Step |
Hyp |
Ref |
Expression |
1 |
|
caures.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
caures.3 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
caures.4 |
⊢ ( 𝜑 → 𝐷 ∈ ( Met ‘ 𝑋 ) ) |
4 |
|
caushft.4 |
⊢ 𝑊 = ( ℤ≥ ‘ ( 𝑀 + 𝑁 ) ) |
5 |
|
caushft.5 |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
6 |
|
caushft.7 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) ) |
7 |
|
caushft.8 |
⊢ ( 𝜑 → 𝐹 ∈ ( Cau ‘ 𝐷 ) ) |
8 |
|
caushft.9 |
⊢ ( 𝜑 → 𝐺 : 𝑊 ⟶ 𝑋 ) |
9 |
|
metxmet |
⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
10 |
3 9
|
syl |
⊢ ( 𝜑 → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
11 |
6
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝑍 ( 𝐹 ‘ 𝑘 ) = ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) ) |
12 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑗 ) ) |
13 |
|
fvoveq1 |
⊢ ( 𝑘 = 𝑗 → ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) = ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) |
14 |
12 13
|
eqeq12d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝐹 ‘ 𝑘 ) = ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) ↔ ( 𝐹 ‘ 𝑗 ) = ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) ) |
15 |
14
|
rspccva |
⊢ ( ( ∀ 𝑘 ∈ 𝑍 ( 𝐹 ‘ 𝑘 ) = ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) ∧ 𝑗 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑗 ) = ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) |
16 |
11 15
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑗 ) = ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) |
17 |
1 10 2 6 16
|
iscau4 |
⊢ ( 𝜑 → ( 𝐹 ∈ ( Cau ‘ 𝐷 ) ↔ ( 𝐹 ∈ ( 𝑋 ↑pm ℂ ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) ∈ 𝑋 ∧ ( ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) < 𝑥 ) ) ) ) |
18 |
7 17
|
mpbid |
⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 ↑pm ℂ ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) ∈ 𝑋 ∧ ( ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) < 𝑥 ) ) ) |
19 |
18
|
simprd |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) ∈ 𝑋 ∧ ( ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) < 𝑥 ) ) |
20 |
1
|
eleq2i |
⊢ ( 𝑗 ∈ 𝑍 ↔ 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
21 |
20
|
biimpi |
⊢ ( 𝑗 ∈ 𝑍 → 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
22 |
|
eluzadd |
⊢ ( ( 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑁 ∈ ℤ ) → ( 𝑗 + 𝑁 ) ∈ ( ℤ≥ ‘ ( 𝑀 + 𝑁 ) ) ) |
23 |
21 5 22
|
syl2anr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝑗 + 𝑁 ) ∈ ( ℤ≥ ‘ ( 𝑀 + 𝑁 ) ) ) |
24 |
23 4
|
eleqtrrdi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝑗 + 𝑁 ) ∈ 𝑊 ) |
25 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → 𝑗 ∈ 𝑍 ) |
26 |
25 1
|
eleqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
27 |
|
eluzelz |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑗 ∈ ℤ ) |
28 |
26 27
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → 𝑗 ∈ ℤ ) |
29 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → 𝑁 ∈ ℤ ) |
30 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) |
31 |
|
eluzsub |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → ( 𝑚 − 𝑁 ) ∈ ( ℤ≥ ‘ 𝑗 ) ) |
32 |
28 29 30 31
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → ( 𝑚 − 𝑁 ) ∈ ( ℤ≥ ‘ 𝑗 ) ) |
33 |
|
simp3 |
⊢ ( ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) ∈ 𝑋 ∧ ( ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) < 𝑥 ) → ( ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) < 𝑥 ) |
34 |
33
|
ralimi |
⊢ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) ∈ 𝑋 ∧ ( ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) < 𝑥 ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) < 𝑥 ) |
35 |
|
fvoveq1 |
⊢ ( 𝑘 = ( 𝑚 − 𝑁 ) → ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) = ( 𝐺 ‘ ( ( 𝑚 − 𝑁 ) + 𝑁 ) ) ) |
36 |
35
|
oveq1d |
⊢ ( 𝑘 = ( 𝑚 − 𝑁 ) → ( ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) = ( ( 𝐺 ‘ ( ( 𝑚 − 𝑁 ) + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) ) |
37 |
36
|
breq1d |
⊢ ( 𝑘 = ( 𝑚 − 𝑁 ) → ( ( ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) < 𝑥 ↔ ( ( 𝐺 ‘ ( ( 𝑚 − 𝑁 ) + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) < 𝑥 ) ) |
38 |
37
|
rspcv |
⊢ ( ( 𝑚 − 𝑁 ) ∈ ( ℤ≥ ‘ 𝑗 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) < 𝑥 → ( ( 𝐺 ‘ ( ( 𝑚 − 𝑁 ) + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) < 𝑥 ) ) |
39 |
32 34 38
|
syl2im |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) ∈ 𝑋 ∧ ( ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) < 𝑥 ) → ( ( 𝐺 ‘ ( ( 𝑚 − 𝑁 ) + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) < 𝑥 ) ) |
40 |
|
eluzelz |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) → 𝑚 ∈ ℤ ) |
41 |
40
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → 𝑚 ∈ ℤ ) |
42 |
41
|
zcnd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → 𝑚 ∈ ℂ ) |
43 |
5
|
zcnd |
⊢ ( 𝜑 → 𝑁 ∈ ℂ ) |
44 |
43
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → 𝑁 ∈ ℂ ) |
45 |
42 44
|
npcand |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → ( ( 𝑚 − 𝑁 ) + 𝑁 ) = 𝑚 ) |
46 |
45
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → ( 𝐺 ‘ ( ( 𝑚 − 𝑁 ) + 𝑁 ) ) = ( 𝐺 ‘ 𝑚 ) ) |
47 |
46
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → ( ( 𝐺 ‘ ( ( 𝑚 − 𝑁 ) + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) = ( ( 𝐺 ‘ 𝑚 ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) ) |
48 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → 𝐷 ∈ ( Met ‘ 𝑋 ) ) |
49 |
8
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → 𝐺 : 𝑊 ⟶ 𝑋 ) |
50 |
4
|
uztrn2 |
⊢ ( ( ( 𝑗 + 𝑁 ) ∈ 𝑊 ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → 𝑚 ∈ 𝑊 ) |
51 |
24 50
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → 𝑚 ∈ 𝑊 ) |
52 |
49 51
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → ( 𝐺 ‘ 𝑚 ) ∈ 𝑋 ) |
53 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝐺 : 𝑊 ⟶ 𝑋 ) |
54 |
53 24
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ∈ 𝑋 ) |
55 |
54
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ∈ 𝑋 ) |
56 |
|
metsym |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( 𝐺 ‘ 𝑚 ) ∈ 𝑋 ∧ ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ∈ 𝑋 ) → ( ( 𝐺 ‘ 𝑚 ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) = ( ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ 𝑚 ) ) ) |
57 |
48 52 55 56
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → ( ( 𝐺 ‘ 𝑚 ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) = ( ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ 𝑚 ) ) ) |
58 |
47 57
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → ( ( 𝐺 ‘ ( ( 𝑚 − 𝑁 ) + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) = ( ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ 𝑚 ) ) ) |
59 |
58
|
breq1d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → ( ( ( 𝐺 ‘ ( ( 𝑚 − 𝑁 ) + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) < 𝑥 ↔ ( ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ 𝑚 ) ) < 𝑥 ) ) |
60 |
39 59
|
sylibd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) ∈ 𝑋 ∧ ( ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) < 𝑥 ) → ( ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ 𝑚 ) ) < 𝑥 ) ) |
61 |
60
|
ralrimdva |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) ∈ 𝑋 ∧ ( ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) < 𝑥 ) → ∀ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ( ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ 𝑚 ) ) < 𝑥 ) ) |
62 |
|
fveq2 |
⊢ ( 𝑛 = ( 𝑗 + 𝑁 ) → ( ℤ≥ ‘ 𝑛 ) = ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ) |
63 |
|
fveq2 |
⊢ ( 𝑛 = ( 𝑗 + 𝑁 ) → ( 𝐺 ‘ 𝑛 ) = ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) |
64 |
63
|
oveq1d |
⊢ ( 𝑛 = ( 𝑗 + 𝑁 ) → ( ( 𝐺 ‘ 𝑛 ) 𝐷 ( 𝐺 ‘ 𝑚 ) ) = ( ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ 𝑚 ) ) ) |
65 |
64
|
breq1d |
⊢ ( 𝑛 = ( 𝑗 + 𝑁 ) → ( ( ( 𝐺 ‘ 𝑛 ) 𝐷 ( 𝐺 ‘ 𝑚 ) ) < 𝑥 ↔ ( ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ 𝑚 ) ) < 𝑥 ) ) |
66 |
62 65
|
raleqbidv |
⊢ ( 𝑛 = ( 𝑗 + 𝑁 ) → ( ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐺 ‘ 𝑛 ) 𝐷 ( 𝐺 ‘ 𝑚 ) ) < 𝑥 ↔ ∀ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ( ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ 𝑚 ) ) < 𝑥 ) ) |
67 |
66
|
rspcev |
⊢ ( ( ( 𝑗 + 𝑁 ) ∈ 𝑊 ∧ ∀ 𝑚 ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑁 ) ) ( ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ 𝑚 ) ) < 𝑥 ) → ∃ 𝑛 ∈ 𝑊 ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐺 ‘ 𝑛 ) 𝐷 ( 𝐺 ‘ 𝑚 ) ) < 𝑥 ) |
68 |
24 61 67
|
syl6an |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) ∈ 𝑋 ∧ ( ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) < 𝑥 ) → ∃ 𝑛 ∈ 𝑊 ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐺 ‘ 𝑛 ) 𝐷 ( 𝐺 ‘ 𝑚 ) ) < 𝑥 ) ) |
69 |
68
|
rexlimdva |
⊢ ( 𝜑 → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) ∈ 𝑋 ∧ ( ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) < 𝑥 ) → ∃ 𝑛 ∈ 𝑊 ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐺 ‘ 𝑛 ) 𝐷 ( 𝐺 ‘ 𝑚 ) ) < 𝑥 ) ) |
70 |
69
|
ralimdv |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) ∈ 𝑋 ∧ ( ( 𝐺 ‘ ( 𝑘 + 𝑁 ) ) 𝐷 ( 𝐺 ‘ ( 𝑗 + 𝑁 ) ) ) < 𝑥 ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑛 ∈ 𝑊 ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐺 ‘ 𝑛 ) 𝐷 ( 𝐺 ‘ 𝑚 ) ) < 𝑥 ) ) |
71 |
19 70
|
mpd |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑛 ∈ 𝑊 ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐺 ‘ 𝑛 ) 𝐷 ( 𝐺 ‘ 𝑚 ) ) < 𝑥 ) |
72 |
2 5
|
zaddcld |
⊢ ( 𝜑 → ( 𝑀 + 𝑁 ) ∈ ℤ ) |
73 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝑊 ) → ( 𝐺 ‘ 𝑚 ) = ( 𝐺 ‘ 𝑚 ) ) |
74 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑊 ) → ( 𝐺 ‘ 𝑛 ) = ( 𝐺 ‘ 𝑛 ) ) |
75 |
4 10 72 73 74 8
|
iscauf |
⊢ ( 𝜑 → ( 𝐺 ∈ ( Cau ‘ 𝐷 ) ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑛 ∈ 𝑊 ∀ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ( ( 𝐺 ‘ 𝑛 ) 𝐷 ( 𝐺 ‘ 𝑚 ) ) < 𝑥 ) ) |
76 |
71 75
|
mpbird |
⊢ ( 𝜑 → 𝐺 ∈ ( Cau ‘ 𝐷 ) ) |