Step | Hyp | Ref
| Expression |
1 | | breq2 4587 |
. . . . . 6
⊢ (𝑥 = 𝑤 → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤)) |
2 | 1 | ralbidv 2969 |
. . . . 5
⊢ (𝑥 = 𝑤 → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤)) |
3 | 2 | rexralbidv 3040 |
. . . 4
⊢ (𝑥 = 𝑤 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤)) |
4 | 3 | cbvralv 3147 |
. . 3
⊢
(∀𝑥 ∈
ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 ↔ ∀𝑤 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤) |
5 | | rphalfcl 11734 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ+
→ (𝑥 / 2) ∈
ℝ+) |
6 | | breq2 4587 |
. . . . . . . . . 10
⊢ (𝑤 = (𝑥 / 2) → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤 ↔ (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2))) |
7 | 6 | ralbidv 2969 |
. . . . . . . . 9
⊢ (𝑤 = (𝑥 / 2) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤 ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2))) |
8 | 7 | rexralbidv 3040 |
. . . . . . . 8
⊢ (𝑤 = (𝑥 / 2) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2))) |
9 | 8 | rspcv 3278 |
. . . . . . 7
⊢ ((𝑥 / 2) ∈ ℝ+
→ (∀𝑤 ∈
ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2))) |
10 | 5, 9 | syl 17 |
. . . . . 6
⊢ (𝑥 ∈ ℝ+
→ (∀𝑤 ∈
ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2))) |
11 | 10 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(∀𝑤 ∈
ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2))) |
12 | | fveq2 6103 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚)) |
13 | 12 | fveq1d 6105 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑚)‘𝑧)) |
14 | 13 | oveq1d 6564 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑚 → (((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧)) = (((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) |
15 | 14 | fveq2d 6107 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑚 → (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) = (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧)))) |
16 | 15 | breq1d 4593 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑚 → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ↔ (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2))) |
17 | 16 | ralbidv 2969 |
. . . . . . . . 9
⊢ (𝑘 = 𝑚 → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2))) |
18 | 17 | cbvralv 3147 |
. . . . . . . 8
⊢
(∀𝑘 ∈
(ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ↔ ∀𝑚 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) |
19 | 18 | biimpi 205 |
. . . . . . 7
⊢
(∀𝑘 ∈
(ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑚 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) |
20 | | uzss 11584 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈
(ℤ≥‘𝑗) → (ℤ≥‘𝑘) ⊆
(ℤ≥‘𝑗)) |
21 | 20 | ad2antlr 759 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) →
(ℤ≥‘𝑘) ⊆ (ℤ≥‘𝑗)) |
22 | | ssralv 3629 |
. . . . . . . . . . . . . 14
⊢
((ℤ≥‘𝑘) ⊆ (ℤ≥‘𝑗) → (∀𝑚 ∈
(ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2))) |
23 | 21, 22 | syl 17 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) → (∀𝑚 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2))) |
24 | | r19.26 3046 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑧 ∈
𝑆 ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ∧ (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) ↔ (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2))) |
25 | | ulmcau.f |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑𝑚
𝑆)) |
26 | 25 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝐹:𝑍⟶(ℂ ↑𝑚
𝑆)) |
27 | 26 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑚 ∈ (ℤ≥‘𝑘)) → 𝐹:𝑍⟶(ℂ ↑𝑚
𝑆)) |
28 | | ulmcau.z |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ 𝑍 =
(ℤ≥‘𝑀) |
29 | 28 | uztrn2 11581 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍) |
30 | 29 | adantll 746 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍) |
31 | 28 | uztrn2 11581 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑘 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ≥‘𝑘)) → 𝑚 ∈ 𝑍) |
32 | 30, 31 | sylan 487 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑚 ∈ (ℤ≥‘𝑘)) → 𝑚 ∈ 𝑍) |
33 | 27, 32 | ffvelrnd 6268 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑚 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑚) ∈ (ℂ ↑𝑚
𝑆)) |
34 | | elmapi 7765 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐹‘𝑚) ∈ (ℂ ↑𝑚
𝑆) → (𝐹‘𝑚):𝑆⟶ℂ) |
35 | 33, 34 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑚 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑚):𝑆⟶ℂ) |
36 | 35 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑚 ∈ (ℤ≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑚)‘𝑧) ∈ ℂ) |
37 | 26 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ (ℂ ↑𝑚
𝑆)) |
38 | 37 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑚 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑗) ∈ (ℂ ↑𝑚
𝑆)) |
39 | | elmapi 7765 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐹‘𝑗) ∈ (ℂ ↑𝑚
