Step | Hyp | Ref
| Expression |
1 | | ftalem3.5 |
. . . 4
⊢ 𝐷 = {𝑦 ∈ ℂ ∣ (abs‘𝑦) ≤ 𝑅} |
2 | | ssrab2 3687 |
. . . 4
⊢ {𝑦 ∈ ℂ ∣
(abs‘𝑦) ≤ 𝑅} ⊆
ℂ |
3 | 1, 2 | eqsstri 3635 |
. . 3
⊢ 𝐷 ⊆
ℂ |
4 | | ftalem3.6 |
. . . . . . . 8
⊢ 𝐽 =
(TopOpen‘ℂfld) |
5 | 4 | cnfldtopon 22586 |
. . . . . . 7
⊢ 𝐽 ∈
(TopOn‘ℂ) |
6 | | resttopon 20965 |
. . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘ℂ)
∧ 𝐷 ⊆ ℂ)
→ (𝐽
↾t 𝐷)
∈ (TopOn‘𝐷)) |
7 | 5, 3, 6 | mp2an 708 |
. . . . . 6
⊢ (𝐽 ↾t 𝐷) ∈ (TopOn‘𝐷) |
8 | 7 | toponunii 20721 |
. . . . 5
⊢ 𝐷 = ∪
(𝐽 ↾t
𝐷) |
9 | | eqid 2622 |
. . . . 5
⊢
(topGen‘ran (,)) = (topGen‘ran (,)) |
10 | | cnxmet 22576 |
. . . . . . . 8
⊢ (abs
∘ − ) ∈ (∞Met‘ℂ) |
11 | 10 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (abs ∘ − )
∈ (∞Met‘ℂ)) |
12 | | 0cn 10032 |
. . . . . . . 8
⊢ 0 ∈
ℂ |
13 | 12 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 0 ∈
ℂ) |
14 | | ftalem3.7 |
. . . . . . . 8
⊢ (𝜑 → 𝑅 ∈
ℝ+) |
15 | 14 | rpxrd 11873 |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈
ℝ*) |
16 | 4 | cnfldtopn 22585 |
. . . . . . . 8
⊢ 𝐽 = (MetOpen‘(abs ∘
− )) |
17 | | eqid 2622 |
. . . . . . . . . . . . . 14
⊢ (abs
∘ − ) = (abs ∘ − ) |
18 | 17 | cnmetdval 22574 |
. . . . . . . . . . . . 13
⊢ ((0
∈ ℂ ∧ 𝑦
∈ ℂ) → (0(abs ∘ − )𝑦) = (abs‘(0 − 𝑦))) |
19 | 12, 18 | mpan 706 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℂ → (0(abs
∘ − )𝑦) =
(abs‘(0 − 𝑦))) |
20 | | df-neg 10269 |
. . . . . . . . . . . . . 14
⊢ -𝑦 = (0 − 𝑦) |
21 | 20 | fveq2i 6194 |
. . . . . . . . . . . . 13
⊢
(abs‘-𝑦) =
(abs‘(0 − 𝑦)) |
22 | | absneg 14017 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℂ →
(abs‘-𝑦) =
(abs‘𝑦)) |
23 | 21, 22 | syl5eqr 2670 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℂ →
(abs‘(0 − 𝑦)) =
(abs‘𝑦)) |
24 | 19, 23 | eqtrd 2656 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ℂ → (0(abs
∘ − )𝑦) =
(abs‘𝑦)) |
25 | 24 | breq1d 4663 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℂ → ((0(abs
∘ − )𝑦) ≤
𝑅 ↔ (abs‘𝑦) ≤ 𝑅)) |
26 | 25 | rabbiia 3185 |
. . . . . . . . 9
⊢ {𝑦 ∈ ℂ ∣ (0(abs
∘ − )𝑦) ≤
𝑅} = {𝑦 ∈ ℂ ∣ (abs‘𝑦) ≤ 𝑅} |
27 | 1, 26 | eqtr4i 2647 |
. . . . . . . 8
⊢ 𝐷 = {𝑦 ∈ ℂ ∣ (0(abs ∘
− )𝑦) ≤ 𝑅} |
28 | 16, 27 | blcld 22310 |
. . . . . . 7
⊢ (((abs
∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ
∧ 𝑅 ∈
ℝ*) → 𝐷 ∈ (Clsd‘𝐽)) |
29 | 11, 13, 15, 28 | syl3anc 1326 |
. . . . . 6
⊢ (𝜑 → 𝐷 ∈ (Clsd‘𝐽)) |
30 | 14 | rpred 11872 |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈ ℝ) |
31 | | fveq2 6191 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑥 → (abs‘𝑦) = (abs‘𝑥)) |
32 | 31 | breq1d 4663 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑥 → ((abs‘𝑦) ≤ 𝑅 ↔ (abs‘𝑥) ≤ 𝑅)) |
33 | 32, 1 | elrab2 3366 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (abs‘𝑥) ≤ 𝑅)) |
34 | 33 | simprbi 480 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝐷 → (abs‘𝑥) ≤ 𝑅) |
35 | 34 | rgen 2922 |
. . . . . . 7
⊢
∀𝑥 ∈
𝐷 (abs‘𝑥) ≤ 𝑅 |
36 | | breq2 4657 |
. . . . . . . . 9
⊢ (𝑠 = 𝑅 → ((abs‘𝑥) ≤ 𝑠 ↔ (abs‘𝑥) ≤ 𝑅)) |
37 | 36 | ralbidv 2986 |
. . . . . . . 8
⊢ (𝑠 = 𝑅 → (∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑠 ↔ ∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑅)) |
38 | 37 | rspcev 3309 |
. . . . . . 7
⊢ ((𝑅 ∈ ℝ ∧
∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑅) → ∃𝑠 ∈ ℝ ∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑠) |
39 | 30, 35, 38 | sylancl 694 |
. . . . . 6
⊢ (𝜑 → ∃𝑠 ∈ ℝ ∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑠) |
40 | | eqid 2622 |
. . . . . . . 8
⊢ (𝐽 ↾t 𝐷) = (𝐽 ↾t 𝐷) |
41 | 4, 40 | cnheibor 22754 |
. . . . . . 7
⊢ (𝐷 ⊆ ℂ → ((𝐽 ↾t 𝐷) ∈ Comp ↔ (𝐷 ∈ (Clsd‘𝐽) ∧ ∃𝑠 ∈ ℝ ∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑠))) |
42 | 3, 41 | ax-mp 5 |
. . . . . 6
⊢ ((𝐽 ↾t 𝐷) ∈ Comp ↔ (𝐷 ∈ (Clsd‘𝐽) ∧ ∃𝑠 ∈ ℝ ∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑠)) |
43 | 29, 39, 42 | sylanbrc 698 |
. . . . 5
⊢ (𝜑 → (𝐽 ↾t 𝐷) ∈ Comp) |
44 | | ftalem.3 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) |
45 | | plycn 24017 |
. . . . . . . . 9
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ)) |
46 | 44, 45 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ)) |
47 | | abscncf 22704 |
. . . . . . . . 9
⊢ abs
∈ (ℂ–cn→ℝ) |
48 | 47 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → abs ∈
(ℂ–cn→ℝ)) |
49 | 46, 48 | cncfco 22710 |
. . . . . . 7
⊢ (𝜑 → (abs ∘ 𝐹) ∈ (ℂ–cn→ℝ)) |
50 | | ssid 3624 |
. . . . . . . 8
⊢ ℂ
⊆ ℂ |
51 | | ax-resscn 9993 |
. . . . . . . 8
⊢ ℝ
⊆ ℂ |
52 | 4 | cnfldtop 22587 |
. . . . . . . . . . 11
⊢ 𝐽 ∈ Top |
53 | 5 | toponunii 20721 |
. . . . . . . . . . . 12
⊢ ℂ =
∪ 𝐽 |
54 | 53 | restid 16094 |
. . . . . . . . . . 11
⊢ (𝐽 ∈ Top → (𝐽 ↾t ℂ) =
𝐽) |
55 | 52, 54 | ax-mp 5 |
. . . . . . . . . 10
⊢ (𝐽 ↾t ℂ) =
𝐽 |
56 | 55 | eqcomi 2631 |
. . . . . . . . 9
⊢ 𝐽 = (𝐽 ↾t
ℂ) |
57 | 4 | tgioo2 22606 |
. . . . . . . . 9
⊢
(topGen‘ran (,)) = (𝐽 ↾t
ℝ) |
58 | 4, 56, 57 | cncfcn 22712 |
. . . . . . . 8
⊢ ((ℂ
⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℂ–cn→ℝ) = (𝐽 Cn (topGen‘ran
(,)))) |
59 | 50, 51, 58 | mp2an 708 |
. . . . . . 7
⊢
(ℂ–cn→ℝ) =
(𝐽 Cn (topGen‘ran
(,))) |
60 | 49, 59 | syl6eleq 2711 |
. . . . . 6
⊢ (𝜑 → (abs ∘ 𝐹) ∈ (𝐽 Cn (topGen‘ran
(,)))) |
61 | 53 | cnrest 21089 |
. . . . . 6
⊢ (((abs
∘ 𝐹) ∈ (𝐽 Cn (topGen‘ran (,)))
∧ 𝐷 ⊆ ℂ)
→ ((abs ∘ 𝐹)
↾ 𝐷) ∈ ((𝐽 ↾t 𝐷) Cn (topGen‘ran
(,)))) |
62 | 60, 3, 61 | sylancl 694 |
. . . . 5
⊢ (𝜑 → ((abs ∘ 𝐹) ↾ 𝐷) ∈ ((𝐽 ↾t 𝐷) Cn (topGen‘ran
(,)))) |
63 | 14 | rpge0d 11876 |
. . . . . . 7
⊢ (𝜑 → 0 ≤ 𝑅) |
64 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑦 = 0 → (abs‘𝑦) =
(abs‘0)) |
65 | | abs0 14025 |
. . . . . . . . . 10
⊢
(abs‘0) = 0 |
66 | 64, 65 | syl6eq 2672 |
. . . . . . . . 9
⊢ (𝑦 = 0 → (abs‘𝑦) = 0) |
67 | 66 | breq1d 4663 |
. . . . . . . 8
⊢ (𝑦 = 0 → ((abs‘𝑦) ≤ 𝑅 ↔ 0 ≤ 𝑅)) |
68 | 67, 1 | elrab2 3366 |
. . . . . . 7
⊢ (0 ∈
𝐷 ↔ (0 ∈ ℂ
∧ 0 ≤ 𝑅)) |
69 | 13, 63, 68 | sylanbrc 698 |
. . . . . 6
⊢ (𝜑 → 0 ∈ 𝐷) |
70 | | ne0i 3921 |
. . . . . 6
⊢ (0 ∈
