Step |
Hyp |
Ref |
Expression |
1 |
|
bfp.2 |
⊢ ( 𝜑 → 𝐷 ∈ ( CMet ‘ 𝑋 ) ) |
2 |
|
bfp.3 |
⊢ ( 𝜑 → 𝑋 ≠ ∅ ) |
3 |
|
bfp.4 |
⊢ ( 𝜑 → 𝐾 ∈ ℝ+ ) |
4 |
|
bfp.5 |
⊢ ( 𝜑 → 𝐾 < 1 ) |
5 |
|
bfp.6 |
⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ 𝑋 ) |
6 |
|
bfp.7 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) ) → ( ( 𝐹 ‘ 𝑥 ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) ≤ ( 𝐾 · ( 𝑥 𝐷 𝑦 ) ) ) |
7 |
|
bfp.8 |
⊢ 𝐽 = ( MetOpen ‘ 𝐷 ) |
8 |
|
bfp.9 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑋 ) |
9 |
|
bfp.10 |
⊢ 𝐺 = seq 1 ( ( 𝐹 ∘ 1st ) , ( ℕ × { 𝐴 } ) ) |
10 |
|
cmetmet |
⊢ ( 𝐷 ∈ ( CMet ‘ 𝑋 ) → 𝐷 ∈ ( Met ‘ 𝑋 ) ) |
11 |
1 10
|
syl |
⊢ ( 𝜑 → 𝐷 ∈ ( Met ‘ 𝑋 ) ) |
12 |
|
metxmet |
⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
13 |
7
|
mopntopon |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
14 |
11 12 13
|
3syl |
⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
15 |
1 2 3 4 5 6 7 8 9
|
bfplem1 |
⊢ ( 𝜑 → 𝐺 ( ⇝𝑡 ‘ 𝐽 ) ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) |
16 |
|
lmcl |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐺 ( ⇝𝑡 ‘ 𝐽 ) ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) → ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ∈ 𝑋 ) |
17 |
14 15 16
|
syl2anc |
⊢ ( 𝜑 → ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ∈ 𝑋 ) |
18 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝐷 ∈ ( Met ‘ 𝑋 ) ) |
19 |
18 12
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
20 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
21 |
|
1zzd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 1 ∈ ℤ ) |
22 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ 𝑘 ) = ( 𝐺 ‘ 𝑘 ) ) |
23 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝐺 ( ⇝𝑡 ‘ 𝐽 ) ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) |
24 |
|
rphalfcl |
⊢ ( 𝑥 ∈ ℝ+ → ( 𝑥 / 2 ) ∈ ℝ+ ) |
25 |
24
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 𝑥 / 2 ) ∈ ℝ+ ) |
26 |
7 19 20 21 22 23 25
|
lmmcvg |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑗 ∈ ℕ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐺 ‘ 𝑘 ) ∈ 𝑋 ∧ ( ( 𝐺 ‘ 𝑘 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ) ) |
27 |
|
simpr |
⊢ ( ( ( 𝐺 ‘ 𝑘 ) ∈ 𝑋 ∧ ( ( 𝐺 ‘ 𝑘 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ) → ( ( 𝐺 ‘ 𝑘 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ) |
28 |
27
|
ralimi |
⊢ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐺 ‘ 𝑘 ) ∈ 𝑋 ∧ ( ( 𝐺 ‘ 𝑘 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐺 ‘ 𝑘 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ) |
29 |
|
nnz |
⊢ ( 𝑗 ∈ ℕ → 𝑗 ∈ ℤ ) |
30 |
29
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → 𝑗 ∈ ℤ ) |
31 |
|
uzid |
⊢ ( 𝑗 ∈ ℤ → 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) ) |
32 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( 𝐺 ‘ 𝑘 ) = ( 𝐺 ‘ 𝑗 ) ) |
33 |
32
|
oveq1d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝐺 ‘ 𝑘 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) = ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) |
34 |
33
|
breq1d |
⊢ ( 𝑘 = 𝑗 → ( ( ( 𝐺 ‘ 𝑘 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ↔ ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ) ) |
35 |
34
|
rspcv |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐺 ‘ 𝑘 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) → ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ) ) |
36 |
30 31 35
|
3syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐺 ‘ 𝑘 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) → ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ) ) |
37 |
30 31
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) ) |
38 |
|
peano2uz |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) → ( 𝑗 + 1 ) ∈ ( ℤ≥ ‘ 𝑗 ) ) |
39 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝑗 + 1 ) → ( 𝐺 ‘ 𝑘 ) = ( 𝐺 ‘ ( 𝑗 + 1 ) ) ) |
40 |
39
|
oveq1d |
⊢ ( 𝑘 = ( 𝑗 + 1 ) → ( ( 𝐺 ‘ 𝑘 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) = ( ( 𝐺 ‘ ( 𝑗 + 1 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) |
41 |
40
|
breq1d |
⊢ ( 𝑘 = ( 𝑗 + 1 ) → ( ( ( 𝐺 ‘ 𝑘 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ↔ ( ( 𝐺 ‘ ( 𝑗 + 1 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ) ) |
42 |
41
|
rspcv |
⊢ ( ( 𝑗 + 1 ) ∈ ( ℤ≥ ‘ 𝑗 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐺 ‘ 𝑘 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) → ( ( 𝐺 ‘ ( 𝑗 + 1 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ) ) |
43 |
37 38 42
|
3syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐺 ‘ 𝑘 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) → ( ( 𝐺 ‘ ( 𝑗 + 1 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ) ) |
44 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
45 |
20 9 44 8 5
|
algrp1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐺 ‘ ( 𝑗 + 1 ) ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) ) |
