Step |
Hyp |
Ref |
Expression |
1 |
|
ulmcau.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
ulmcau.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
ulmcau.s |
⊢ ( 𝜑 → 𝑆 ∈ 𝑉 ) |
4 |
|
ulmcau.f |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ( ℂ ↑m 𝑆 ) ) |
5 |
|
breq2 |
⊢ ( 𝑥 = 𝑤 → ( ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) ) ) < 𝑥 ↔ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) ) ) < 𝑤 ) ) |
6 |
5
|
ralbidv |
⊢ ( 𝑥 = 𝑤 → ( ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) ) ) < 𝑥 ↔ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) ) ) < 𝑤 ) ) |
7 |
6
|
rexralbidv |
⊢ ( 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 ) ) |
8 |
7
|
cbvralvw |
⊢ ( ∀ 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 ) |
9 |
|
rphalfcl |
⊢ ( 𝑥 ∈ ℝ+ → ( 𝑥 / 2 ) ∈ ℝ+ ) |
10 |
|
breq2 |
⊢ ( 𝑤 = ( 𝑥 / 2 ) → ( ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) ) ) < 𝑤 ↔ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) ) ) < ( 𝑥 / 2 ) ) ) |
11 |
10
|
ralbidv |
⊢ ( 𝑤 = ( 𝑥 / 2 ) → ( ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) ) ) < 𝑤 ↔ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) ) ) < ( 𝑥 / 2 ) ) ) |
12 |
11
|
rexralbidv |
⊢ ( 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 ) ) ) |
13 |
12
|
rspcv |
⊢ ( ( 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 ) ) ) |
14 |
9 13
|
syl |
⊢ ( 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 ) ) ) |
15 |
14
|
adantl |
⊢ ( ( 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 ) ) ) |
16 |
|
fveq2 |
⊢ ( 𝑘 = 𝑚 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑚 ) ) |
17 |
16
|
fveq1d |
⊢ ( 𝑘 = 𝑚 → ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) = ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑧 ) ) |
18 |
17
|
fvoveq1d |
⊢ ( 𝑘 = 𝑚 → ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) ) ) = ( abs ‘ ( ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) ) ) ) |
19 |
18
|
breq1d |
⊢ ( 𝑘 = 𝑚 → ( ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) ) ) < ( 𝑥 / 2 ) ↔ ( abs ‘ ( ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) ) ) < ( 𝑥 / 2 ) ) ) |
20 |
19
|
ralbidv |
⊢ ( 𝑘 = 𝑚 → ( ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) ) ) < ( 𝑥 / 2 ) ↔ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) ) ) < ( 𝑥 / 2 ) ) ) |
21 |
20
|
cbvralvw |
⊢ ( ∀ 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 ) ) |
22 |
21
|
biimpi |
⊢ ( ∀ 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 ) ) |
23 |
|
uzss |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) → ( ℤ≥ ‘ 𝑘 ) ⊆ ( ℤ≥ ‘ 𝑗 ) ) |
24 |
23
|
ad2antlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) ) ) < ( 𝑥 / 2 ) ) → ( ℤ≥ ‘ 𝑘 ) ⊆ ( ℤ≥ ‘ 𝑗 ) ) |
25 |
|
ssralv |
⊢ ( ( 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 ) ) ) |
26 |
24 25
|
syl |
⊢ ( ( ( ( ( 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 ) ) ) |
27 |
|
r19.26 |
⊢ ( ∀ 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 ) ) ) |
28 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝐹 : 𝑍 ⟶ ( ℂ ↑m 𝑆 ) ) |
29 |
28
|
ad3antrrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ) → 𝐹 : 𝑍 ⟶ ( ℂ ↑m 𝑆 ) ) |
30 |
1
|
uztrn2 |
⊢ ( ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝑘 ∈ 𝑍 ) |
31 |
30
|
adantll |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝑘 ∈ 𝑍 ) |
32 |
1
|
uztrn2 |
⊢ ( ( 𝑘 ∈ 𝑍 ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ) → 𝑚 ∈ 𝑍 ) |
33 |
31 32
|
sylan |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ) → 𝑚 ∈ 𝑍 ) |
34 |
29 33
|
ffvelrnd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ) → ( 𝐹 ‘ 𝑚 ) ∈ ( ℂ ↑m 𝑆 ) ) |