𝐷 → 𝐷 ≠ ∅) |
71 | 69, 70 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝐷 ≠ ∅) |
72 | 8, 9, 43, 62, 71 | evth2 22759 |
. . . 4
⊢ (𝜑 → ∃𝑧 ∈ 𝐷 ∀𝑥 ∈ 𝐷 (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) ≤ (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥)) |
73 | | fvres 6207 |
. . . . . . . . 9
⊢ (𝑧 ∈ 𝐷 → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) = ((abs ∘ 𝐹)‘𝑧)) |
74 | 73 | ad2antlr 763 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) = ((abs ∘ 𝐹)‘𝑧)) |
75 | | plyf 23954 |
. . . . . . . . . . 11
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) |
76 | 44, 75 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹:ℂ⟶ℂ) |
77 | 76 | ad2antrr 762 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝐹:ℂ⟶ℂ) |
78 | | simplr 792 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝑧 ∈ 𝐷) |
79 | 3, 78 | sseldi 3601 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝑧 ∈ ℂ) |
80 | | fvco3 6275 |
. . . . . . . . 9
⊢ ((𝐹:ℂ⟶ℂ ∧
𝑧 ∈ ℂ) →
((abs ∘ 𝐹)‘𝑧) = (abs‘(𝐹‘𝑧))) |
81 | 77, 79, 80 | syl2anc 693 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → ((abs ∘ 𝐹)‘𝑧) = (abs‘(𝐹‘𝑧))) |
82 | 74, 81 | eqtrd 2656 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) = (abs‘(𝐹‘𝑧))) |
83 | | fvres 6207 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐷 → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) = ((abs ∘ 𝐹)‘𝑥)) |
84 | 83 | adantl 482 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) = ((abs ∘ 𝐹)‘𝑥)) |
85 | | simpr 477 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷) |
86 | 3, 85 | sseldi 3601 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ℂ) |
87 | | fvco3 6275 |
. . . . . . . . 9
⊢ ((𝐹:ℂ⟶ℂ ∧
𝑥 ∈ ℂ) →
((abs ∘ 𝐹)‘𝑥) = (abs‘(𝐹‘𝑥))) |
88 | 77, 86, 87 | syl2anc 693 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → ((abs ∘ 𝐹)‘𝑥) = (abs‘(𝐹‘𝑥))) |
89 | 84, 88 | eqtrd 2656 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) = (abs‘(𝐹‘𝑥))) |
90 | 82, 89 | breq12d 4666 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → ((((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) ≤ (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) ↔ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))) |
91 | 90 | ralbidva 2985 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (∀𝑥 ∈ 𝐷 (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) ≤ (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) ↔ ∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))) |
92 | 91 | rexbidva 3049 |
. . . 4
⊢ (𝜑 → (∃𝑧 ∈ 𝐷 ∀𝑥 ∈ 𝐷 (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) ≤ (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) ↔ ∃𝑧 ∈ 𝐷 ∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))) |
93 | 72, 92 | mpbid 222 |
. . 3
⊢ (𝜑 → ∃𝑧 ∈ 𝐷 ∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))) |
94 | | ssrexv 3667 |
. . 3
⊢ (𝐷 ⊆ ℂ →
(∃𝑧 ∈ 𝐷 ∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) → ∃𝑧 ∈ ℂ ∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))) |