46 |
45
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( 𝐺 ‘ ( 𝑗 + 1 ) ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) ) |
47 |
46
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝐺 ‘ ( 𝑗 + 1 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) |
48 |
47
|
breq1d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝐺 ‘ ( 𝑗 + 1 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ↔ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ) ) |
49 |
43 48
|
sylibd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐺 ‘ 𝑘 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ) ) |
50 |
36 49
|
jcad |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐺 ‘ 𝑘 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) → ( ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ) ) ) |
51 |
11
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → 𝐷 ∈ ( Met ‘ 𝑋 ) ) |
52 |
20 9 44 8 5
|
algrf |
⊢ ( 𝜑 → 𝐺 : ℕ ⟶ 𝑋 ) |
53 |
52
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝐺 : ℕ ⟶ 𝑋 ) |
54 |
53
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( 𝐺 ‘ 𝑗 ) ∈ 𝑋 ) |
55 |
17
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ∈ 𝑋 ) |
56 |
|
metcl |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( 𝐺 ‘ 𝑗 ) ∈ 𝑋 ∧ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ∈ 𝑋 ) → ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ∈ ℝ ) |
57 |
51 54 55 56
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ∈ ℝ ) |
58 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → 𝐹 : 𝑋 ⟶ 𝑋 ) |
59 |
58 54
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) ∈ 𝑋 ) |
60 |
|
metcl |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) ∈ 𝑋 ∧ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ∈ 𝑋 ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ∈ ℝ ) |
61 |
51 59 55 60
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ∈ ℝ ) |
62 |
|
rpre |
⊢ ( 𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ ) |
63 |
62
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → 𝑥 ∈ ℝ ) |
64 |
|
lt2halves |
⊢ ( ( ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ∈ ℝ ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( ( ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ) → ( ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) + ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) < 𝑥 ) ) |
65 |
57 61 63 64
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ( ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ) → ( ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) + ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) < 𝑥 ) ) |
66 |
5 17
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ∈ 𝑋 ) |
67 |
|
metcl |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ∈ 𝑋 ∧ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ∈ 𝑋 ) → ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ∈ ℝ ) |
68 |
11 66 17 67
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ∈ ℝ ) |
69 |
68
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ∈ ℝ ) |
70 |
58 55
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ∈ 𝑋 ) |
71 |
|
metcl |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) ∈ 𝑋 ∧ ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ∈ 𝑋 ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ∈ ℝ ) |
72 |
51 59 70 71
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ∈ ℝ ) |
73 |
72 61
|
readdcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) + ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ∈ ℝ ) |
74 |
57 61
|
readdcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) + ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ∈ ℝ ) |
75 |
|
mettri2 |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) ∈ 𝑋 ∧ ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ∈ 𝑋 ∧ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ∈ 𝑋 ) ) → ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ≤ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) + ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ) |
76 |
51 59 70 55 75
|
syl13anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ≤ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) + ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ) |
77 |
3
|
rpred |
⊢ ( 𝜑 → 𝐾 ∈ ℝ ) |
78 |
77
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → 𝐾 ∈ ℝ ) |
79 |
78 57
|
remulcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( 𝐾 · ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ∈ ℝ ) |
80 |
54 55
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝐺 ‘ 𝑗 ) ∈ 𝑋 ∧ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ∈ 𝑋 ) ) |
81 |
6
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝐹 ‘ 𝑥 ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) ≤ ( 𝐾 · ( 𝑥 𝐷 𝑦 ) ) ) |
82 |
81
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝐹 ‘ 𝑥 ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) ≤ ( 𝐾 · ( 𝑥 𝐷 𝑦 ) ) ) |
