Step |
Hyp |
Ref |
Expression |
1 |
|
cau3lem.1 |
⊢ 𝑍 ⊆ ℤ |
2 |
|
cau3lem.2 |
⊢ ( 𝜏 → 𝜓 ) |
3 |
|
cau3lem.3 |
⊢ ( ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑗 ) → ( 𝜓 ↔ 𝜒 ) ) |
4 |
|
cau3lem.4 |
⊢ ( ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑚 ) → ( 𝜓 ↔ 𝜃 ) ) |
5 |
|
cau3lem.5 |
⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜓 ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑘 ) ) ) = ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) ) |
6 |
|
cau3lem.6 |
⊢ ( ( 𝜑 ∧ 𝜃 ∧ 𝜒 ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) = ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) ) |
7 |
|
cau3lem.7 |
⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜃 ) ∧ ( 𝜒 ∧ 𝑥 ∈ ℝ ) ) → ( ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < ( 𝑥 / 2 ) ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
8 |
|
breq2 |
⊢ ( 𝑥 = 𝑧 → ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ↔ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑧 ) ) |
9 |
8
|
anbi2d |
⊢ ( 𝑥 = 𝑧 → ( ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ↔ ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑧 ) ) ) |
10 |
9
|
rexralbidv |
⊢ ( 𝑥 = 𝑧 → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ↔ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑧 ) ) ) |
11 |
10
|
cbvralvw |
⊢ ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ↔ ∀ 𝑧 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑧 ) ) |
12 |
|
rphalfcl |
⊢ ( 𝑥 ∈ ℝ+ → ( 𝑥 / 2 ) ∈ ℝ+ ) |
13 |
|
breq2 |
⊢ ( 𝑧 = ( 𝑥 / 2 ) → ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑧 ↔ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) |
14 |
13
|
anbi2d |
⊢ ( 𝑧 = ( 𝑥 / 2 ) → ( ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑧 ) ↔ ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ) |
15 |
14
|
rexralbidv |
⊢ ( 𝑧 = ( 𝑥 / 2 ) → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑧 ) ↔ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ) |
16 |
15
|
rspcv |
⊢ ( ( 𝑥 / 2 ) ∈ ℝ+ → ( ∀ 𝑧 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑧 ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ) |
17 |
12 16
|
syl |
⊢ ( 𝑥 ∈ ℝ+ → ( ∀ 𝑧 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑧 ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ) |
18 |
17
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ∀ 𝑧 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑧 ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ) |
19 |
2
|
ralimi |
⊢ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜏 → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜓 ) |
20 |
|
r19.26 |
⊢ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜓 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ↔ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜓 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) |
21 |
|
fveq2 |
⊢ ( 𝑘 = 𝑚 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑚 ) ) |
22 |
21 4
|
syl |
⊢ ( 𝑘 = 𝑚 → ( 𝜓 ↔ 𝜃 ) ) |
23 |
21
|
fvoveq1d |
⊢ ( 𝑘 = 𝑚 → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) = ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) ) |
24 |
23
|
breq1d |
⊢ ( 𝑘 = 𝑚 → ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ↔ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) |
25 |
22 24
|
anbi12d |
⊢ ( 𝑘 = 𝑚 → ( ( 𝜓 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ↔ ( 𝜃 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ) |
26 |
25
|
cbvralvw |
⊢ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜓 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ↔ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜃 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) |
27 |
26
|
biimpi |
⊢ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜓 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜃 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) |
28 |
27
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜓 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜃 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ) |
29 |
20 28
|
syl5bir |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) → ( ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜓 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜃 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ) |
30 |
29
|
expdimp |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜓 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜃 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ) |
31 |
1
|
sseli |
⊢ ( 𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ ) |
32 |
|
uzid |
⊢ ( 𝑗 ∈ ℤ → 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) ) |
33 |
31 32
|
syl |
⊢ ( 𝑗 ∈ 𝑍 → 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) ) |
34 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑗 ) ) |
35 |
34 3
|
syl |
⊢ ( 𝑘 = 𝑗 → ( 𝜓 ↔ 𝜒 ) ) |
36 |
35
|
rspcva |
⊢ ( ( 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜓 ) → 𝜒 ) |
37 |
33 36
|
sylan |
⊢ ( ( 𝑗 ∈ 𝑍 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜓 ) → 𝜒 ) |
38 |
37
|
adantll |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜓 ) → 𝜒 ) |
39 |
30 38
|
jctild |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜓 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) → ( 𝜒 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜃 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ) ) |
40 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜒 ∧ 𝜃 ) ) ∧ 𝜓 ) → 𝜑 ) |
41 |
|
simplrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜒 ∧ 𝜃 ) ) ∧ 𝜓 ) → 𝜃 ) |
42 |
|
simplrl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜒 ∧ 𝜃 ) ) ∧ 𝜓 ) → 𝜒 ) |
43 |
40 41 42 6
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜒 ∧ 𝜃 ) ) ∧ 𝜓 ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) = ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) ) |
44 |
43
|
breq1d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜒 ∧ 𝜃 ) ) ∧ 𝜓 ) → ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ↔ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < ( 𝑥 / 2 ) ) ) |
45 |
44
|
anbi2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜒 ∧ 𝜃 ) ) ∧ 𝜓 ) → ( ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ↔ ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < ( 𝑥 / 2 ) ) ) ) |
46 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜒 ∧ 𝜃 ) ) ∧ 𝜓 ) → 𝜓 ) |
47 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜒 ∧ 𝜃 ) ) ∧ 𝜓 ) → 𝑥 ∈ ℝ+ ) |
48 |
47
|
rpred |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜒 ∧ 𝜃 ) ) ∧ 𝜓 ) → 𝑥 ∈ ℝ ) |
49 |
40 46 41 42 48 7
|
syl122anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜒 ∧ 𝜃 ) ) ∧ 𝜓 ) → ( ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < ( 𝑥 / 2 ) ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
50 |
45 49
|
sylbid |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜒 ∧ 𝜃 ) ) ∧ 𝜓 ) → ( ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
51 |
50
|
expd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜒 ∧ 𝜃 ) ) ∧ 𝜓 ) → ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) → ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
52 |
51
|
impr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜒 ∧ 𝜃 ) ) ∧ ( 𝜓 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) → ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
53 |
52
|
an32s |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜓 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ∧ ( 𝜒 ∧ 𝜃 ) ) → ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
54 |
53
|
anassrs |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜓 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ∧ 𝜒 ) ∧ 𝜃 ) → ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
55 |
54
|
expimpd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜓 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ∧ 𝜒 ) → ( ( 𝜃 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
56 |
55
|
ralimdv |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜓 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ∧ 𝜒 ) → ( ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜃 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
57 |
56
|
impr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜓 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ∧ ( 𝜒 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜃 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ) → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) |
58 |
57
|
an32s |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜒 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜃 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ) ∧ ( 𝜓 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) |
59 |
58
|
expr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜒 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜃 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ) ∧ 𝜓 ) → ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
60 |
|
uzss |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) → ( ℤ≥ ‘ 𝑘 ) ⊆ ( ℤ≥ ‘ 𝑗 ) ) |
61 |
|
ssralv |
⊢ ( ( ℤ≥ ‘ 𝑘 ) ⊆ ( ℤ≥ ‘ 𝑗 ) → ( ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
62 |
60 61
|
syl |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) → ( ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
63 |
59 62
|
sylan9 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜒 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜃 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ) ∧ 𝜓 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
64 |
63
|
an32s |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜒 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜃 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝜓 ) → ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
65 |
64
|
expimpd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜒 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜃 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( ( 𝜓 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
66 |
65
|
ralimdva |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝜒 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜃 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜓 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
67 |
66
|
ex |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ( 𝜒 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜃 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜓 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
68 |
67
|
com23 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜓 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) → ( ( 𝜒 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜃 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
69 |
68
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜓 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) → ( ( 𝜒 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜃 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
70 |
20 69
|
syl5bir |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) → ( ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜓 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) → ( ( 𝜒 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜃 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
71 |
70
|
expdimp |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜓 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) → ( ( 𝜒 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜃 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑚 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
72 |
39 71
|
mpdd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜓 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
73 |
19 72
|
sylan2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜏 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
74 |
73
|
imdistanda |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) → ( ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜏 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜏 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
75 |
|
r19.26 |
⊢ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ↔ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜏 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) ) |
76 |
|
r19.26 |
⊢ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ↔ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜏 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
77 |
74 75 76
|
3imtr4g |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ 𝑍 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
78 |
77
|
reximdva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < ( 𝑥 / 2 ) ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
79 |
18 78
|
syld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ∀ 𝑧 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑧 ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
80 |
79
|
ralrimdva |
⊢ ( 𝜑 → ( ∀ 𝑧 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑧 ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
81 |
11 80
|
syl5bi |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
82 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( ℤ≥ ‘ 𝑘 ) = ( ℤ≥ ‘ 𝑗 ) ) |
83 |
34
|
fvoveq1d |
⊢ ( 𝑘 = 𝑗 → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) = ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) ) |
84 |
83
|
breq1d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ↔ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
85 |
82 84
|
raleqbidv |
⊢ ( 𝑘 = 𝑗 → ( ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ↔ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
86 |
85
|
rspcv |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
87 |
86
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜓 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 → ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
88 |
|
fveq2 |
⊢ ( 𝑚 = 𝑘 → ( 𝐹 ‘ 𝑚 ) = ( 𝐹 ‘ 𝑘 ) ) |
89 |
88
|
oveq2d |
⊢ ( 𝑚 = 𝑘 → ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) = ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑘 ) ) ) |
90 |
89
|
fveq2d |
⊢ ( 𝑚 = 𝑘 → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) = ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑘 ) ) ) ) |
91 |
90
|
breq1d |
⊢ ( 𝑚 = 𝑘 → ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ↔ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑘 ) ) ) < 𝑥 ) ) |
92 |
91
|
cbvralvw |
⊢ ( ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ↔ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑘 ) ) ) < 𝑥 ) |
93 |
36
|
anim2i |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜓 ) ) → ( 𝜑 ∧ 𝜒 ) ) |
94 |
93
|
anassrs |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜓 ) → ( 𝜑 ∧ 𝜒 ) ) |
95 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜓 ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜓 ) |
96 |
5
|
breq1d |
⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜓 ) → ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑘 ) ) ) < 𝑥 ↔ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ) |
97 |
96
|
3expia |
⊢ ( ( 𝜑 ∧ 𝜒 ) → ( 𝜓 → ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑘 ) ) ) < 𝑥 ↔ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ) ) |
98 |
97
|
ralimdv |
⊢ ( ( 𝜑 ∧ 𝜒 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜓 → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑘 ) ) ) < 𝑥 ↔ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ) ) |
99 |
94 95 98
|
sylc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜓 ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑘 ) ) ) < 𝑥 ↔ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ) |
100 |
|
ralbi |
⊢ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑘 ) ) ) < 𝑥 ↔ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑘 ) ) ) < 𝑥 ↔ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ) |
101 |
99 100
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜓 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑘 ) ) ) < 𝑥 ↔ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ) |
102 |
92 101
|
syl5bb |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜓 ) → ( ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ↔ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ) |
103 |
87 102
|
sylibd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜓 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ) |
104 |
19 103
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜏 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ) |
105 |
104
|
imdistanda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜏 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜏 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ) ) |
106 |
33 105
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜏 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜏 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ) ) |
107 |
|
r19.26 |
⊢ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ↔ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝜏 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ) |
108 |
106 76 107
|
3imtr4g |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ) ) |
109 |
108
|
reximdva |
⊢ ( 𝜑 → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ) ) |
110 |
109
|
ralimdv |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ) ) |
111 |
81 110
|
impbid |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑗 ) ) ) < 𝑥 ) ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝜏 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑘 ) ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |