝑆)) ∈
(∞Met‘𝑆)) |
219 | 218 | ad3antrrr 766 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑤 ∈ ℝ+ ∧
∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) → ((abs ∘ − )
↾ (𝑆 × 𝑆)) ∈
(∞Met‘𝑆)) |
220 | 21 | cnfldtop 22587 |
. . . . . . . . . . . . . . . . 17
⊢
(TopOpen‘ℂfld) ∈ Top |
221 | | resttop 20964 |
. . . . . . . . . . . . . . . . 17
⊢
(((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ {ℝ, ℂ})
→ ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top) |
222 | 220, 6, 221 | sylancr 695 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 →
((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top) |
223 | 21 | cnfldtopon 22586 |
. . . . . . . . . . . . . . . . . . 19
⊢
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ) |
224 | | resttopon 20965 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
∧ 𝑆 ⊆ ℂ)
→ ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)) |
225 | 223, 13, 224 | sylancr 695 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 →
((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)) |
226 | | toponuni 20719 |
. . . . . . . . . . . . . . . . . 18
⊢
(((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆) → 𝑆 = ∪
((TopOpen‘ℂfld) ↾t 𝑆)) |
227 | 225, 226 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑆 = ∪
((TopOpen‘ℂfld) ↾t 𝑆)) |
228 | 125, 227 | sseqtrd 3641 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑋 ⊆ ∪
((TopOpen‘ℂfld) ↾t 𝑆)) |
229 | | eqid 2622 |
. . . . . . . . . . . . . . . . 17
⊢ ∪ ((TopOpen‘ℂfld)
↾t 𝑆) =
∪ ((TopOpen‘ℂfld)
↾t 𝑆) |
230 | 229 | ntrss2 20861 |
. . . . . . . . . . . . . . . 16
⊢
((((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top ∧ 𝑋 ⊆ ∪ ((TopOpen‘ℂfld)
↾t 𝑆))
→ ((int‘((TopOpen‘ℂfld) ↾t
𝑆))‘𝑋) ⊆ 𝑋) |
231 | 222, 228,
230 | syl2anc 693 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 →
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋) ⊆ 𝑋) |
232 | 231, 27 | eqssd 3620 |
. . . . . . . . . . . . . 14
⊢ (𝜑 →
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋) = 𝑋) |
233 | 229 | isopn3 20870 |
. . . . . . . . . . . . . . 15
⊢
((((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top ∧ 𝑋 ⊆ ∪ ((TopOpen‘ℂfld)
↾t 𝑆))
→ (𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆) ↔
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋) = 𝑋)) |
234 | 222, 228,
233 | syl2anc 693 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆) ↔
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋) = 𝑋)) |
235 | 232, 234 | mpbird 247 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆)) |
236 | | eqid 2622 |
. . . . . . . . . . . . . . 15
⊢ ((abs
∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (𝑆 × 𝑆)) |
237 | 21 | cnfldtopn 22585 |
. . . . . . . . . . . . . . 15
⊢
(TopOpen‘ℂfld) = (MetOpen‘(abs ∘
− )) |
238 | | eqid 2622 |
. . . . . . . . . . . . . . 15
⊢
(MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))) = (MetOpen‘((abs ∘ − )
↾ (𝑆 × 𝑆))) |
239 | 236, 237,
238 | metrest 22329 |
. . . . . . . . . . . . . 14
⊢ (((abs
∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) →
((TopOpen‘ℂfld) ↾t 𝑆) = (MetOpen‘((abs ∘ − )
↾ (𝑆 × 𝑆)))) |
240 | 216, 13, 239 | sylancr 695 |
. . . . . . . . . . . . 13
⊢ (𝜑 →
((TopOpen‘ℂfld) ↾t 𝑆) = (MetOpen‘((abs ∘ − )
↾ (𝑆 × 𝑆)))) |
241 | 235, 240 | eleqtrd 2703 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑋 ∈ (MetOpen‘((abs ∘ −
) ↾ (𝑆 × 𝑆)))) |
242 | 241 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑋 ∈ (MetOpen‘((abs ∘ −
) ↾ (𝑆 × 𝑆)))) |
243 | 242 | ad3antrrr 766 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑤 ∈ ℝ+ ∧
∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) → 𝑋 ∈ (MetOpen‘((abs ∘ −
) ↾ (𝑆 × 𝑆)))) |
244 | 88 | ad2antrr 762 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑤 ∈ ℝ+ ∧
∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) → 𝑧 ∈ 𝑋) |
245 | | simprl 794 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑤 ∈ ℝ+ ∧
∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) → 𝑤 ∈ ℝ+) |
246 | 238 | mopni3 22299 |
. . . . . . . . . 10
⊢ (((((abs
∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆) ∧ 𝑋 ∈ (MetOpen‘((abs ∘ −
) ↾ (𝑆 × 𝑆))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ ℝ+) →
∃𝑢 ∈
ℝ+ (𝑢 <
𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) |
247 | 219, 243,
244, 245, 246 | syl31anc 1329 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑤 ∈ ℝ+ ∧
∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) → ∃𝑢 ∈ ℝ+
(𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾
(𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) |
248 | 215, 247 | reximddv 3018 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑤 ∈ ℝ+ ∧
∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) → ∃𝑢 ∈ ℝ+
∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟)) |
249 | 156, 248 | rexlimddv 3035 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) → ∃𝑢 ∈ ℝ+ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟)) |
250 | 249 | rexlimdvaa 3032 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) →
(∃𝑛 ∈ 𝑍 (∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) → ∃𝑢 ∈ ℝ+ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟))) |
251 | 101, 250 | syl5 34 |
. . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) →
((∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) → ∃𝑢 ∈ ℝ+ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟))) |
252 | 80, 98, 251 | mp2and 715 |
. . . 4
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) →
∃𝑢 ∈
ℝ+ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟)) |
253 | 252 | ralrimiva 2966 |
. . 3
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ∀𝑟 ∈ ℝ+ ∃𝑢 ∈ ℝ+
∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟)) |
254 | 8 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐺:𝑋⟶ℂ) |
255 | | simpr 477 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋) |
256 | 254, 132,
255 | dvlem 23660 |
. . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑦 ∈ (𝑋 ∖ {𝑧})) → (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)) ∈ ℂ) |
257 | 256, 204 | fmptd 6385 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧))):(𝑋 ∖ {𝑧})⟶ℂ) |
258 | 132 | ssdifssd 3748 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑋 ∖ {𝑧}) ⊆ ℂ) |
259 | 132, 255 | sseldd 3604 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ ℂ) |
260 | 257, 258,
259 | ellimc3 23643 |
. . 3
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ((𝐻‘𝑧) ∈ ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧))) limℂ 𝑧) ↔ ((𝐻‘𝑧) ∈ ℂ ∧ ∀𝑟 ∈ ℝ+
∃𝑢 ∈
ℝ+ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟)))) |
261 | 31, 253, 260 | mpbir2and 957 |
. 2
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝐻‘𝑧) ∈ ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧))) limℂ 𝑧)) |
262 | 20, 21, 204, 131, 254, 130 | eldv 23662 |
. 2
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑧(𝑆 D 𝐺)(𝐻‘𝑧) ↔ (𝑧 ∈
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋) ∧ (𝐻‘𝑧) ∈ ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧))) limℂ 𝑧)))) |
263 | 28, 261, 262 | mpbir2and 957 |
1
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧(𝑆 D 𝐺)(𝐻‘𝑧)) |