| Step |
Hyp |
Ref |
Expression |
| 1 |
|
cau3.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
| 2 |
|
abscl |
⊢ ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
| 3 |
2
|
ralimi |
⊢ ( ∀ 𝑘 ∈ 𝑍 ( 𝐹 ‘ 𝑘 ) ∈ ℂ → ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
| 4 |
1
|
r19.29uz |
⊢ ( ( ∀ 𝑘 ∈ 𝑍 ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ) |
| 5 |
4
|
ex |
⊢ ( ∀ 𝑘 ∈ 𝑍 ( 𝐹 ‘ 𝑘 ) ∈ ℂ → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ) ) |
| 6 |
5
|
ralimdv |
⊢ ( ∀ 𝑘 ∈ 𝑍 ( 𝐹 ‘ 𝑘 ) ∈ ℂ → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ) ) |
| 7 |
1
|
caubnd2 |
⊢ ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) → ∃ 𝑧 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) |
| 8 |
6 7
|
syl6 |
⊢ ( ∀ 𝑘 ∈ 𝑍 ( 𝐹 ‘ 𝑘 ) ∈ ℂ → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 → ∃ 𝑧 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) ) |
| 9 |
|
fzssuz |
⊢ ( 𝑀 ... 𝑗 ) ⊆ ( ℤ≥ ‘ 𝑀 ) |
| 10 |
9 1
|
sseqtrri |
⊢ ( 𝑀 ... 𝑗 ) ⊆ 𝑍 |
| 11 |
|
ssralv |
⊢ ( ( 𝑀 ... 𝑗 ) ⊆ 𝑍 → ( ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ → ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) ) |
| 12 |
10 11
|
ax-mp |
⊢ ( ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ → ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
| 13 |
|
fzfi |
⊢ ( 𝑀 ... 𝑗 ) ∈ Fin |
| 14 |
|
fimaxre3 |
⊢ ( ( ( 𝑀 ... 𝑗 ) ∈ Fin ∧ ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) → ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ 𝑥 ) |
| 15 |
13 14
|
mpan |
⊢ ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ → ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ 𝑥 ) |
| 16 |
|
peano2re |
⊢ ( 𝑥 ∈ ℝ → ( 𝑥 + 1 ) ∈ ℝ ) |
| 17 |
16
|
adantl |
⊢ ( ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( 𝑥 + 1 ) ∈ ℝ ) |
| 18 |
|
ltp1 |
⊢ ( 𝑥 ∈ ℝ → 𝑥 < ( 𝑥 + 1 ) ) |
| 19 |
18
|
adantl |
⊢ ( ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ 𝑥 ∈ ℝ ) → 𝑥 < ( 𝑥 + 1 ) ) |
| 20 |
16
|
adantl |
⊢ ( ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( 𝑥 + 1 ) ∈ ℝ ) |
| 21 |
|
lelttr |
⊢ ( ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ ( 𝑥 + 1 ) ∈ ℝ ) → ( ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ 𝑥 ∧ 𝑥 < ( 𝑥 + 1 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 𝑥 + 1 ) ) ) |
| 22 |
20 21
|
mpd3an3 |
⊢ ( ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ 𝑥 ∧ 𝑥 < ( 𝑥 + 1 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 𝑥 + 1 ) ) ) |
| 23 |
19 22
|
mpan2d |
⊢ ( ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ 𝑥 → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 𝑥 + 1 ) ) ) |
| 24 |
23
|
expcom |
⊢ ( 𝑥 ∈ ℝ → ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ → ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ 𝑥 → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 𝑥 + 1 ) ) ) ) |
| 25 |
24
|
ralimdv |
⊢ ( 𝑥 ∈ ℝ → ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ → ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ 𝑥 → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 𝑥 + 1 ) ) ) ) |
| 26 |
25
|
impcom |
⊢ ( ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ 𝑥 → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 𝑥 + 1 ) ) ) |
| 27 |
|
ralim |
⊢ ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ 𝑥 → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 𝑥 + 1 ) ) → ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ 𝑥 → ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 𝑥 + 1 ) ) ) |
| 28 |
26 27
|
syl |
⊢ ( ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ 𝑥 → ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 𝑥 + 1 ) ) ) |
| 29 |
|
brralrspcev |
⊢ ( ( ( 𝑥 + 1 ) ∈ ℝ ∧ ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < ( 𝑥 + 1 ) ) → ∃ 𝑤 ∈ ℝ ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ) |
| 30 |
17 28 29
|
syl6an |
⊢ ( ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ 𝑥 → ∃ 𝑤 ∈ ℝ ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ) ) |
| 31 |
30
|
rexlimdva |
⊢ ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ → ( ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ 𝑥 → ∃ 𝑤 ∈ ℝ ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ) ) |
| 32 |
15 31
|
mpd |
⊢ ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ → ∃ 𝑤 ∈ ℝ ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ) |
| 33 |
12 32
|
syl |
⊢ ( ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ → ∃ 𝑤 ∈ ℝ ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ) |
| 34 |
|
max1 |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ) → 𝑤 ≤ if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ) |
| 35 |
34
|
3adant3 |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) → 𝑤 ≤ if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ) |
| 36 |
|
simp3 |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
| 37 |
|
simp1 |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) → 𝑤 ∈ ℝ ) |
| 38 |
|
ifcl |
⊢ ( ( 𝑧 ∈ ℝ ∧ 𝑤 ∈ ℝ ) → if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ∈ ℝ ) |
| 39 |
38
|
ancoms |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ) → if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ∈ ℝ ) |
| 40 |
39
|
3adant3 |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) → if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ∈ ℝ ) |
| 41 |
|
ltletr |
⊢ ( ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ∈ ℝ ) → ( ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∧ 𝑤 ≤ if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ) ) |
