Step | Hyp | Ref
| Expression |
1 | | efopn.j |
. . . . . . . 8
⊢ 𝐽 =
(TopOpen‘ℂfld) |
2 | 1 | cnfldtopon 22586 |
. . . . . . 7
⊢ 𝐽 ∈
(TopOn‘ℂ) |
3 | | toponss 20731 |
. . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘ℂ)
∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ ℂ) |
4 | 2, 3 | mpan 706 |
. . . . . 6
⊢ (𝑆 ∈ 𝐽 → 𝑆 ⊆ ℂ) |
5 | 4 | sselda 3603 |
. . . . 5
⊢ ((𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ ℂ) |
6 | | cnxmet 22576 |
. . . . . 6
⊢ (abs
∘ − ) ∈ (∞Met‘ℂ) |
7 | | pirp 24213 |
. . . . . . 7
⊢ π
∈ ℝ+ |
8 | 1 | cnfldtopn 22585 |
. . . . . . . 8
⊢ 𝐽 = (MetOpen‘(abs ∘
− )) |
9 | 8 | mopni3 22299 |
. . . . . . 7
⊢ ((((abs
∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) ∧ π ∈ ℝ+)
→ ∃𝑟 ∈
ℝ+ (𝑟 <
π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆)) |
10 | 7, 9 | mpan2 707 |
. . . . . 6
⊢ (((abs
∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → ∃𝑟 ∈ ℝ+ (𝑟 < π ∧ (𝑥(ball‘(abs ∘ −
))𝑟) ⊆ 𝑆)) |
11 | 6, 10 | mp3an1 1411 |
. . . . 5
⊢ ((𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → ∃𝑟 ∈ ℝ+ (𝑟 < π ∧ (𝑥(ball‘(abs ∘ −
))𝑟) ⊆ 𝑆)) |
12 | | imass2 5501 |
. . . . . . . 8
⊢ ((𝑥(ball‘(abs ∘ −
))𝑟) ⊆ 𝑆 → (exp “ (𝑥(ball‘(abs ∘ −
))𝑟)) ⊆ (exp “
𝑆)) |
13 | | imassrn 5477 |
. . . . . . . . . . . . . 14
⊢ (exp
“ (𝑥(ball‘(abs
∘ − ))𝑟))
⊆ ran exp |
14 | | eff 14812 |
. . . . . . . . . . . . . . 15
⊢
exp:ℂ⟶ℂ |
15 | | frn 6053 |
. . . . . . . . . . . . . . 15
⊢
(exp:ℂ⟶ℂ → ran exp ⊆
ℂ) |
16 | 14, 15 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ ran exp
⊆ ℂ |
17 | 13, 16 | sstri 3612 |
. . . . . . . . . . . . 13
⊢ (exp
“ (𝑥(ball‘(abs
∘ − ))𝑟))
⊆ ℂ |
18 | | sseqin2 3817 |
. . . . . . . . . . . . 13
⊢ ((exp
“ (𝑥(ball‘(abs
∘ − ))𝑟))
⊆ ℂ ↔ (ℂ ∩ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))) = (exp “ (𝑥(ball‘(abs ∘ −
))𝑟))) |
19 | 17, 18 | mpbi 220 |
. . . . . . . . . . . 12
⊢ (ℂ
∩ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))) = (exp “ (𝑥(ball‘(abs ∘ −
))𝑟)) |
20 | | rpxr 11840 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑟 ∈ ℝ+
→ 𝑟 ∈
ℝ*) |
21 | | blssm 22223 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((abs
∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ*) → (𝑥(ball‘(abs ∘ −
))𝑟) ⊆
ℂ) |
22 | 6, 21 | mp3an1 1411 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ*)
→ (𝑥(ball‘(abs
∘ − ))𝑟)
⊆ ℂ) |
23 | 20, 22 | sylan2 491 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
→ (𝑥(ball‘(abs
∘ − ))𝑟)
⊆ ℂ) |
24 | 23 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) ∧
𝑧 ∈ ℂ) →
(𝑥(ball‘(abs ∘
− ))𝑟) ⊆
ℂ) |
25 | 24 | sselda 3603 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑦 ∈ ℂ) |
26 | | simp-4l 806 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑥 ∈ ℂ) |
27 | 25, 26 | subcld 10392 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦 − 𝑥) ∈ ℂ) |