35 |
|
elmapi |
⊢ ( ( 𝐹 ‘ 𝑚 ) ∈ ( ℂ ↑m 𝑆 ) → ( 𝐹 ‘ 𝑚 ) : 𝑆 ⟶ ℂ ) |
36 |
34 35
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ) → ( 𝐹 ‘ 𝑚 ) : 𝑆 ⟶ ℂ ) |
37 |
36
|
ffvelrnda |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ) ∧ 𝑧 ∈ 𝑆 ) → ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑧 ) ∈ ℂ ) |
38 |
28
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑗 ) ∈ ( ℂ ↑m 𝑆 ) ) |
39 |
38
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ) → ( 𝐹 ‘ 𝑗 ) ∈ ( ℂ ↑m 𝑆 ) ) |
40 |
|
elmapi |
⊢ ( ( 𝐹 ‘ 𝑗 ) ∈ ( ℂ ↑m 𝑆 ) → ( 𝐹 ‘ 𝑗 ) : 𝑆 ⟶ ℂ ) |
41 |
39 40
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ) → ( 𝐹 ‘ 𝑗 ) : 𝑆 ⟶ ℂ ) |
42 |
41
|
ffvelrnda |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ) ∧ 𝑧 ∈ 𝑆 ) → ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) ∈ ℂ ) |
43 |
37 42
|
abssubd |
⊢ ( ( ( ( ( ( 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 ) ) ) ) |
44 |
43
|
breq1d |
⊢ ( ( ( ( ( ( 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 |
44
|
biimpd |
⊢ ( ( ( ( ( ( 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 ) ) ) |
46 |
|
ffvelrn |
⊢ ( ( 𝐹 : 𝑍 ⟶ ( ℂ ↑m 𝑆 ) ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ( ℂ ↑m 𝑆 ) ) |
47 |
28 30 46
|
syl2an |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ( ℂ ↑m 𝑆 ) ) |
48 |
47
|
anassrs |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ( ℂ ↑m 𝑆 ) ) |
49 |
48
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ( ℂ ↑m 𝑆 ) ) |
50 |
|
elmapi |
⊢ ( ( 𝐹 ‘ 𝑘 ) ∈ ( ℂ ↑m 𝑆 ) → ( 𝐹 ‘ 𝑘 ) : 𝑆 ⟶ ℂ ) |
51 |
49 50
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ) → ( 𝐹 ‘ 𝑘 ) : 𝑆 ⟶ ℂ ) |
52 |
51
|
ffvelrnda |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ) ∧ 𝑧 ∈ 𝑆 ) → ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ∈ ℂ ) |
53 |
|
rpre |
⊢ ( 𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ ) |
54 |
53
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) → 𝑥 ∈ ℝ ) |
55 |
54
|
ad3antrrr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ) ∧ 𝑧 ∈ 𝑆 ) → 𝑥 ∈ ℝ ) |
56 |
|
abs3lem |
⊢ ( ( ( ( ( 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 ) ) |
57 |
52 37 42 55 56
|
syl22anc |
⊢ ( ( ( ( ( ( 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 ) ) |
58 |
45 57
|
sylan2d |
⊢ ( ( ( ( ( ( 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 ) ) |
59 |
58
|
ralimdva |
⊢ ( ( ( ( ( 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 ) ) |
60 |
27 59
|
syl5bir |
⊢ ( ( ( ( ( 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
|
expdimp |
⊢ ( ( ( ( ( ( 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 ) ) |
62 |
61
|
an32s |
⊢ ( ( ( ( ( ( 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 ) ) |
63 |
62
|
ralimdva |
⊢ ( ( ( ( ( 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 ) ) |
64 |
26 63
|
syld |
⊢ ( ( ( ( ( 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 ) ) |
65 |
64
|
impancom |
⊢ ( ( ( ( ( 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 ) ) |
66 |
65
|
an32s |
⊢ ( ( ( ( ( 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 ) ) |
67 |
66
|
ralimdva |
⊢ ( ( ( ( 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
|
ex |
⊢ ( ( ( 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 ) ) ) |
69 |
68
|
com23 |
⊢ ( ( ( 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 ) ) ) |
70 |
22 69
|
mpdi |
⊢ ( ( ( 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 ) ) |
71 |
70
|
reximdva |
⊢ ( ( 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 ) ) |
72 |
15 71
|
syld |
⊢ ( ( 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 ) ) |
73 |
72
|
ralrimdva |
⊢ ( 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 ) ) |