| 42 |
36 37 40 41
|
syl3anc |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) → ( ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∧ 𝑤 ≤ if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ) ) |
| 43 |
35 42
|
mpan2d |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) → ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ) ) |
| 44 |
|
max2 |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ) → 𝑧 ≤ if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ) |
| 45 |
44
|
3adant3 |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) → 𝑧 ≤ if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ) |
| 46 |
|
simp2 |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) → 𝑧 ∈ ℝ ) |
| 47 |
|
ltletr |
⊢ ( ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ∈ ℝ ) → ( ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ∧ 𝑧 ≤ if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ) ) |
| 48 |
36 46 40 47
|
syl3anc |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) → ( ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ∧ 𝑧 ≤ if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ) ) |
| 49 |
45 48
|
mpan2d |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) → ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ) ) |
| 50 |
43 49
|
jaod |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) → ( ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∨ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ) ) |
| 51 |
50
|
3expia |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ) → ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ → ( ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∨ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ) ) ) |
| 52 |
51
|
ralimdv |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ) → ( ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ → ∀ 𝑘 ∈ 𝑍 ( ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∨ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ) ) ) |
| 53 |
|
ralim |
⊢ ( ∀ 𝑘 ∈ 𝑍 ( ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∨ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ) → ( ∀ 𝑘 ∈ 𝑍 ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∨ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) → ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ) ) |
| 54 |
52 53
|
syl6 |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ) → ( ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ → ( ∀ 𝑘 ∈ 𝑍 ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∨ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) → ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ) ) ) |
| 55 |
|
brralrspcev |
⊢ ( ( if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ∈ ℝ ∧ ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ) → ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑦 ) |
| 56 |
55
|
ex |
⊢ ( if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) ∈ ℝ → ( ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑦 ) ) |
| 57 |
39 56
|
syl |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ) → ( ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < if ( 𝑤 ≤ 𝑧 , 𝑧 , 𝑤 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑦 ) ) |
| 58 |
54 57
|
syl6d |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ) → ( ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ → ( ∀ 𝑘 ∈ 𝑍 ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∨ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑦 ) ) ) |
| 59 |
|
uzssz |
⊢ ( ℤ≥ ‘ 𝑀 ) ⊆ ℤ |
| 60 |
1 59
|
eqsstri |
⊢ 𝑍 ⊆ ℤ |
| 61 |
60
|
sseli |
⊢ ( 𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ ) |
| 62 |
60
|
sseli |
⊢ ( 𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ ) |
| 63 |
|
uztric |
⊢ ( ( 𝑘 ∈ ℤ ∧ 𝑗 ∈ ℤ ) → ( 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ∨ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) |
| 64 |
61 62 63
|
syl2anr |
⊢ ( ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍 ) → ( 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ∨ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) |
| 65 |
|
simpr |
⊢ ( ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍 ) → 𝑘 ∈ 𝑍 ) |
| 66 |
65 1
|
eleqtrdi |
⊢ ( ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍 ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
| 67 |
|
elfzuzb |
⊢ ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) ↔ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ) ) |
| 68 |
67
|
baib |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) ↔ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ) ) |
| 69 |
66 68
|
syl |
⊢ ( ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍 ) → ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) ↔ 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ) ) |
| 70 |
69
|
orbi1d |
⊢ ( ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) ∨ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ↔ ( 𝑗 ∈ ( ℤ≥ ‘ 𝑘 ) ∨ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ) |
| 71 |
64 70
|
mpbird |
⊢ ( ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍 ) → ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) ∨ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) |
| 72 |
71
|
ex |
⊢ ( 𝑗 ∈ 𝑍 → ( 𝑘 ∈ 𝑍 → ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) ∨ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ) |
| 73 |
|
pm3.48 |