𝑆) → (𝐹‘𝑗):𝑆⟶ℂ) |
40 | 38, 39 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑚 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑗):𝑆⟶ℂ) |
41 | 40 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑚 ∈ (ℤ≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑗)‘𝑧) ∈ ℂ) |
42 | 36, 41 | abssubd 14040 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑚 ∈ (ℤ≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) = (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧)))) |
43 | 42 | breq1d 4593 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑚 ∈ (ℤ≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → ((abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ↔ (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑥 / 2))) |
44 | 43 | biimpd 218 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑚 ∈ (ℤ≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → ((abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑥 / 2))) |
45 | | ffvelrn 6265 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝐹:𝑍⟶(ℂ ↑𝑚
𝑆) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (ℂ ↑𝑚
𝑆)) |
46 | 26, 29, 45 | syl2an 493 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗))) → (𝐹‘𝑘) ∈ (ℂ ↑𝑚
𝑆)) |
47 | 46 | anassrs 678 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ (ℂ ↑𝑚
𝑆)) |
48 | 47 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑚 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑘) ∈ (ℂ ↑𝑚
𝑆)) |
49 | | elmapi 7765 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐹‘𝑘) ∈ (ℂ ↑𝑚
𝑆) → (𝐹‘𝑘):𝑆⟶ℂ) |
50 | 48, 49 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑚 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑘):𝑆⟶ℂ) |
51 | 50 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑚 ∈ (ℤ≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ) |
52 | | rpre 11715 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℝ) |
53 | 52 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) → 𝑥 ∈ ℝ) |
54 | 53 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑚 ∈ (ℤ≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → 𝑥 ∈ ℝ) |
55 | | abs3lem 13926 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝐹‘𝑘)‘𝑧) ∈ ℂ ∧ ((𝐹‘𝑚)‘𝑧) ∈ ℂ) ∧ (((𝐹‘𝑗)‘𝑧) ∈ ℂ ∧ 𝑥 ∈ ℝ)) →
(((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ∧ (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑥 / 2)) → (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
56 | 51, 36, 41, 54, 55 | syl22anc 1319 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑚 ∈ (ℤ≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → (((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ∧ (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑥 / 2)) → (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
57 | 44, 56 | sylan2d 498 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑚 ∈ (ℤ≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → (((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ∧ (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) → (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
58 | 57 | ralimdva 2945 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑚 ∈ (ℤ≥‘𝑘)) → (∀𝑧 ∈ 𝑆 ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ∧ (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) → ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
59 | 24, 58 | syl5bir 232 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑚 ∈ (ℤ≥‘𝑘)) → ((∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) → ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
60 | 59 | expdimp 452 |
. . . . . . . . . . . . . . 15
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑚 ∈ (ℤ≥‘𝑘)) ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
61 | 60 | an32s 842 |
. . . . . . . . . . . . . 14
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) ∧ 𝑚 ∈ (ℤ≥‘𝑘)) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
62 | 61 | ralimdva 2945 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) → (∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
63 | 23, 62 | syld 46 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) → (∀𝑚 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
64 | 63 | impancom 455 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ ∀𝑚 ∈
(ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
65 | 64 | an32s 842 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈ 𝑍) ∧ ∀𝑚 ∈
(ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
66 | 65 | ralimdva 2945 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) → (∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
67 | 66 | ex 449 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) → (∀𝑚 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → (∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))) |
68 | 67 | com23 84 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → (∀𝑚 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))) |