28 | 27 | subid1d 10381 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦 − 𝑥) − 0) = (𝑦 − 𝑥)) |
29 | 28 | fveq2d 6195 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (abs‘((𝑦 − 𝑥) − 0)) = (abs‘(𝑦 − 𝑥))) |
30 | | 0cn 10032 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 0 ∈
ℂ |
31 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (abs
∘ − ) = (abs ∘ − ) |
32 | 31 | cnmetdval 22574 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑦 − 𝑥) ∈ ℂ ∧ 0 ∈ ℂ)
→ ((𝑦 − 𝑥)(abs ∘ − )0) =
(abs‘((𝑦 −
𝑥) −
0))) |
33 | 27, 30, 32 | sylancl 694 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦 − 𝑥)(abs ∘ − )0) =
(abs‘((𝑦 −
𝑥) −
0))) |
34 | 31 | cnmetdval 22574 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑦(abs ∘ − )𝑥) = (abs‘(𝑦 − 𝑥))) |
35 | 25, 26, 34 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦(abs ∘ − )𝑥) = (abs‘(𝑦 − 𝑥))) |
36 | 29, 33, 35 | 3eqtr4d 2666 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦 − 𝑥)(abs ∘ − )0) = (𝑦(abs ∘ − )𝑥)) |
37 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) |
38 | 6 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (abs ∘ − )
∈ (∞Met‘ℂ)) |
39 | | simpllr 799 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) ∧
𝑧 ∈ ℂ) →
𝑟 ∈
ℝ+) |
40 | 39 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑟 ∈ ℝ+) |
41 | 40 | rpxrd 11873 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 𝑟 ∈ ℝ*) |
42 | | elbl3 22197 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((abs
∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑟 ∈ ℝ*) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟) ↔ (𝑦(abs ∘ − )𝑥) < 𝑟)) |
43 | 38, 41, 26, 25, 42 | syl22anc 1327 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟) ↔ (𝑦(abs ∘ − )𝑥) < 𝑟)) |
44 | 37, 43 | mpbid 222 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦(abs ∘ − )𝑥) < 𝑟) |
45 | 36, 44 | eqbrtrd 4675 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦 − 𝑥)(abs ∘ − )0) < 𝑟) |
46 | | 0cnd 10033 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → 0 ∈
ℂ) |
47 | | elbl3 22197 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((abs
∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑟 ∈ ℝ*) ∧ (0 ∈
ℂ ∧ (𝑦 −
𝑥) ∈ ℂ)) →
((𝑦 − 𝑥) ∈ (0(ball‘(abs
∘ − ))𝑟) ↔
((𝑦 − 𝑥)(abs ∘ − )0) <
𝑟)) |
48 | 38, 41, 46, 27, 47 | syl22anc 1327 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((𝑦 − 𝑥) ∈ (0(ball‘(abs ∘ −
))𝑟) ↔ ((𝑦 − 𝑥)(abs ∘ − )0) < 𝑟)) |
49 | 45, 48 | mpbird 247 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (𝑦 − 𝑥) ∈ (0(ball‘(abs ∘ −
))𝑟)) |
50 | | efsub 14830 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) →
(exp‘(𝑦 − 𝑥)) = ((exp‘𝑦) / (exp‘𝑥))) |
51 | 25, 26, 50 | syl2anc 693 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → (exp‘(𝑦 − 𝑥)) = ((exp‘𝑦) / (exp‘𝑥))) |
52 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 = (𝑦 − 𝑥) → (exp‘𝑤) = (exp‘(𝑦 − 𝑥))) |
53 | 52 | eqeq1d 2624 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 = (𝑦 − 𝑥) → ((exp‘𝑤) = ((exp‘𝑦) / (exp‘𝑥)) ↔ (exp‘(𝑦 − 𝑥)) = ((exp‘𝑦) / (exp‘𝑥)))) |
54 | 53 | rspcev 3309 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑦 − 𝑥) ∈ (0(ball‘(abs ∘ −
))𝑟) ∧
(exp‘(𝑦 − 𝑥)) = ((exp‘𝑦) / (exp‘𝑥))) → ∃𝑤 ∈ (0(ball‘(abs
∘ − ))𝑟)(exp‘𝑤) = ((exp‘𝑦) / (exp‘𝑥))) |
55 | 49, 51, 54 | syl2anc 693 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ∃𝑤 ∈ (0(ball‘(abs
∘ − ))𝑟)(exp‘𝑤) = ((exp‘𝑦) / (exp‘𝑥))) |
56 | | oveq1 6657 |
. . . . . . . . . . . . . . . . . . 19
⊢
((exp‘𝑦) =
𝑧 → ((exp‘𝑦) / (exp‘𝑥)) = (𝑧 / (exp‘𝑥))) |
57 | 56 | eqeq2d 2632 |
. . . . . . . . . . . . . . . . . 18
⊢
((exp‘𝑦) =
𝑧 → ((exp‘𝑤) = ((exp‘𝑦) / (exp‘𝑥)) ↔ (exp‘𝑤) = (𝑧 / (exp‘𝑥)))) |
58 | 57 | rexbidv 3052 |
. . . . . . . . . . . . . . . . 17
⊢
((exp‘𝑦) =
𝑧 → (∃𝑤 ∈ (0(ball‘(abs
∘ − ))𝑟)(exp‘𝑤) = ((exp‘𝑦) / (exp‘𝑥)) ↔ ∃𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)(exp‘𝑤) = (𝑧 / (exp‘𝑥)))) |
59 | 55, 58 | syl5ibcom 235 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) → ((exp‘𝑦) = 𝑧 → ∃𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)(exp‘𝑤) = (𝑧 / (exp‘𝑥)))) |
60 | 59 | rexlimdva 3031 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) ∧
𝑧 ∈ ℂ) →
(∃𝑦 ∈ (𝑥(ball‘(abs ∘ −
))𝑟)(exp‘𝑦) = 𝑧 → ∃𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)(exp‘𝑤) = (𝑧 / (exp‘𝑥)))) |
61 | | eqcom 2629 |
. . . . . . . . . . . . . . . . . 18
⊢
((exp‘𝑤) =
(𝑧 / (exp‘𝑥)) ↔ (𝑧 / (exp‘𝑥)) = (exp‘𝑤)) |
62 | | simplr 792 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → 𝑧 ∈
ℂ) |
63 | | simp-4l 806 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → 𝑥 ∈
ℂ) |
64 | | efcl 14813 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ ℂ →
(exp‘𝑥) ∈
ℂ) |
65 | 63, 64 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) →
(exp‘𝑥) ∈
ℂ) |
66 | 39 | rpxrd 11873 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) ∧
𝑧 ∈ ℂ) →
𝑟 ∈
ℝ*) |
67 | | blssm 22223 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((abs
∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ
∧ 𝑟 ∈
ℝ*) → (0(ball‘(abs ∘ − ))𝑟) ⊆
ℂ) |
68 | 6, 30, 67 | mp3an12 1414 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑟 ∈ ℝ*
→ (0(ball‘(abs ∘ − ))𝑟) ⊆ ℂ) |
69 | 66, 68 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) ∧
𝑧 ∈ ℂ) →
(0(ball‘(abs ∘ − ))𝑟) ⊆ ℂ) |
70 | 69 | sselda 3603 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → 𝑤 ∈
ℂ) |
71 | | efcl 14813 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 ∈ ℂ →
(exp‘𝑤) ∈
ℂ) |
72 | 70, 71 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) →
(exp‘𝑤) ∈
ℂ) |