⊢ ( ( ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ) ∧ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) ) → ( ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) ∨ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∨ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) ) ) |
| 74 |
72 73
|
syl9 |
⊢ ( 𝑗 ∈ 𝑍 → ( ( ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ) ∧ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) ) → ( 𝑘 ∈ 𝑍 → ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∨ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) ) ) ) |
| 75 |
74
|
alimdv |
⊢ ( 𝑗 ∈ 𝑍 → ( ∀ 𝑘 ( ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ) ∧ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) ) → ∀ 𝑘 ( 𝑘 ∈ 𝑍 → ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∨ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) ) ) ) |
| 76 |
|
df-ral |
⊢ ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ↔ ∀ 𝑘 ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ) ) |
| 77 |
|
df-ral |
⊢ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ↔ ∀ 𝑘 ( 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) ) |
| 78 |
76 77
|
anbi12i |
⊢ ( ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) ↔ ( ∀ 𝑘 ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ) ∧ ∀ 𝑘 ( 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) ) ) |
| 79 |
|
19.26 |
⊢ ( ∀ 𝑘 ( ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ) ∧ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) ) ↔ ( ∀ 𝑘 ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ) ∧ ∀ 𝑘 ( 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) ) ) |
| 80 |
78 79
|
bitr4i |
⊢ ( ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) ↔ ∀ 𝑘 ( ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ) ∧ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) ) ) |
| 81 |
|
df-ral |
⊢ ( ∀ 𝑘 ∈ 𝑍 ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∨ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) ↔ ∀ 𝑘 ( 𝑘 ∈ 𝑍 → ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∨ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) ) ) |
| 82 |
75 80 81
|
3imtr4g |
⊢ ( 𝑗 ∈ 𝑍 → ( ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) → ∀ 𝑘 ∈ 𝑍 ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∨ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) ) ) |
| 83 |
82
|
3impib |
⊢ ( ( 𝑗 ∈ 𝑍 ∧ ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) → ∀ 𝑘 ∈ 𝑍 ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∨ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) ) |
| 84 |
83
|
imim1i |
⊢ ( ( ∀ 𝑘 ∈ 𝑍 ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∨ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑦 ) → ( ( 𝑗 ∈ 𝑍 ∧ ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑦 ) ) |
| 85 |
84
|
3expd |
⊢ ( ( ∀ 𝑘 ∈ 𝑍 ( ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 ∨ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑦 ) → ( 𝑗 ∈ 𝑍 → ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 → ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑦 ) ) ) ) |
| 86 |
58 85
|
syl6 |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ) → ( ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ → ( 𝑗 ∈ 𝑍 → ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 → ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑦 ) ) ) ) ) |
| 87 |
86
|
com23 |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ) → ( 𝑗 ∈ 𝑍 → ( ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ → ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 → ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑦 ) ) ) ) ) |
| 88 |
87
|
expimpd |
⊢ ( 𝑤 ∈ ℝ → ( ( 𝑧 ∈ ℝ ∧ 𝑗 ∈ 𝑍 ) → ( ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ → ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 → ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑦 ) ) ) ) ) |
| 89 |
88
|
com3r |
⊢ ( ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ → ( 𝑤 ∈ ℝ → ( ( 𝑧 ∈ ℝ ∧ 𝑗 ∈ 𝑍 ) → ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 → ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑦 ) ) ) ) ) |
| 90 |
89
|
com34 |
⊢ ( ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ → ( 𝑤 ∈ ℝ → ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 → ( ( 𝑧 ∈ ℝ ∧ 𝑗 ∈ 𝑍 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 → ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑦 ) ) ) ) ) |
| 91 |
90
|
rexlimdv |
⊢ ( ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ → ( ∃ 𝑤 ∈ ℝ ∀ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑤 → ( ( 𝑧 ∈ ℝ ∧ 𝑗 ∈ 𝑍 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 → ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑦 ) ) ) ) |
| 92 |
33 91
|
mpd |
⊢ ( ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ → ( ( 𝑧 ∈ ℝ ∧ 𝑗 ∈ 𝑍 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 → ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑦 ) ) ) |
| 93 |
92
|
rexlimdvv |
⊢ ( ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ → ( ∃ 𝑧 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑧 → ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑦 ) ) |
| 94 |
3 8 93
|
sylsyld |
⊢ ( ∀ 𝑘 ∈ 𝑍 ( 𝐹 ‘ 𝑘 ) ∈ ℂ → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 → ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑦 ) ) |
| 95 |
94
|
imp |
⊢ ( ( ∀ 𝑘 ∈ 𝑍 ( 𝐹 ‘ 𝑘 ) ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ 𝑍 ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) < 𝑦 ) |