83 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝐺 ‘ 𝑗 ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) ) |
84 |
83
|
oveq1d |
⊢ ( 𝑥 = ( 𝐺 ‘ 𝑗 ) → ( ( 𝐹 ‘ 𝑥 ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) ) |
85 |
|
oveq1 |
⊢ ( 𝑥 = ( 𝐺 ‘ 𝑗 ) → ( 𝑥 𝐷 𝑦 ) = ( ( 𝐺 ‘ 𝑗 ) 𝐷 𝑦 ) ) |
86 |
85
|
oveq2d |
⊢ ( 𝑥 = ( 𝐺 ‘ 𝑗 ) → ( 𝐾 · ( 𝑥 𝐷 𝑦 ) ) = ( 𝐾 · ( ( 𝐺 ‘ 𝑗 ) 𝐷 𝑦 ) ) ) |
87 |
84 86
|
breq12d |
⊢ ( 𝑥 = ( 𝐺 ‘ 𝑗 ) → ( ( ( 𝐹 ‘ 𝑥 ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) ≤ ( 𝐾 · ( 𝑥 𝐷 𝑦 ) ) ↔ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) ≤ ( 𝐾 · ( ( 𝐺 ‘ 𝑗 ) 𝐷 𝑦 ) ) ) ) |
88 |
|
fveq2 |
⊢ ( 𝑦 = ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) |
89 |
88
|
oveq2d |
⊢ ( 𝑦 = ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ) |
90 |
|
oveq2 |
⊢ ( 𝑦 = ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) → ( ( 𝐺 ‘ 𝑗 ) 𝐷 𝑦 ) = ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) |
91 |
90
|
oveq2d |
⊢ ( 𝑦 = ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) → ( 𝐾 · ( ( 𝐺 ‘ 𝑗 ) 𝐷 𝑦 ) ) = ( 𝐾 · ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ) |
92 |
89 91
|
breq12d |
⊢ ( 𝑦 = ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) → ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) ≤ ( 𝐾 · ( ( 𝐺 ‘ 𝑗 ) 𝐷 𝑦 ) ) ↔ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ≤ ( 𝐾 · ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ) ) |
93 |
87 92
|
rspc2v |
⊢ ( ( ( 𝐺 ‘ 𝑗 ) ∈ 𝑋 ∧ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ∈ 𝑋 ) → ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝐹 ‘ 𝑥 ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) ≤ ( 𝐾 · ( 𝑥 𝐷 𝑦 ) ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ≤ ( 𝐾 · ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ) ) |
94 |
80 82 93
|
sylc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ≤ ( 𝐾 · ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ) |
95 |
|
1red |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → 1 ∈ ℝ ) |
96 |
|
metge0 |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( 𝐺 ‘ 𝑗 ) ∈ 𝑋 ∧ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ∈ 𝑋 ) → 0 ≤ ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) |
97 |
51 54 55 96
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → 0 ≤ ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) |
98 |
|
1re |
⊢ 1 ∈ ℝ |
99 |
|
ltle |
⊢ ( ( 𝐾 ∈ ℝ ∧ 1 ∈ ℝ ) → ( 𝐾 < 1 → 𝐾 ≤ 1 ) ) |
100 |
77 98 99
|
sylancl |
⊢ ( 𝜑 → ( 𝐾 < 1 → 𝐾 ≤ 1 ) ) |
101 |
4 100
|
mpd |
⊢ ( 𝜑 → 𝐾 ≤ 1 ) |
102 |
101
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → 𝐾 ≤ 1 ) |
103 |
78 95 57 97 102
|
lemul1ad |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( 𝐾 · ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ≤ ( 1 · ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ) |
104 |
57
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ∈ ℂ ) |
105 |
104
|
mulid2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( 1 · ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) = ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) |
106 |
103 105
|
breqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( 𝐾 · ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ≤ ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) |
107 |
72 79 57 94 106
|
letrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ≤ ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) |
108 |
72 57 61 107
|
leadd1dd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) + ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ≤ ( ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) + ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ) |
109 |
69 73 74 76 108
|
letrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ≤ ( ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) + ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ) |
110 |
|
lelttr |
⊢ ( ( ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ∈ ℝ ∧ ( ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) + ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( ( ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ≤ ( ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) + ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ∧ ( ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) + ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) < 𝑥 ) → ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < 𝑥 ) ) |
111 |
69 74 63 110
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ( ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ≤ ( ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) + ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ∧ ( ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) + ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) < 𝑥 ) → ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < 𝑥 ) ) |
112 |