69 | 19, 68 | mpdi 44 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
70 | 69 | reximdva 3000 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
71 | 11, 70 | syld 46 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(∀𝑤 ∈
ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
72 | 71 | ralrimdva 2952 |
. . 3
⊢ (𝜑 → (∀𝑤 ∈ ℝ+
∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
73 | 4, 72 | syl5bi 231 |
. 2
⊢ (𝜑 → (∀𝑥 ∈ ℝ+
∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
74 | | eluzelz 11573 |
. . . . . . . . 9
⊢ (𝑗 ∈
(ℤ≥‘𝑀) → 𝑗 ∈ ℤ) |
75 | 74, 28 | eleq2s 2706 |
. . . . . . . 8
⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ) |
76 | | uzid 11578 |
. . . . . . . 8
⊢ (𝑗 ∈ ℤ → 𝑗 ∈
(ℤ≥‘𝑗)) |
77 | 75, 76 | syl 17 |
. . . . . . 7
⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ≥‘𝑗)) |
78 | 77 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑗)) |
79 | | fveq2 6103 |
. . . . . . . 8
⊢ (𝑘 = 𝑗 → (ℤ≥‘𝑘) =
(ℤ≥‘𝑗)) |
80 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) |
81 | 80 | fveq1d 6105 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑗)‘𝑧)) |
82 | 81 | oveq1d 6564 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑗 → (((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧)) = (((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) |
83 | 82 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑗 → (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) = (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧)))) |
84 | 83 | breq1d 4593 |
. . . . . . . . 9
⊢ (𝑘 = 𝑗 → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
85 | 84 | ralbidv 2969 |
. . . . . . . 8
⊢ (𝑘 = 𝑗 → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
86 | 79, 85 | raleqbidv 3129 |
. . . . . . 7
⊢ (𝑘 = 𝑗 → (∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ ∀𝑚 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
87 | 86 | rspcv 3278 |
. . . . . 6
⊢ (𝑗 ∈
(ℤ≥‘𝑗) → (∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 → ∀𝑚 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
88 | 78, 87 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 → ∀𝑚 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
89 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘)) |
90 | 89 | fveq1d 6105 |
. . . . . . . . . . 11
⊢ (𝑚 = 𝑘 → ((𝐹‘𝑚)‘𝑧) = ((𝐹‘𝑘)‘𝑧)) |
91 | 90 | oveq2d 6565 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑘 → (((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧)) = (((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))) |
92 | 91 | fveq2d 6107 |
. . . . . . . . 9
⊢ (𝑚 = 𝑘 → (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) = (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧)))) |
93 | 92 | breq1d 4593 |
. . . . . . . 8
⊢ (𝑚 = 𝑘 → ((abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))) < 𝑥)) |
94 | 93 | ralbidv 2969 |
. . . . . . 7
⊢ (𝑚 = 𝑘 → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))) < 𝑥)) |
95 | 94 | cbvralv 3147 |
. . . . . 6
⊢
(∀𝑚 ∈
(ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))) < 𝑥) |
96 | 25 | ffvelrnda 6267 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ (ℂ ↑𝑚
𝑆)) |
97 | 96 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑗) ∈ (ℂ ↑𝑚
𝑆)) |
98 | 97, 39 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑗):𝑆⟶ℂ) |
99 | 98 | ffvelrnda 6267 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑗)‘𝑧) ∈ ℂ) |
100 | 25, 29, 45 | syl2an 493 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗))) → (𝐹‘𝑘) ∈ (ℂ ↑𝑚
𝑆)) |
101 | 100 | anassrs 678 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ (ℂ ↑𝑚
𝑆)) |
102 | 101, 49 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘):𝑆⟶ℂ) |
103 | 102 | ffvelrnda 6267 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ) |
104 | 99, 103 | abssubd 14040 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))) = (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧)))) |
105 | 104 | breq1d 4593 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → ((abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥)) |
106 | 105 | ralbidva 2968 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥)) |
107 | 106 | ralbidva 2968 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))) < 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥)) |
108 | 95, 107 | syl5bb 271 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑚 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥)) |
109 | 88, 108 | sylibd 228 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 → ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥)) |
110 | 109 | reximdva 3000 |
. . 3
⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥)) |
111 | 110 | ralimdv 2946 |
. 2
⊢ (𝜑 → (∀𝑥 ∈ ℝ+
∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥)) |
112 | 73, 111 | impbid 201 |
1
⊢ (𝜑 → (∀𝑥 ∈ ℝ+
∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |