Step |
Hyp |
Ref |
Expression |
1 |
|
ulmbdd.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
ulmbdd.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
ulmbdd.f |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ( ℂ ↑m 𝑆 ) ) |
4 |
|
ulmbdd.b |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ∃ 𝑥 ∈ ℝ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ) |
5 |
|
ulmbdd.u |
⊢ ( 𝜑 → 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ) |
6 |
|
eqidd |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) = ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) |
7 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑆 ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) |
8 |
|
1rp |
⊢ 1 ∈ ℝ+ |
9 |
8
|
a1i |
⊢ ( 𝜑 → 1 ∈ ℝ+ ) |
10 |
1 2 3 6 7 5 9
|
ulmi |
⊢ ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) |
11 |
1
|
r19.2uz |
⊢ ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 → ∃ 𝑘 ∈ 𝑍 ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) |
12 |
|
r19.26 |
⊢ ( ∀ 𝑧 ∈ 𝑆 ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ↔ ( ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) |
13 |
|
peano2re |
⊢ ( 𝑥 ∈ ℝ → ( 𝑥 + 1 ) ∈ ℝ ) |
14 |
13
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) → ( 𝑥 + 1 ) ∈ ℝ ) |
15 |
|
ulmcl |
⊢ ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → 𝐺 : 𝑆 ⟶ ℂ ) |
16 |
5 15
|
syl |
⊢ ( 𝜑 → 𝐺 : 𝑆 ⟶ ℂ ) |
17 |
16
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → 𝐺 : 𝑆 ⟶ ℂ ) |
18 |
|
simprl |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → 𝑧 ∈ 𝑆 ) |
19 |
17 18
|
ffvelrnd |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( 𝐺 ‘ 𝑧 ) ∈ ℂ ) |
20 |
19
|
abscld |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ∈ ℝ ) |
21 |
3
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → 𝐹 : 𝑍 ⟶ ( ℂ ↑m 𝑆 ) ) |
22 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → 𝑘 ∈ 𝑍 ) |
23 |
21 22
|
ffvelrnd |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ( ℂ ↑m 𝑆 ) ) |
24 |
|
elmapi |
⊢ ( ( 𝐹 ‘ 𝑘 ) ∈ ( ℂ ↑m 𝑆 ) → ( 𝐹 ‘ 𝑘 ) : 𝑆 ⟶ ℂ ) |
25 |
23 24
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( 𝐹 ‘ 𝑘 ) : 𝑆 ⟶ ℂ ) |
26 |
25 18
|
ffvelrnd |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ∈ ℂ ) |
27 |
26
|
abscld |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ∈ ℝ ) |
28 |
19 26
|
subcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( ( 𝐺 ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ∈ ℂ ) |
29 |
28
|
abscld |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( abs ‘ ( ( 𝐺 ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ) ∈ ℝ ) |
30 |
27 29
|
readdcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) + ( abs ‘ ( ( 𝐺 ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ) ) ∈ ℝ ) |
31 |
14
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( 𝑥 + 1 ) ∈ ℝ ) |
32 |
26 19
|
pncan3d |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) + ( ( 𝐺 ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ) = ( 𝐺 ‘ 𝑧 ) ) |
33 |
32
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) + ( ( 𝐺 ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ) ) = ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ) |
34 |
26 28
|
abstrid |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) + ( ( 𝐺 ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ) ) ≤ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) + ( abs ‘ ( ( 𝐺 ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ) ) ) |
35 |
33 34
|
eqbrtrrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) + ( abs ‘ ( ( 𝐺 ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ) ) ) |
36 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → 𝑥 ∈ ℝ ) |
37 |
|
1re |
⊢ 1 ∈ ℝ |
38 |
37
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → 1 ∈ ℝ ) |
39 |
|
simprrl |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ) |
40 |
19 26
|
abssubd |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( abs ‘ ( ( 𝐺 ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ) = ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) ) |
41 |
|
simprrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) |
42 |
40 41
|
eqbrtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( abs ‘ ( ( 𝐺 ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ) < 1 ) |
43 |
|
ltle |
⊢ ( ( ( abs ‘ ( ( 𝐺 ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ) ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( abs ‘ ( ( 𝐺 ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ) < 1 → ( abs ‘ ( ( 𝐺 ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ) ≤ 1 ) ) |
44 |
29 37 43
|
sylancl |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( ( abs ‘ ( ( 𝐺 ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ) < 1 → ( abs ‘ ( ( 𝐺 ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ) ≤ 1 ) ) |
45 |
42 44
|
mpd |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( abs ‘ ( ( 𝐺 ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ) ≤ 1 ) |
46 |
27 29 36 38 39 45
|
le2addd |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) + ( abs ‘ ( ( 𝐺 ‘ 𝑧 ) − ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ) ) ≤ ( 𝑥 + 1 ) ) |
47 |
20 30 31 35 46
|
letrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑧 ∈ 𝑆 ∧ ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) ) ) → ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ ( 𝑥 + 1 ) ) |
48 |
47
|
expr |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) ∧ 𝑧 ∈ 𝑆 ) → ( ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) → ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ ( 𝑥 + 1 ) ) ) |
49 |
48
|
ralimdva |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) → ( ∀ 𝑧 ∈ 𝑆 ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) → ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ ( 𝑥 + 1 ) ) ) |
50 |
|
brralrspcev |
⊢ ( ( ( 𝑥 + 1 ) ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ ( 𝑥 + 1 ) ) → ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ 𝑦 ) |
51 |
14 49 50
|
syl6an |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) → ( ∀ 𝑧 ∈ 𝑆 ( ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ 𝑦 ) ) |
52 |
12 51
|
syl5bir |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) → ( ( ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ 𝑦 ) ) |
53 |
52
|
expd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑥 ∈ ℝ ) → ( ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 → ( ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 → ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ 𝑦 ) ) ) |
54 |
53
|
rexlimdva |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ∃ 𝑥 ∈ ℝ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) ≤ 𝑥 → ( ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 → ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ 𝑦 ) ) ) |
55 |
4 54
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 → ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ 𝑦 ) ) |
56 |
|
breq2 |
⊢ ( 𝑦 = 𝑥 → ( ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ 𝑦 ↔ ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ 𝑥 ) ) |
57 |
56
|
ralbidv |
⊢ ( 𝑦 = 𝑥 → ( ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ 𝑦 ↔ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ 𝑥 ) ) |
58 |
57
|
cbvrexvw |
⊢ ( ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ 𝑦 ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ 𝑥 ) |
59 |
55 58
|
syl6ib |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 → ∃ 𝑥 ∈ ℝ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ 𝑥 ) ) |
60 |
59
|
rexlimdva |
⊢ ( 𝜑 → ( ∃ 𝑘 ∈ 𝑍 ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 → ∃ 𝑥 ∈ ℝ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ 𝑥 ) ) |
61 |
11 60
|
syl5 |
⊢ ( 𝜑 → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 1 → ∃ 𝑥 ∈ ℝ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ 𝑥 ) ) |
62 |
10 61
|
mpd |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ 𝑥 ) |