| Step |
Hyp |
Ref |
Expression |
| 1 |
|
fveq2 |
⊢ ( 𝑧 = 𝑤 → ( [,] ‘ 𝑧 ) = ( [,] ‘ 𝑤 ) ) |
| 2 |
1
|
sseq1d |
⊢ ( 𝑧 = 𝑤 → ( ( [,] ‘ 𝑧 ) ⊆ 𝐴 ↔ ( [,] ‘ 𝑤 ) ⊆ 𝐴 ) ) |
| 3 |
2
|
elrab |
⊢ ( 𝑤 ∈ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ↔ ( 𝑤 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∧ ( [,] ‘ 𝑤 ) ⊆ 𝐴 ) ) |
| 4 |
|
simprr |
⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ ( 𝑤 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∧ ( [,] ‘ 𝑤 ) ⊆ 𝐴 ) ) → ( [,] ‘ 𝑤 ) ⊆ 𝐴 ) |
| 5 |
|
fvex |
⊢ ( [,] ‘ 𝑤 ) ∈ V |
| 6 |
5
|
elpw |
⊢ ( ( [,] ‘ 𝑤 ) ∈ 𝒫 𝐴 ↔ ( [,] ‘ 𝑤 ) ⊆ 𝐴 ) |
| 7 |
4 6
|
sylibr |
⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ ( 𝑤 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∧ ( [,] ‘ 𝑤 ) ⊆ 𝐴 ) ) → ( [,] ‘ 𝑤 ) ∈ 𝒫 𝐴 ) |
| 8 |
3 7
|
sylan2b |
⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) → ( [,] ‘ 𝑤 ) ∈ 𝒫 𝐴 ) |
| 9 |
8
|
ralrimiva |
⊢ ( 𝐴 ∈ ( topGen ‘ ran (,) ) → ∀ 𝑤 ∈ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ( [,] ‘ 𝑤 ) ∈ 𝒫 𝐴 ) |
| 10 |
|
iccf |
⊢ [,] : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ* |
| 11 |
|
ffun |
⊢ ( [,] : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ* → Fun [,] ) |
| 12 |
10 11
|
ax-mp |
⊢ Fun [,] |
| 13 |
|
ssrab2 |
⊢ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ⊆ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) |
| 14 |
|
oveq1 |
⊢ ( 𝑥 = 𝑟 → ( 𝑥 / ( 2 ↑ 𝑦 ) ) = ( 𝑟 / ( 2 ↑ 𝑦 ) ) ) |
| 15 |
|
oveq1 |
⊢ ( 𝑥 = 𝑟 → ( 𝑥 + 1 ) = ( 𝑟 + 1 ) ) |
| 16 |
15
|
oveq1d |
⊢ ( 𝑥 = 𝑟 → ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) = ( ( 𝑟 + 1 ) / ( 2 ↑ 𝑦 ) ) ) |
| 17 |
14 16
|
opeq12d |
⊢ ( 𝑥 = 𝑟 → ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ = ⟨ ( 𝑟 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑟 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) |
| 18 |
|
oveq2 |
⊢ ( 𝑦 = 𝑠 → ( 2 ↑ 𝑦 ) = ( 2 ↑ 𝑠 ) ) |
| 19 |
18
|
oveq2d |
⊢ ( 𝑦 = 𝑠 → ( 𝑟 / ( 2 ↑ 𝑦 ) ) = ( 𝑟 / ( 2 ↑ 𝑠 ) ) ) |
| 20 |
18
|
oveq2d |
⊢ ( 𝑦 = 𝑠 → ( ( 𝑟 + 1 ) / ( 2 ↑ 𝑦 ) ) = ( ( 𝑟 + 1 ) / ( 2 ↑ 𝑠 ) ) ) |
| 21 |
19 20
|
opeq12d |
⊢ ( 𝑦 = 𝑠 → ⟨ ( 𝑟 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑟 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ = ⟨ ( 𝑟 / ( 2 ↑ 𝑠 ) ) , ( ( 𝑟 + 1 ) / ( 2 ↑ 𝑠 ) ) ⟩ ) |
| 22 |
17 21
|
cbvmpov |
⊢ ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) = ( 𝑟 ∈ ℤ , 𝑠 ∈ ℕ0 ↦ ⟨ ( 𝑟 / ( 2 ↑ 𝑠 ) ) , ( ( 𝑟 + 1 ) / ( 2 ↑ 𝑠 ) ) ⟩ ) |
| 23 |
22
|
dyadf |
⊢ ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) : ( ℤ × ℕ0 ) ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) |
| 24 |
|
frn |
⊢ ( ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) : ( ℤ × ℕ0 ) ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) → ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ⊆ ( ≤ ∩ ( ℝ × ℝ ) ) ) |
| 25 |
23 24
|
ax-mp |
⊢ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ⊆ ( ≤ ∩ ( ℝ × ℝ ) ) |
| 26 |
|
inss2 |
⊢ ( ≤ ∩ ( ℝ × ℝ ) ) ⊆ ( ℝ × ℝ ) |
| 27 |
|
rexpssxrxp |
⊢ ( ℝ × ℝ ) ⊆ ( ℝ* × ℝ* ) |
| 28 |
26 27
|
sstri |
⊢ ( ≤ ∩ ( ℝ × ℝ ) ) ⊆ ( ℝ* × ℝ* ) |
| 29 |
25 28
|
sstri |
⊢ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ⊆ ( ℝ* × ℝ* ) |
| 30 |
13 29
|
sstri |
⊢ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ⊆ ( ℝ* × ℝ* ) |
| 31 |
10
|
fdmi |
⊢ dom [,] = ( ℝ* × ℝ* ) |
| 32 |
30 31
|
sseqtrri |
⊢ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ⊆ dom [,] |
| 33 |
|
funimass4 |
⊢ ( ( Fun [,] ∧ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ⊆ dom [,] ) → ( ( [,] “ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ⊆ 𝒫 𝐴 ↔ ∀ 𝑤 ∈ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ( [,] ‘ 𝑤 ) ∈ 𝒫 𝐴 ) ) |
| 34 |
12 32 33
|
mp2an |
⊢ ( ( [,] “ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ⊆ 𝒫 𝐴 ↔ ∀ 𝑤 ∈ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ( [,] ‘ 𝑤 ) ∈ 𝒫 𝐴 ) |
| 35 |
9 34
|
sylibr |
⊢ ( 𝐴 ∈ ( topGen ‘ ran (,) ) → ( [,] “ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ⊆ 𝒫 𝐴 ) |
| 36 |
|
sspwuni |