73 | | efne0 14827 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ ℂ →
(exp‘𝑥) ≠
0) |
74 | 63, 73 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) →
(exp‘𝑥) ≠
0) |
75 | 62, 65, 72, 74 | divmuld 10823 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → ((𝑧 / (exp‘𝑥)) = (exp‘𝑤) ↔ ((exp‘𝑥) · (exp‘𝑤)) = 𝑧)) |
76 | 61, 75 | syl5bb 272 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) →
((exp‘𝑤) = (𝑧 / (exp‘𝑥)) ↔ ((exp‘𝑥) · (exp‘𝑤)) = 𝑧)) |
77 | 63, 70 | pncan2d 10394 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → ((𝑥 + 𝑤) − 𝑥) = 𝑤) |
78 | 70 | subid1d 10381 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → (𝑤 − 0) = 𝑤) |
79 | 77, 78 | eqtr4d 2659 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → ((𝑥 + 𝑤) − 𝑥) = (𝑤 − 0)) |
80 | 79 | fveq2d 6195 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) →
(abs‘((𝑥 + 𝑤) − 𝑥)) = (abs‘(𝑤 − 0))) |
81 | 63, 70 | addcld 10059 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → (𝑥 + 𝑤) ∈ ℂ) |
82 | 31 | cnmetdval 22574 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑥 + 𝑤) ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑥 + 𝑤)(abs ∘ − )𝑥) = (abs‘((𝑥 + 𝑤) − 𝑥))) |
83 | 81, 63, 82 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → ((𝑥 + 𝑤)(abs ∘ − )𝑥) = (abs‘((𝑥 + 𝑤) − 𝑥))) |
84 | 31 | cnmetdval 22574 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑤 ∈ ℂ ∧ 0 ∈
ℂ) → (𝑤(abs
∘ − )0) = (abs‘(𝑤 − 0))) |
85 | 70, 30, 84 | sylancl 694 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → (𝑤(abs ∘ − )0) =
(abs‘(𝑤 −
0))) |
86 | 80, 83, 85 | 3eqtr4d 2666 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → ((𝑥 + 𝑤)(abs ∘ − )𝑥) = (𝑤(abs ∘ − )0)) |
87 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → 𝑤 ∈ (0(ball‘(abs
∘ − ))𝑟)) |
88 | 6 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → (abs ∘
− ) ∈ (∞Met‘ℂ)) |
89 | 39 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → 𝑟 ∈
ℝ+) |
90 | 89 | rpxrd 11873 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → 𝑟 ∈
ℝ*) |
91 | | 0cnd 10033 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → 0 ∈
ℂ) |
92 | | elbl3 22197 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((abs
∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑟 ∈ ℝ*) ∧ (0 ∈
ℂ ∧ 𝑤 ∈
ℂ)) → (𝑤 ∈
(0(ball‘(abs ∘ − ))𝑟) ↔ (𝑤(abs ∘ − )0) < 𝑟)) |
93 | 88, 90, 91, 70, 92 | syl22anc 1327 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → (𝑤 ∈ (0(ball‘(abs
∘ − ))𝑟) ↔
(𝑤(abs ∘ − )0)
< 𝑟)) |
94 | 87, 93 | mpbid 222 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → (𝑤(abs ∘ − )0) <
𝑟) |
95 | 86, 94 | eqbrtrd 4675 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → ((𝑥 + 𝑤)(abs ∘ − )𝑥) < 𝑟) |
96 | | elbl3 22197 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((abs
∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑟 ∈ ℝ*) ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝑤) ∈ ℂ)) → ((𝑥 + 𝑤) ∈ (𝑥(ball‘(abs ∘ − ))𝑟) ↔ ((𝑥 + 𝑤)(abs ∘ − )𝑥) < 𝑟)) |
97 | 88, 90, 63, 81, 96 | syl22anc 1327 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → ((𝑥 + 𝑤) ∈ (𝑥(ball‘(abs ∘ − ))𝑟) ↔ ((𝑥 + 𝑤)(abs ∘ − )𝑥) < 𝑟)) |
98 | 95, 97 | mpbird 247 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → (𝑥 + 𝑤) ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) |
99 | | efadd 14824 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℂ) →
(exp‘(𝑥 + 𝑤)) = ((exp‘𝑥) · (exp‘𝑤))) |
100 | 63, 70, 99 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) →
(exp‘(𝑥 + 𝑤)) = ((exp‘𝑥) · (exp‘𝑤))) |
101 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = (𝑥 + 𝑤) → (exp‘𝑦) = (exp‘(𝑥 + 𝑤))) |
102 | 101 | eqeq1d 2624 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = (𝑥 + 𝑤) → ((exp‘𝑦) = ((exp‘𝑥) · (exp‘𝑤)) ↔ (exp‘(𝑥 + 𝑤)) = ((exp‘𝑥) · (exp‘𝑤)))) |
103 | 102 | rspcev 3309 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑥 + 𝑤) ∈ (𝑥(ball‘(abs ∘ − ))𝑟) ∧ (exp‘(𝑥 + 𝑤)) = ((exp‘𝑥) · (exp‘𝑤))) → ∃𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)(exp‘𝑦) = ((exp‘𝑥) · (exp‘𝑤))) |
104 | 98, 100, 103 | syl2anc 693 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) → ∃𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)(exp‘𝑦) = ((exp‘𝑥) · (exp‘𝑤))) |
105 | | eqeq2 2633 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((exp‘𝑥)
· (exp‘𝑤)) =
𝑧 → ((exp‘𝑦) = ((exp‘𝑥) · (exp‘𝑤)) ↔ (exp‘𝑦) = 𝑧)) |
106 | 105 | rexbidv 3052 |
. . . . . . . . . . . . . . . . . 18
⊢
(((exp‘𝑥)
· (exp‘𝑤)) =
𝑧 → (∃𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)(exp‘𝑦) = ((exp‘𝑥) · (exp‘𝑤)) ↔ ∃𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)(exp‘𝑦) = 𝑧)) |
107 | 104, 106 | syl5ibcom 235 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) →
(((exp‘𝑥) ·
(exp‘𝑤)) = 𝑧 → ∃𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)(exp‘𝑦) = 𝑧)) |
108 | 76, 107 | sylbid 230 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑥 ∈
ℂ ∧ 𝑟 ∈
ℝ+) ∧ 𝑟 < π) ∧ 𝑧 ∈ ℂ) ∧ 𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)) →
((exp‘𝑤) = (𝑧 / (exp‘𝑥)) → ∃𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)(exp‘𝑦) = 𝑧)) |
109 | 108 | rexlimdva 3031 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) ∧
𝑧 ∈ ℂ) →
(∃𝑤 ∈
(0(ball‘(abs ∘ − ))𝑟)(exp‘𝑤) = (𝑧 / (exp‘𝑥)) → ∃𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)(exp‘𝑦) = 𝑧)) |
110 | 60, 109 | impbid 202 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) ∧
𝑧 ∈ ℂ) →
(∃𝑦 ∈ (𝑥(ball‘(abs ∘ −
))𝑟)(exp‘𝑦) = 𝑧 ↔ ∃𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)(exp‘𝑤) = (𝑧 / (exp‘𝑥)))) |
111 | | ffn 6045 |
. . . . . . . . . . . . . . . 16
⊢
(exp:ℂ⟶ℂ → exp Fn ℂ) |