74 |
8 73
|
syl5bi |
⊢ ( 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 ) ) |
75 |
|
eluzelz |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑗 ∈ ℤ ) |
76 |
75 1
|
eleq2s |
⊢ ( 𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ ) |
77 |
|
uzid |
⊢ ( 𝑗 ∈ ℤ → 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) ) |
78 |
76 77
|
syl |
⊢ ( 𝑗 ∈ 𝑍 → 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) ) |
79 |
78
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) ) |
80 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( ℤ≥ ‘ 𝑘 ) = ( ℤ≥ ‘ 𝑗 ) ) |
81 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑗 ) ) |
82 |
81
|
fveq1d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) = ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) ) |
83 |
82
|
fvoveq1d |
⊢ ( 𝑘 = 𝑗 → ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑧 ) ) ) = ( abs ‘ ( ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑧 ) ) ) ) |
84 |
83
|
breq1d |
⊢ ( 𝑘 = 𝑗 → ( ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑧 ) ) ) < 𝑥 ↔ ( abs ‘ ( ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑧 ) ) ) < 𝑥 ) ) |
85 |
84
|
ralbidv |
⊢ ( 𝑘 = 𝑗 → ( ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑧 ) ) ) < 𝑥 ↔ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑧 ) ) ) < 𝑥 ) ) |
86 |
80 85
|
raleqbidv |
⊢ ( 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 |
⊢ ( 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 |
79 87
|
syl |
⊢ ( ( 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 |
⊢ ( 𝑚 = 𝑘 → ( 𝐹 ‘ 𝑚 ) = ( 𝐹 ‘ 𝑘 ) ) |
90 |
89
|
fveq1d |
⊢ ( 𝑚 = 𝑘 → ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑧 ) = ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) |
91 |
90
|
oveq2d |
⊢ ( 𝑚 = 𝑘 → ( ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑧 ) ) = ( ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ) |
92 |
91
|
fveq2d |
⊢ ( 𝑚 = 𝑘 → ( abs ‘ ( ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑧 ) ) ) = ( abs ‘ ( ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ) ) |
93 |
92
|
breq1d |
⊢ ( 𝑚 = 𝑘 → ( ( abs ‘ ( ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑧 ) ) ) < 𝑥 ↔ ( abs ‘ ( ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ) < 𝑥 ) ) |
94 |
93
|
ralbidv |
⊢ ( 𝑚 = 𝑘 → ( ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑧 ) ) ) < 𝑥 ↔ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ) < 𝑥 ) ) |
95 |
94
|
cbvralvw |
⊢ ( ∀ 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 |
4
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑗 ) ∈ ( ℂ ↑m 𝑆 ) ) |
97 |
96
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( 𝐹 ‘ 𝑗 ) ∈ ( ℂ ↑m 𝑆 ) ) |
98 |
97 40
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( 𝐹 ‘ 𝑗 ) : 𝑆 ⟶ ℂ ) |
99 |
98
|
ffvelrnda |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑧 ∈ 𝑆 ) → ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑧 ) ∈ ℂ ) |
100 |
4 30 46
|
syl2an |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ( ℂ ↑m 𝑆 ) ) |
101 |
100
|
anassrs |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ( ℂ ↑m 𝑆 ) ) |
102 |
101 50
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( 𝐹 ‘ 𝑘 ) : 𝑆 ⟶ ℂ ) |
103 |
102
|
ffvelrnda |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑧 ∈ 𝑆 ) → ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ∈ ℂ ) |
104 |
99 103
|
abssubd |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( 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 |
⊢ ( 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 |
74 111
|
impbid |
⊢ ( 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 ) ) |