⊢ ( ( [,] “ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ⊆ 𝒫 𝐴 ↔ ∪ ( [,] “ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ⊆ 𝐴 ) |
| 37 |
35 36
|
sylib |
⊢ ( 𝐴 ∈ ( topGen ‘ ran (,) ) → ∪ ( [,] “ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ⊆ 𝐴 ) |
| 38 |
|
eqid |
⊢ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) = ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) |
| 39 |
38
|
rexmet |
⊢ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ∈ ( ∞Met ‘ ℝ ) |
| 40 |
|
eqid |
⊢ ( MetOpen ‘ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ) = ( MetOpen ‘ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ) |
| 41 |
38 40
|
tgioo |
⊢ ( topGen ‘ ran (,) ) = ( MetOpen ‘ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ) |
| 42 |
41
|
mopni2 |
⊢ ( ( ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ∈ ( ∞Met ‘ ℝ ) ∧ 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) → ∃ 𝑟 ∈ ℝ+ ( 𝑤 ( ball ‘ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ) 𝑟 ) ⊆ 𝐴 ) |
| 43 |
39 42
|
mp3an1 |
⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) → ∃ 𝑟 ∈ ℝ+ ( 𝑤 ( ball ‘ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ) 𝑟 ) ⊆ 𝐴 ) |
| 44 |
|
elssuni |
⊢ ( 𝐴 ∈ ( topGen ‘ ran (,) ) → 𝐴 ⊆ ∪ ( topGen ‘ ran (,) ) ) |
| 45 |
|
uniretop |
⊢ ℝ = ∪ ( topGen ‘ ran (,) ) |
| 46 |
44 45
|
sseqtrrdi |
⊢ ( 𝐴 ∈ ( topGen ‘ ran (,) ) → 𝐴 ⊆ ℝ ) |
| 47 |
46
|
sselda |
⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) → 𝑤 ∈ ℝ ) |
| 48 |
|
rpre |
⊢ ( 𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ ) |
| 49 |
38
|
bl2ioo |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑟 ∈ ℝ ) → ( 𝑤 ( ball ‘ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ) 𝑟 ) = ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ) |
| 50 |
47 48 49
|
syl2an |
⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ+ ) → ( 𝑤 ( ball ‘ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ) 𝑟 ) = ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ) |
| 51 |
50
|
sseq1d |
⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ+ ) → ( ( 𝑤 ( ball ‘ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ) 𝑟 ) ⊆ 𝐴 ↔ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) |
| 52 |
|
2re |
⊢ 2 ∈ ℝ |
| 53 |
|
1lt2 |
⊢ 1 < 2 |
| 54 |
|
expnlbnd |
⊢ ( ( 𝑟 ∈ ℝ+ ∧ 2 ∈ ℝ ∧ 1 < 2 ) → ∃ 𝑛 ∈ ℕ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) |
| 55 |
52 53 54
|
mp3an23 |
⊢ ( 𝑟 ∈ ℝ+ → ∃ 𝑛 ∈ ℕ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) |
| 56 |
55
|
ad2antrl |
⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) → ∃ 𝑛 ∈ ℕ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) |
| 57 |
47
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → 𝑤 ∈ ℝ ) |
| 58 |
|
2nn |
⊢ 2 ∈ ℕ |
| 59 |
|
nnnn0 |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℕ0 ) |
| 60 |
59
|
ad2antrl |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → 𝑛 ∈ ℕ0 ) |
| 61 |
|
nnexpcl |
⊢ ( ( 2 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) → ( 2 ↑ 𝑛 ) ∈ ℕ ) |
| 62 |
58 60 61
|
sylancr |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 2 ↑ 𝑛 ) ∈ ℕ ) |
| 63 |
62
|
nnred |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 2 ↑ 𝑛 ) ∈ ℝ ) |
| 64 |
57 63
|
remulcld |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 𝑤 · ( 2 ↑ 𝑛 ) ) ∈ ℝ ) |
| 65 |
|
fllelt |
⊢ ( ( 𝑤 · ( 2 ↑ 𝑛 ) ) ∈ ℝ → ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ≤ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ∧ ( 𝑤 · ( 2 ↑ 𝑛 ) ) < ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) ) ) |
| 66 |
64 65
|
syl |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ≤ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ∧ ( 𝑤 · ( 2 ↑ 𝑛 ) ) < ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) ) ) |
| 67 |
66
|
simpld |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ≤ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) |
| 68 |
|
reflcl |
⊢ ( ( 𝑤 · ( 2 ↑ 𝑛 ) ) ∈ ℝ → ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ∈ ℝ ) |
| 69 |
64 68
|