112 | 14, 111 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ exp Fn
ℂ |
113 | | fvelimab 6253 |
. . . . . . . . . . . . . . 15
⊢ ((exp Fn
ℂ ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ ℂ) → (𝑧 ∈ (exp “ (𝑥(ball‘(abs ∘ −
))𝑟)) ↔ ∃𝑦 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)(exp‘𝑦) = 𝑧)) |
114 | 112, 24, 113 | sylancr 695 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) ∧
𝑧 ∈ ℂ) →
(𝑧 ∈ (exp “
(𝑥(ball‘(abs ∘
− ))𝑟)) ↔
∃𝑦 ∈ (𝑥(ball‘(abs ∘ −
))𝑟)(exp‘𝑦) = 𝑧)) |
115 | | fvelimab 6253 |
. . . . . . . . . . . . . . 15
⊢ ((exp Fn
ℂ ∧ (0(ball‘(abs ∘ − ))𝑟) ⊆ ℂ) → ((𝑧 / (exp‘𝑥)) ∈ (exp “ (0(ball‘(abs
∘ − ))𝑟))
↔ ∃𝑤 ∈
(0(ball‘(abs ∘ − ))𝑟)(exp‘𝑤) = (𝑧 / (exp‘𝑥)))) |
116 | 112, 69, 115 | sylancr 695 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) ∧
𝑧 ∈ ℂ) →
((𝑧 / (exp‘𝑥)) ∈ (exp “
(0(ball‘(abs ∘ − ))𝑟)) ↔ ∃𝑤 ∈ (0(ball‘(abs ∘ −
))𝑟)(exp‘𝑤) = (𝑧 / (exp‘𝑥)))) |
117 | 110, 114,
116 | 3bitr4d 300 |
. . . . . . . . . . . . 13
⊢ ((((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) ∧
𝑧 ∈ ℂ) →
(𝑧 ∈ (exp “
(𝑥(ball‘(abs ∘
− ))𝑟)) ↔ (𝑧 / (exp‘𝑥)) ∈ (exp “ (0(ball‘(abs
∘ − ))𝑟)))) |
118 | 117 | rabbi2dva 3821 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) →
(ℂ ∩ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟))) = {𝑧 ∈ ℂ ∣ (𝑧 / (exp‘𝑥)) ∈ (exp “ (0(ball‘(abs
∘ − ))𝑟))}) |
119 | 19, 118 | syl5eqr 2670 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) →
(exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) = {𝑧 ∈ ℂ ∣ (𝑧 / (exp‘𝑥)) ∈ (exp “ (0(ball‘(abs
∘ − ))𝑟))}) |
120 | | eqid 2622 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) = (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) |
121 | 120 | mptpreima 5628 |
. . . . . . . . . . 11
⊢ (◡(𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) “ (exp “ (0(ball‘(abs
∘ − ))𝑟))) =
{𝑧 ∈ ℂ ∣
(𝑧 / (exp‘𝑥)) ∈ (exp “
(0(ball‘(abs ∘ − ))𝑟))} |
122 | 119, 121 | syl6eqr 2674 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) →
(exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) = (◡(𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) “ (exp “ (0(ball‘(abs
∘ − ))𝑟)))) |
123 | 64 | ad2antrr 762 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) →
(exp‘𝑥) ∈
ℂ) |
124 | 73 | ad2antrr 762 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) →
(exp‘𝑥) ≠
0) |
125 | 120 | divccncf 22709 |
. . . . . . . . . . . . 13
⊢
(((exp‘𝑥)
∈ ℂ ∧ (exp‘𝑥) ≠ 0) → (𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) ∈ (ℂ–cn→ℂ)) |
126 | 123, 124,
125 | syl2anc 693 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) →
(𝑧 ∈ ℂ ↦
(𝑧 / (exp‘𝑥))) ∈ (ℂ–cn→ℂ)) |
127 | 1 | cncfcn1 22713 |
. . . . . . . . . . . 12
⊢
(ℂ–cn→ℂ) =
(𝐽 Cn 𝐽) |
128 | 126, 127 | syl6eleq 2711 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) →