109 111
|
mpand |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ( ( ( 𝐺 ‘ 𝑗 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) + ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑗 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) < 𝑥 → ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < 𝑥 ) ) |
113 |
50 65 112
|
3syld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐺 ‘ 𝑘 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) → ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < 𝑥 ) ) |
114 |
28 113
|
syl5 |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐺 ‘ 𝑘 ) ∈ 𝑋 ∧ ( ( 𝐺 ‘ 𝑘 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ) → ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < 𝑥 ) ) |
115 |
114
|
rexlimdva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ∃ 𝑗 ∈ ℕ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐺 ‘ 𝑘 ) ∈ 𝑋 ∧ ( ( 𝐺 ‘ 𝑘 ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < ( 𝑥 / 2 ) ) → ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < 𝑥 ) ) |
116 |
26 115
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < 𝑥 ) |
117 |
|
ltle |
⊢ ( ( ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < 𝑥 → ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ≤ 𝑥 ) ) |
118 |
68 62 117
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) < 𝑥 → ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ≤ 𝑥 ) ) |
119 |
116 118
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ≤ 𝑥 ) |
120 |
62
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝑥 ∈ ℝ ) |
121 |
120
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝑥 ∈ ℂ ) |
122 |
121
|
addid2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 0 + 𝑥 ) = 𝑥 ) |
123 |
119 122
|
breqtrrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ≤ ( 0 + 𝑥 ) ) |
124 |
123
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ+ ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ≤ ( 0 + 𝑥 ) ) |
125 |
|
0re |
⊢ 0 ∈ ℝ |
126 |
|
alrple |
⊢ ( ( ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ∈ ℝ ∧ 0 ∈ ℝ ) → ( ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ≤ 0 ↔ ∀ 𝑥 ∈ ℝ+ ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ≤ ( 0 + 𝑥 ) ) ) |
127 |
68 125 126
|
sylancl |
⊢ ( 𝜑 → ( ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ≤ 0 ↔ ∀ 𝑥 ∈ ℝ+ ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ≤ ( 0 + 𝑥 ) ) ) |
128 |
124 127
|
mpbird |
⊢ ( 𝜑 → ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ≤ 0 ) |
129 |
|
metge0 |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ∈ 𝑋 ∧ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ∈ 𝑋 ) → 0 ≤ ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) |
130 |
11 66 17 129
|
syl3anc |
⊢ ( 𝜑 → 0 ≤ ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) |
131 |
|
letri3 |
⊢ ( ( ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ∈ ℝ ∧ 0 ∈ ℝ ) → ( ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) = 0 ↔ ( ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ≤ 0 ∧ 0 ≤ ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ) ) |
132 |
68 125 131
|
sylancl |
⊢ ( 𝜑 → ( ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) = 0 ↔ ( ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ≤ 0 ∧ 0 ≤ ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) ) ) |
133 |
128 130 132
|
mpbir2and |
⊢ ( 𝜑 → ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) = 0 ) |
134 |
|
meteq0 |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ∈ 𝑋 ∧ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ∈ 𝑋 ) → ( ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) = 0 ↔ ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) = ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) |
135 |
11 66 17 134
|
syl3anc |
⊢ ( 𝜑 → ( ( ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) 𝐷 ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) = 0 ↔ ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) = ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) |
136 |
133 135
|
mpbid |
⊢ ( 𝜑 → ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) = ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) |
137 |
|
fveq2 |
⊢ ( 𝑧 = ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) |
138 |
|
id |
⊢ ( 𝑧 = ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) → 𝑧 = ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) |
139 |
137 138
|
eqeq12d |
⊢ ( 𝑧 = ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) → ( ( 𝐹 ‘ 𝑧 ) = 𝑧 ↔ ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) = ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) ) |
140 |
139
|
rspcev |
⊢ ( ( ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ∈ 𝑋 ∧ ( 𝐹 ‘ ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) = ( ( ⇝𝑡 ‘ 𝐽 ) ‘ 𝐺 ) ) → ∃ 𝑧 ∈ 𝑋 ( 𝐹 ‘ 𝑧 ) = 𝑧 ) |
141 |
17 136 140
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑧 ∈ 𝑋 ( 𝐹 ‘ 𝑧 ) = 𝑧 ) |