syl |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ∈ ℝ ) |
| 70 |
62
|
nngt0d |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → 0 < ( 2 ↑ 𝑛 ) ) |
| 71 |
|
ledivmul2 |
⊢ ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ ( ( 2 ↑ 𝑛 ) ∈ ℝ ∧ 0 < ( 2 ↑ 𝑛 ) ) ) → ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) ≤ 𝑤 ↔ ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ≤ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ) |
| 72 |
69 57 63 70 71
|
syl112anc |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) ≤ 𝑤 ↔ ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ≤ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ) |
| 73 |
67 72
|
mpbird |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) ≤ 𝑤 ) |
| 74 |
|
peano2re |
⊢ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ∈ ℝ → ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) ∈ ℝ ) |
| 75 |
69 74
|
syl |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) ∈ ℝ ) |
| 76 |
75 62
|
nndivred |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ∈ ℝ ) |
| 77 |
66
|
simprd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 𝑤 · ( 2 ↑ 𝑛 ) ) < ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) ) |
| 78 |
|
ltmuldiv |
⊢ ( ( 𝑤 ∈ ℝ ∧ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) ∈ ℝ ∧ ( ( 2 ↑ 𝑛 ) ∈ ℝ ∧ 0 < ( 2 ↑ 𝑛 ) ) ) → ( ( 𝑤 · ( 2 ↑ 𝑛 ) ) < ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) ↔ 𝑤 < ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ) ) |
| 79 |
57 75 63 70 78
|
syl112anc |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( 𝑤 · ( 2 ↑ 𝑛 ) ) < ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) ↔ 𝑤 < ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ) ) |
| 80 |
77 79
|
mpbid |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → 𝑤 < ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ) |
| 81 |
57 76 80
|
ltled |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → 𝑤 ≤ ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ) |
| 82 |
69 62
|
nndivred |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) ∈ ℝ ) |
| 83 |
|
elicc2 |
⊢ ( ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) ∈ ℝ ∧ ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ∈ ℝ ) → ( 𝑤 ∈ ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) [,] ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ) ↔ ( 𝑤 ∈ ℝ ∧ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) ≤ 𝑤 ∧ 𝑤 ≤ ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ) ) ) |
| 84 |
82 76 83
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 𝑤 ∈ ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) [,] ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ) ↔ ( 𝑤 ∈ ℝ ∧ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) ≤ 𝑤 ∧ 𝑤 ≤ ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ) ) ) |
| 85 |
57 73 81 84
|
mpbir3and |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → 𝑤 ∈ ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) [,] ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ) ) |
| 86 |
64
|
flcld |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ∈ ℤ ) |
| 87 |
22
|
dyadval |
⊢ ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ∈ ℤ ∧ 𝑛 ∈ ℕ0 ) → ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) = ⟨ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) , ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ⟩ ) |
| 88 |
86 60 87
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) = ⟨ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) , ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ⟩ ) |
| 89 |
88
|
fveq2d |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) ) = ( [,] ‘ ⟨ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) , ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ⟩ ) ) |
| 90 |
|
df-ov |
⊢ ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) [,] ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ) = ( [,] ‘ ⟨ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) , ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ⟩ ) |