(𝑧 ∈ ℂ ↦
(𝑧 / (exp‘𝑥))) ∈ (𝐽 Cn 𝐽)) |
129 | 1 | efopnlem2 24403 |
. . . . . . . . . . . 12
⊢ ((𝑟 ∈ ℝ+
∧ 𝑟 < π) →
(exp “ (0(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽) |
130 | 129 | adantll 750 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) →
(exp “ (0(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽) |
131 | | cnima 21069 |
. . . . . . . . . . 11
⊢ (((𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) ∈ (𝐽 Cn 𝐽) ∧ (exp “ (0(ball‘(abs
∘ − ))𝑟))
∈ 𝐽) → (◡(𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) “ (exp “ (0(ball‘(abs
∘ − ))𝑟)))
∈ 𝐽) |
132 | 128, 130,
131 | syl2anc 693 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) →
(◡(𝑧 ∈ ℂ ↦ (𝑧 / (exp‘𝑥))) “ (exp “ (0(ball‘(abs
∘ − ))𝑟)))
∈ 𝐽) |
133 | 122, 132 | eqeltrd 2701 |
. . . . . . . . 9
⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) →
(exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ∈ 𝐽) |
134 | | blcntr 22218 |
. . . . . . . . . . . 12
⊢ (((abs
∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+) → 𝑥 ∈ (𝑥(ball‘(abs ∘ − ))𝑟)) |
135 | 6, 134 | mp3an1 1411 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
→ 𝑥 ∈ (𝑥(ball‘(abs ∘ −
))𝑟)) |
136 | | ffun 6048 |
. . . . . . . . . . . . 13
⊢
(exp:ℂ⟶ℂ → Fun exp) |
137 | 14, 136 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ Fun
exp |
138 | 14 | fdmi 6052 |
. . . . . . . . . . . . 13
⊢ dom exp =
ℂ |
139 | 23, 138 | syl6sseqr 3652 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
→ (𝑥(ball‘(abs
∘ − ))𝑟)
⊆ dom exp) |
140 | | funfvima2 6493 |
. . . . . . . . . . . 12
⊢ ((Fun exp
∧ (𝑥(ball‘(abs
∘ − ))𝑟)
⊆ dom exp) → (𝑥
∈ (𝑥(ball‘(abs
∘ − ))𝑟) →
(exp‘𝑥) ∈ (exp
“ (𝑥(ball‘(abs
∘ − ))𝑟)))) |
141 | 137, 139,
140 | sylancr 695 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
→ (𝑥 ∈ (𝑥(ball‘(abs ∘ −
))𝑟) →
(exp‘𝑥) ∈ (exp
“ (𝑥(ball‘(abs
∘ − ))𝑟)))) |
142 | 135, 141 | mpd 15 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
→ (exp‘𝑥) ∈
(exp “ (𝑥(ball‘(abs ∘ − ))𝑟))) |
143 | 142 | adantr 481 |
. . . . . . . . 9
⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) →
(exp‘𝑥) ∈ (exp
“ (𝑥(ball‘(abs
∘ − ))𝑟))) |
144 | | eleq2 2690 |
. . . . . . . . . . . 12
⊢ (𝑦 = (exp “ (𝑥(ball‘(abs ∘ −
))𝑟)) →
((exp‘𝑥) ∈ 𝑦 ↔ (exp‘𝑥) ∈ (exp “ (𝑥(ball‘(abs ∘ −
))𝑟)))) |
145 | | sseq1 3626 |
. . . . . . . . . . . 12
⊢ (𝑦 = (exp “ (𝑥(ball‘(abs ∘ −
))𝑟)) → (𝑦 ⊆ (exp “ 𝑆) ↔ (exp “ (𝑥(ball‘(abs ∘ −
))𝑟)) ⊆ (exp “
𝑆))) |
146 | 144, 145 | anbi12d 747 |
. . . . . . . . . . 11
⊢ (𝑦 = (exp “ (𝑥(ball‘(abs ∘ −
))𝑟)) →
(((exp‘𝑥) ∈
𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)) ↔ ((exp‘𝑥) ∈ (exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ∧ (exp “ (𝑥(ball‘(abs ∘ −
))𝑟)) ⊆ (exp “
𝑆)))) |
147 | 146 | rspcev 3309 |
. . . . . . . . . 10