95 | 3, 93, 94 | mpsyl 68 |
. 2
⊢ (𝜑 → ∃𝑧 ∈ ℂ ∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))) |
96 | 69 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 0 ∈ 𝐷) |
97 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑥 = 0 → (𝐹‘𝑥) = (𝐹‘0)) |
98 | 97 | fveq2d 6195 |
. . . . . . . . 9
⊢ (𝑥 = 0 → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘0))) |
99 | 98 | breq2d 4665 |
. . . . . . . 8
⊢ (𝑥 = 0 → ((abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) ↔ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0)))) |
100 | 99 | rspcv 3305 |
. . . . . . 7
⊢ (0 ∈
𝐷 → (∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) → (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0)))) |
101 | 96, 100 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) → (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0)))) |
102 | 76 | ad2antrr 762 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝐹:ℂ⟶ℂ) |
103 | | ffvelrn 6357 |
. . . . . . . . . . 11
⊢ ((𝐹:ℂ⟶ℂ ∧ 0
∈ ℂ) → (𝐹‘0) ∈ ℂ) |
104 | 102, 12, 103 | sylancl 694 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝐹‘0) ∈ ℂ) |
105 | 104 | abscld 14175 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘(𝐹‘0)) ∈ ℝ) |
106 | | simpr 477 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝑥 ∈ (ℂ ∖ 𝐷)) |
107 | 106 | eldifad 3586 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝑥 ∈ ℂ) |
108 | 102, 107 | ffvelrnd 6360 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝐹‘𝑥) ∈ ℂ) |
109 | 108 | abscld 14175 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘(𝐹‘𝑥)) ∈ ℝ) |
110 | | ftalem3.8 |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑥 ∈ ℂ (𝑅 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥)))) |
111 | 110 | ad2antrr 762 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → ∀𝑥 ∈ ℂ (𝑅 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥)))) |
112 | 106 | eldifbd 3587 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → ¬ 𝑥 ∈ 𝐷) |
113 | 33 | baib 944 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℂ → (𝑥 ∈ 𝐷 ↔ (abs‘𝑥) ≤ 𝑅)) |
114 | 107, 113 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝑥 ∈ 𝐷 ↔ (abs‘𝑥) ≤ 𝑅)) |
115 | 112, 114 | mtbid 314 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → ¬ (abs‘𝑥) ≤ 𝑅) |
116 | 30 | ad2antrr 762 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝑅 ∈ ℝ) |
117 | 107 | abscld 14175 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘𝑥) ∈ ℝ) |
118 | 116, 117 | ltnled 10184 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝑅 < (abs‘𝑥) ↔ ¬ (abs‘𝑥) ≤ 𝑅)) |
119 | 115, 118 | mpbird 247 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝑅 < (abs‘𝑥)) |
120 | | rsp 2929 |
. . . . . . . . . 10
⊢
(∀𝑥 ∈
ℂ (𝑅 <
(abs‘𝑥) →
(abs‘(𝐹‘0))
< (abs‘(𝐹‘𝑥))) → (𝑥 ∈ ℂ → (𝑅 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥))))) |
121 | 111, 107,
119, 120 | syl3c 66 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥))) |
122 | 105, 109,
121 | ltled 10185 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘(𝐹‘0)) ≤ (abs‘(𝐹‘𝑥))) |
123 | | simplr 792 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝑧 ∈ ℂ) |
124 | 102, 123 | ffvelrnd 6360 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝐹‘𝑧) ∈ ℂ) |
125 | 124 | abscld 14175 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘(𝐹‘𝑧)) ∈ ℝ) |
126 | | letr 10131 |
. . . . . . . . 9
⊢
(((abs‘(𝐹‘𝑧)) ∈ ℝ ∧ (abs‘(𝐹‘0)) ∈ ℝ ∧
(abs‘(𝐹‘𝑥)) ∈ ℝ) →
(((abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0)) ∧
(abs‘(𝐹‘0))
≤ (abs‘(𝐹‘𝑥))) → (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))) |
127 | 125, 105,
109, 126 | syl3anc 1326 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (((abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0)) ∧ (abs‘(𝐹‘0)) ≤
(abs‘(𝐹‘𝑥))) → (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))) |
128 | 122, 127 | mpan2d 710 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → ((abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0)) → (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))) |
129 | 128 | ralrimdva 2969 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0)) → ∀𝑥 ∈ (ℂ ∖ 𝐷)(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))) |
130 | 101, 129 | syld 47 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) → ∀𝑥 ∈ (ℂ ∖ 𝐷)(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))) |
131 | 130 | ancld 576 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) → (∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) ∧ ∀𝑥 ∈ (ℂ ∖ 𝐷)(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))) |
132 | | ralunb 3794 |
. . . . 5
⊢
(∀𝑥 ∈
(𝐷 ∪ (ℂ ∖
𝐷))(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) ↔ (∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) ∧ ∀𝑥 ∈ (ℂ ∖ 𝐷)(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))) |
133 | | undif2 4044 |
. . . . . . 7
⊢ (𝐷 ∪ (ℂ ∖ 𝐷)) = (𝐷 ∪ ℂ) |
134 | | ssequn1 3783 |
. . . . . . . 8
⊢ (𝐷 ⊆ ℂ ↔ (𝐷 ∪ ℂ) =
ℂ) |
135 | 3, 134 | mpbi 220 |
. . . . . . 7
⊢ (𝐷 ∪ ℂ) =
ℂ |
136 | 133, 135 | eqtri 2644 |
. . . . . 6
⊢ (𝐷 ∪ (ℂ ∖ 𝐷)) = ℂ |
137 | 136 | raleqi 3142 |
. . . . 5
⊢
(∀𝑥 ∈
(𝐷 ∪ (ℂ ∖
𝐷))(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) ↔ ∀𝑥 ∈ ℂ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))) |
138 | 132, 137 | bitr3i 266 |
. . . 4
⊢
((∀𝑥 ∈
𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) ∧ ∀𝑥 ∈ (ℂ ∖ 𝐷)(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))) ↔ ∀𝑥 ∈ ℂ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))) |
139 | 131, 138 | syl6ib 241 |
. . 3
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) → ∀𝑥 ∈ ℂ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))) |
140 | 139 | reximdva 3017 |
. 2
⊢ (𝜑 → (∃𝑧 ∈ ℂ ∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) → ∃𝑧 ∈ ℂ ∀𝑥 ∈ ℂ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))) |
141 | 95, 140 | mpd 15 |
1
⊢ (𝜑 → ∃𝑧 ∈ ℂ ∀𝑥 ∈ ℂ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))) |