| 91 |
89 90
|
eqtr4di |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) ) = ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) [,] ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ) ) |
| 92 |
85 91
|
eleqtrrd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → 𝑤 ∈ ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) ) ) |
| 93 |
|
ffn |
⊢ ( ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) : ( ℤ × ℕ0 ) ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) → ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) Fn ( ℤ × ℕ0 ) ) |
| 94 |
23 93
|
ax-mp |
⊢ ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) Fn ( ℤ × ℕ0 ) |
| 95 |
|
fnovrn |
⊢ ( ( ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) Fn ( ℤ × ℕ0 ) ∧ ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ∈ ℤ ∧ 𝑛 ∈ ℕ0 ) → ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ) |
| 96 |
94 95
|
mp3an1 |
⊢ ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ∈ ℤ ∧ 𝑛 ∈ ℕ0 ) → ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ) |
| 97 |
86 60 96
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ) |
| 98 |
|
simplrl |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → 𝑟 ∈ ℝ+ ) |
| 99 |
98
|
rpred |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → 𝑟 ∈ ℝ ) |
| 100 |
57 99
|
resubcld |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 𝑤 − 𝑟 ) ∈ ℝ ) |
| 101 |
100
|
rexrd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 𝑤 − 𝑟 ) ∈ ℝ* ) |
| 102 |
57 99
|
readdcld |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 𝑤 + 𝑟 ) ∈ ℝ ) |
| 103 |
102
|
rexrd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 𝑤 + 𝑟 ) ∈ ℝ* ) |
| 104 |
82 99
|
readdcld |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) + 𝑟 ) ∈ ℝ ) |
| 105 |
69
|
recnd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ∈ ℂ ) |
| 106 |
|
1cnd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → 1 ∈ ℂ ) |
| 107 |
63
|
recnd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 2 ↑ 𝑛 ) ∈ ℂ ) |
| 108 |
62
|
nnne0d |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 2 ↑ 𝑛 ) ≠ 0 ) |
| 109 |
105 106 107 108
|
divdird |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) = ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) + ( 1 / ( 2 ↑ 𝑛 ) ) ) ) |
| 110 |
62
|
nnrecred |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 1 / ( 2 ↑ 𝑛 ) ) ∈ ℝ ) |
| 111 |
|
simprr |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) |
| 112 |
110 99 82 111
|
ltadd2dd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) + ( 1 / ( 2 ↑ 𝑛 ) ) ) < ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) + 𝑟 ) ) |
| 113 |
109 112
|
eqbrtrd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) < ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) + 𝑟 ) ) |
| 114 |
57 76 104 80 113
|
lttrd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → 𝑤 < ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) + 𝑟 ) ) |
| 115 |
57 99 82
|
ltsubaddd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( 𝑤 − 𝑟 ) < ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) ↔ 𝑤 < ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) + 𝑟 ) ) ) |
| 116 |
114 115
|
mpbird |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 𝑤 − 𝑟 ) < ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) ) |
| 117 |
57 110
|
readdcld |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 𝑤 + ( 1 / ( 2 ↑ 𝑛 ) ) ) ∈ ℝ ) |
| 118 |
82 57 110 73
|
leadd1dd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) + ( 1 / ( 2 ↑ 𝑛 ) ) ) ≤ ( 𝑤 + ( 1 / ( 2 ↑ 𝑛 ) ) ) ) |
| 119 |
109 118
|
eqbrtrd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ≤ ( 𝑤 + ( 1 / ( 2 ↑ 𝑛 ) ) ) ) |
| 120 |
110 99 57 111
|
ltadd2dd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 𝑤 + ( 1 / ( 2 ↑ 𝑛 ) ) ) < ( 𝑤 + 𝑟 ) ) |
| 121 |
76 117 102 119 120
|
lelttrd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) < ( 𝑤 + 𝑟 ) ) |
| 122 |