⊢ (((exp
“ (𝑥(ball‘(abs
∘ − ))𝑟))
∈ 𝐽 ∧
((exp‘𝑥) ∈ (exp
“ (𝑥(ball‘(abs
∘ − ))𝑟)) ∧
(exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ (exp “ 𝑆))) → ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))) |
148 | 147 | expr 643 |
. . . . . . . . 9
⊢ (((exp
“ (𝑥(ball‘(abs
∘ − ))𝑟))
∈ 𝐽 ∧
(exp‘𝑥) ∈ (exp
“ (𝑥(ball‘(abs
∘ − ))𝑟)))
→ ((exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ (exp “ 𝑆) → ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)))) |
149 | 133, 143,
148 | syl2anc 693 |
. . . . . . . 8
⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) →
((exp “ (𝑥(ball‘(abs ∘ − ))𝑟)) ⊆ (exp “ 𝑆) → ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)))) |
150 | 12, 149 | syl5 34 |
. . . . . . 7
⊢ (((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
∧ 𝑟 < π) →
((𝑥(ball‘(abs ∘
− ))𝑟) ⊆ 𝑆 → ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)))) |
151 | 150 | expimpd 629 |
. . . . . 6
⊢ ((𝑥 ∈ ℂ ∧ 𝑟 ∈ ℝ+)
→ ((𝑟 < π ∧
(𝑥(ball‘(abs ∘
− ))𝑟) ⊆ 𝑆) → ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)))) |
152 | 151 | rexlimdva 3031 |
. . . . 5
⊢ (𝑥 ∈ ℂ →
(∃𝑟 ∈
ℝ+ (𝑟 <
π ∧ (𝑥(ball‘(abs ∘ − ))𝑟) ⊆ 𝑆) → ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)))) |
153 | 5, 11, 152 | sylc 65 |
. . . 4
⊢ ((𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))) |
154 | 153 | ralrimiva 2966 |
. . 3
⊢ (𝑆 ∈ 𝐽 → ∀𝑥 ∈ 𝑆 ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))) |
155 | | eleq1 2689 |
. . . . . . 7
⊢ (𝑧 = (exp‘𝑥) → (𝑧 ∈ 𝑦 ↔ (exp‘𝑥) ∈ 𝑦)) |
156 | 155 | anbi1d 741 |
. . . . . 6
⊢ (𝑧 = (exp‘𝑥) → ((𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)) ↔ ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)))) |
157 | 156 | rexbidv 3052 |
. . . . 5
⊢ (𝑧 = (exp‘𝑥) → (∃𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)) ↔ ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)))) |
158 | 157 | ralima 6498 |
. . . 4
⊢ ((exp Fn
ℂ ∧ 𝑆 ⊆
ℂ) → (∀𝑧
∈ (exp “ 𝑆)∃𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)) ↔ ∀𝑥 ∈ 𝑆 ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)))) |
159 | 112, 4, 158 | sylancr 695 |
. . 3
⊢ (𝑆 ∈ 𝐽 → (∀𝑧 ∈ (exp “ 𝑆)∃𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)) ↔ ∀𝑥 ∈ 𝑆 ∃𝑦 ∈ 𝐽 ((exp‘𝑥) ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)))) |
160 | 154, 159 | mpbird 247 |
. 2
⊢ (𝑆 ∈ 𝐽 → ∀𝑧 ∈ (exp “ 𝑆)∃𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))) |
161 | 1 | cnfldtop 22587 |
. . 3
⊢ 𝐽 ∈ Top |
162 | | eltop2 20779 |
. . 3
⊢ (𝐽 ∈ Top → ((exp “
𝑆) ∈ 𝐽 ↔ ∀𝑧 ∈ (exp “ 𝑆)∃𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆)))) |
163 | 161, 162 | ax-mp 5 |
. 2
⊢ ((exp
“ 𝑆) ∈ 𝐽 ↔ ∀𝑧 ∈ (exp “ 𝑆)∃𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ∧ 𝑦 ⊆ (exp “ 𝑆))) |
164 | 160, 163 | sylibr 224 |
1
⊢ (𝑆 ∈ 𝐽 → (exp “ 𝑆) ∈ 𝐽) |