|
iccssioo |
⊢ ( ( ( ( 𝑤 − 𝑟 ) ∈ ℝ* ∧ ( 𝑤 + 𝑟 ) ∈ ℝ* ) ∧ ( ( 𝑤 − 𝑟 ) < ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) ∧ ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) < ( 𝑤 + 𝑟 ) ) ) → ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) [,] ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ) ⊆ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ) |
| 123 |
101 103 116 121 122
|
syl22anc |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) [,] ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ) ⊆ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ) |
| 124 |
91 123
|
eqsstrd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) ) ⊆ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ) |
| 125 |
|
simplrr |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) |
| 126 |
124 125
|
sstrd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) ) ⊆ 𝐴 ) |
| 127 |
|
fveq2 |
⊢ ( 𝑧 = ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) → ( [,] ‘ 𝑧 ) = ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) ) ) |
| 128 |
127
|
sseq1d |
⊢ ( 𝑧 = ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) → ( ( [,] ‘ 𝑧 ) ⊆ 𝐴 ↔ ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) ) ⊆ 𝐴 ) ) |
| 129 |
128
|
elrab |
⊢ ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) ∈ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ↔ ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∧ ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) ) ⊆ 𝐴 ) ) |
| 130 |
97 126 129
|
sylanbrc |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) ∈ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) |
| 131 |
|
funfvima2 |
⊢ ( ( Fun [,] ∧ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ⊆ dom [,] ) → ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) ∈ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } → ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) ) ∈ ( [,] “ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) ) |
| 132 |
12 32 131
|
mp2an |
⊢ ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) ∈ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } → ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) ) ∈ ( [,] “ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) |
| 133 |
130 132
|
syl |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) ) ∈ ( [,] “ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) |
| 134 |
|
elunii |
⊢ ( ( 𝑤 ∈ ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) ) ∧ ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) 𝑛 ) ) ∈ ( [,] “ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) → 𝑤 ∈ ∪ ( [,] “ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) |
| 135 |
92 133 134
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → 𝑤 ∈ ∪ ( [,] “ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) |
| 136 |
56 135
|
rexlimddv |
⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) → 𝑤 ∈ ∪ ( [,] “ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) |
| 137 |
136
|
expr |
⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ+ ) → ( ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 → 𝑤 ∈ ∪ ( [,] “ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) ) |
| 138 |
51 137
|
sylbid |
⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ+ ) → ( ( 𝑤 ( ball ‘ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ) 𝑟 ) ⊆ 𝐴 → 𝑤 ∈ ∪ ( [,] “ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) ) |
| 139 |
138
|
rexlimdva |
⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) → ( ∃ 𝑟 ∈ ℝ+ ( 𝑤 ( ball ‘ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ) 𝑟 ) ⊆ 𝐴 → 𝑤 ∈ ∪ ( [,] “ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) ) |
| 140 |
43 139
|
mpd |
⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) → 𝑤 ∈ ∪ ( [,] “ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) |
| 141 |
37 140
|
eqelssd |
⊢ ( 𝐴 ∈ ( topGen ‘ ran (,) ) → ∪ ( [,] “ { 𝑧 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) = 𝐴 ) |