| Step |
Hyp |
Ref |
Expression |
| 1 |
|
dyadmbl.1 |
⊢ 𝐹 = ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) |
| 2 |
|
fveq2 |
⊢ ( 𝑧 = 𝑤 → ( [,] ‘ 𝑧 ) = ( [,] ‘ 𝑤 ) ) |
| 3 |
2
|
sseq1d |
⊢ ( 𝑧 = 𝑤 → ( ( [,] ‘ 𝑧 ) ⊆ 𝐴 ↔ ( [,] ‘ 𝑤 ) ⊆ 𝐴 ) ) |
| 4 |
3
|
elrab |
⊢ ( 𝑤 ∈ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ↔ ( 𝑤 ∈ ran 𝐹 ∧ ( [,] ‘ 𝑤 ) ⊆ 𝐴 ) ) |
| 5 |
|
simprr |
⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ ( 𝑤 ∈ ran 𝐹 ∧ ( [,] ‘ 𝑤 ) ⊆ 𝐴 ) ) → ( [,] ‘ 𝑤 ) ⊆ 𝐴 ) |
| 6 |
|
fvex |
⊢ ( [,] ‘ 𝑤 ) ∈ V |
| 7 |
6
|
elpw |
⊢ ( ( [,] ‘ 𝑤 ) ∈ 𝒫 𝐴 ↔ ( [,] ‘ 𝑤 ) ⊆ 𝐴 ) |
| 8 |
5 7
|
sylibr |
⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ ( 𝑤 ∈ ran 𝐹 ∧ ( [,] ‘ 𝑤 ) ⊆ 𝐴 ) ) → ( [,] ‘ 𝑤 ) ∈ 𝒫 𝐴 ) |
| 9 |
4 8
|
sylan2b |
⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) → ( [,] ‘ 𝑤 ) ∈ 𝒫 𝐴 ) |
| 10 |
9
|
ralrimiva |
⊢ ( 𝐴 ∈ ( topGen ‘ ran (,) ) → ∀ 𝑤 ∈ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ( [,] ‘ 𝑤 ) ∈ 𝒫 𝐴 ) |
| 11 |
|
iccf |
⊢ [,] : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ* |
| 12 |
|
ffun |
⊢ ( [,] : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ* → Fun [,] ) |
| 13 |
11 12
|
ax-mp |
⊢ Fun [,] |
| 14 |
|
ssrab2 |
⊢ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ⊆ ran 𝐹 |
| 15 |
1
|
dyadf |
⊢ 𝐹 : ( ℤ × ℕ0 ) ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) |
| 16 |
|
frn |
⊢ ( 𝐹 : ( ℤ × ℕ0 ) ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) → ran 𝐹 ⊆ ( ≤ ∩ ( ℝ × ℝ ) ) ) |
| 17 |
15 16
|
ax-mp |
⊢ ran 𝐹 ⊆ ( ≤ ∩ ( ℝ × ℝ ) ) |
| 18 |
|
inss2 |
⊢ ( ≤ ∩ ( ℝ × ℝ ) ) ⊆ ( ℝ × ℝ ) |
| 19 |
|
rexpssxrxp |
⊢ ( ℝ × ℝ ) ⊆ ( ℝ* × ℝ* ) |
| 20 |
18 19
|
sstri |
⊢ ( ≤ ∩ ( ℝ × ℝ ) ) ⊆ ( ℝ* × ℝ* ) |
| 21 |
17 20
|
sstri |
⊢ ran 𝐹 ⊆ ( ℝ* × ℝ* ) |
| 22 |
14 21
|
sstri |
⊢ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ⊆ ( ℝ* × ℝ* ) |
| 23 |
11
|
fdmi |
⊢ dom [,] = ( ℝ* × ℝ* ) |
| 24 |
22 23
|
sseqtrri |
⊢ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ⊆ dom [,] |
| 25 |
|
funimass4 |
⊢ ( ( Fun [,] ∧ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ⊆ dom [,] ) → ( ( [,] “ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ⊆ 𝒫 𝐴 ↔ ∀ 𝑤 ∈ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ( [,] ‘ 𝑤 ) ∈ 𝒫 𝐴 ) ) |
| 26 |
13 24 25
|
mp2an |
⊢ ( ( [,] “ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ⊆ 𝒫 𝐴 ↔ ∀ 𝑤 ∈ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ( [,] ‘ 𝑤 ) ∈ 𝒫 𝐴 ) |
| 27 |
10 26
|
sylibr |
⊢ ( 𝐴 ∈ ( topGen ‘ ran (,) ) → ( [,] “ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ⊆ 𝒫 𝐴 ) |
| 28 |
|
sspwuni |
⊢ ( ( [,] “ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ⊆ 𝒫 𝐴 ↔ ∪ ( [,] “ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ⊆ 𝐴 ) |
| 29 |
27 28
|
sylib |
⊢ ( 𝐴 ∈ ( topGen ‘ ran (,) ) → ∪ ( [,] “ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ⊆ 𝐴 ) |
| 30 |
|
eqid |
⊢ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) = ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) |
| 31 |
30
|
rexmet |
⊢ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ∈ ( ∞Met ‘ ℝ ) |
| 32 |
|
eqid |
⊢ ( MetOpen ‘ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ) = ( MetOpen ‘ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ) |
| 33 |
30 32
|
tgioo |
⊢ ( topGen ‘ ran (,) ) = ( MetOpen ‘ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ) |
| 34 |
33
|
mopni2 |
⊢ ( ( ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ∈ ( ∞Met ‘ ℝ ) ∧ 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) → ∃ 𝑟 ∈ ℝ+ ( 𝑤 ( ball ‘ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ) 𝑟 ) ⊆ 𝐴 ) |
| 35 |
31 34
|
mp3an1 |
⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) → ∃ 𝑟 ∈ ℝ+ ( 𝑤 ( ball ‘ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ) 𝑟 ) ⊆ 𝐴 ) |
| 36 |
|
elssuni |
⊢ ( 𝐴 ∈ ( topGen ‘ ran (,) ) → 𝐴 ⊆ ∪ ( topGen ‘ ran (,) ) ) |
| 37 |
|
uniretop |
⊢ ℝ = ∪ ( topGen ‘ ran (,) ) |
| 38 |
36 37
|
sseqtrrdi |
⊢ ( 𝐴 ∈ ( topGen ‘ ran (,) ) → 𝐴 ⊆ ℝ ) |
| 39 |
38
|
sselda |
⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) → 𝑤 ∈ ℝ ) |
| 40 |
|
rpre |
⊢ ( 𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ ) |
| 41 |
30
|
bl2ioo |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑟 ∈ ℝ ) → ( 𝑤 ( ball ‘ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ) 𝑟 ) = ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ) |
| 42 |
39 40 41
|
syl2an |
⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ+ ) → ( 𝑤 ( ball ‘ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ) 𝑟 ) = ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ) |
| 43 |
42
|
sseq1d |
⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ+ ) → ( ( 𝑤 ( ball ‘ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ) 𝑟 ) ⊆ 𝐴 ↔ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) |
| 44 |
|
2re |
⊢ 2 ∈ ℝ |
| 45 |
|
1lt2 |
⊢ 1 < 2 |
| 46 |
|
expnlbnd |
⊢ ( ( 𝑟 ∈ ℝ+ ∧ 2 ∈ ℝ ∧ 1 < 2 ) → ∃ 𝑛 ∈ ℕ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) |
| 47 |
44 45 46
|
mp3an23 |
⊢ ( 𝑟 ∈ ℝ+ → ∃ 𝑛 ∈ ℕ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) |
| 48 |
47
|
ad2antrl |
⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) → ∃ 𝑛 ∈ ℕ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) |
| 49 |
39
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → 𝑤 ∈ ℝ ) |
| 50 |
|
2nn |
⊢ 2 ∈ ℕ |
| 51 |
|
nnnn0 |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℕ0 ) |
| 52 |
51
|
ad2antrl |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → 𝑛 ∈ ℕ0 ) |
| 53 |
|
nnexpcl |
⊢ ( ( 2 ∈ ℕ ∧ 𝑛 ∈ ℕ0 ) → ( 2 ↑ 𝑛 ) ∈ ℕ ) |
| 54 |
50 52 53
|
sylancr |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 2 ↑ 𝑛 ) ∈ ℕ ) |
| 55 |
54
|
nnred |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 2 ↑ 𝑛 ) ∈ ℝ ) |
| 56 |
49 55
|
remulcld |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 𝑤 · ( 2 ↑ 𝑛 ) ) ∈ ℝ ) |
| 57 |
|
fllelt |
⊢ ( ( 𝑤 · ( 2 ↑ 𝑛 ) ) ∈ ℝ → ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ≤ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ∧ ( 𝑤 · ( 2 ↑ 𝑛 ) ) < ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) ) ) |
| 58 |
56 57
|
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 ) ) ) |
| 59 |
58
|
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 ) ) ) |
| 60 |
|
reflcl |
⊢ ( ( 𝑤 · ( 2 ↑ 𝑛 ) ) ∈ ℝ → ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ∈ ℝ ) |
| 61 |
56 60
|
syl |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ∈ ℝ ) |
| 62 |
54
|
nngt0d |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → 0 < ( 2 ↑ 𝑛 ) ) |
| 63 |
|
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 ) ) ) ) |
| 64 |
61 49 55 62 63
|
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 ) ) ) ) |
| 65 |
59 64
|
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 ) |
| 66 |
|
peano2re |
⊢ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ∈ ℝ → ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) ∈ ℝ ) |
| 67 |
61 66
|
syl |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) ∈ ℝ ) |
| 68 |
67 54
|
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 ) |
| 69 |
58
|
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 ) ) |
| 70 |
|
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 ) ) ) ) |
| 71 |
49 67 55 62 70
|
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 ) ) ) ) |
| 72 |
69 71
|
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 ) ) ) |
| 73 |
49 68 72
|
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 ) ) ) |
| 74 |
61 54
|
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 ) |
| 75 |
|
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 ) ) ) ) ) |
| 76 |
74 68 75
|
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 ) ) ) ) ) |
| 77 |
49 65 73 76
|
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 ) ) ) ) |
| 78 |
56
|
flcld |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ∈ ℤ ) |
| 79 |
1
|
dyadval |
⊢ ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ∈ ℤ ∧ 𝑛 ∈ ℕ0 ) → ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) 𝐹 𝑛 ) = ⟨ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) , ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ⟩ ) |
| 80 |
78 52 79
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) 𝐹 𝑛 ) = ⟨ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) , ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ⟩ ) |
| 81 |
80
|
fveq2d |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) 𝐹 𝑛 ) ) = ( [,] ‘ ⟨ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) , ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ⟩ ) ) |
| 82 |
|
df-ov |
⊢ ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) [,] ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ) = ( [,] ‘ ⟨ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) , ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ⟩ ) |
| 83 |
81 82
|
eqtr4di |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) 𝐹 𝑛 ) ) = ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) / ( 2 ↑ 𝑛 ) ) [,] ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) + 1 ) / ( 2 ↑ 𝑛 ) ) ) ) |
| 84 |
77 83
|
eleqtrrd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → 𝑤 ∈ ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) 𝐹 𝑛 ) ) ) |
| 85 |
|
fveq2 |
⊢ ( 𝑧 = ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) 𝐹 𝑛 ) → ( [,] ‘ 𝑧 ) = ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) 𝐹 𝑛 ) ) ) |
| 86 |
85
|
sseq1d |
⊢ ( 𝑧 = ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) 𝐹 𝑛 ) → ( ( [,] ‘ 𝑧 ) ⊆ 𝐴 ↔ ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) 𝐹 𝑛 ) ) ⊆ 𝐴 ) ) |
| 87 |
|
ffn |
⊢ ( 𝐹 : ( ℤ × ℕ0 ) ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) → 𝐹 Fn ( ℤ × ℕ0 ) ) |
| 88 |
15 87
|
ax-mp |
⊢ 𝐹 Fn ( ℤ × ℕ0 ) |
| 89 |
|
fnovrn |
⊢ ( ( 𝐹 Fn ( ℤ × ℕ0 ) ∧ ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ∈ ℤ ∧ 𝑛 ∈ ℕ0 ) → ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) 𝐹 𝑛 ) ∈ ran 𝐹 ) |
| 90 |
88 78 52 89
|
mp3an2i |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) 𝐹 𝑛 ) ∈ ran 𝐹 ) |
| 91 |
|
simplrl |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → 𝑟 ∈ ℝ+ ) |
| 92 |
91
|
rpred |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → 𝑟 ∈ ℝ ) |
| 93 |
49 92
|
resubcld |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 𝑤 − 𝑟 ) ∈ ℝ ) |
| 94 |
93
|
rexrd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 𝑤 − 𝑟 ) ∈ ℝ* ) |
| 95 |
49 92
|
readdcld |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 𝑤 + 𝑟 ) ∈ ℝ ) |
| 96 |
95
|
rexrd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 𝑤 + 𝑟 ) ∈ ℝ* ) |
| 97 |
74 92
|
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 ) |
| 98 |
61
|
recnd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) ∈ ℂ ) |
| 99 |
|
1cnd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → 1 ∈ ℂ ) |
| 100 |
55
|
recnd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 2 ↑ 𝑛 ) ∈ ℂ ) |
| 101 |
54
|
nnne0d |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 2 ↑ 𝑛 ) ≠ 0 ) |
| 102 |
98 99 100 101
|
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 ) ) ) ) |
| 103 |
54
|
nnrecred |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 1 / ( 2 ↑ 𝑛 ) ) ∈ ℝ ) |
| 104 |
|
simprr |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) |
| 105 |
103 92 74 104
|
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 ) ) |
| 106 |
102 105
|
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 ) ) |
| 107 |
49 68 97 72 106
|
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 ) ) |
| 108 |
49 92 74
|
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 ) ) ) |
| 109 |
107 108
|
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 ) ) ) |
| 110 |
49 103
|
readdcld |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( 𝑤 + ( 1 / ( 2 ↑ 𝑛 ) ) ) ∈ ℝ ) |
| 111 |
74 49 103 65
|
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 ) ) ) ) |
| 112 |
102 111
|
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 ) ) ) ) |
| 113 |
103 92 49 104
|
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 ) ) |
| 114 |
68 110 95 112 113
|
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 ) ) |
| 115 |
|
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 ) ) ) |
| 116 |
94 96 109 114 115
|
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 ) ) ) |
| 117 |
83 116
|
eqsstrd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) 𝐹 𝑛 ) ) ⊆ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ) |
| 118 |
|
simplrr |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) |
| 119 |
117 118
|
sstrd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) 𝐹 𝑛 ) ) ⊆ 𝐴 ) |
| 120 |
86 90 119
|
elrabd |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) 𝐹 𝑛 ) ∈ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) |
| 121 |
|
funfvima2 |
⊢ ( ( Fun [,] ∧ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ⊆ dom [,] ) → ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) 𝐹 𝑛 ) ∈ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } → ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) 𝐹 𝑛 ) ) ∈ ( [,] “ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) ) |
| 122 |
13 24 121
|
mp2an |
⊢ ( ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) 𝐹 𝑛 ) ∈ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } → ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) 𝐹 𝑛 ) ) ∈ ( [,] “ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) |
| 123 |
120 122
|
syl |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) 𝐹 𝑛 ) ) ∈ ( [,] “ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) |
| 124 |
|
elunii |
⊢ ( ( 𝑤 ∈ ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) 𝐹 𝑛 ) ) ∧ ( [,] ‘ ( ( ⌊ ‘ ( 𝑤 · ( 2 ↑ 𝑛 ) ) ) 𝐹 𝑛 ) ) ∈ ( [,] “ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) → 𝑤 ∈ ∪ ( [,] “ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) |
| 125 |
84 123 124
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / ( 2 ↑ 𝑛 ) ) < 𝑟 ) ) → 𝑤 ∈ ∪ ( [,] “ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) |
| 126 |
48 125
|
rexlimddv |
⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 ) ) → 𝑤 ∈ ∪ ( [,] “ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) |
| 127 |
126
|
expr |
⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ+ ) → ( ( ( 𝑤 − 𝑟 ) (,) ( 𝑤 + 𝑟 ) ) ⊆ 𝐴 → 𝑤 ∈ ∪ ( [,] “ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) ) |
| 128 |
43 127
|
sylbid |
⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ+ ) → ( ( 𝑤 ( ball ‘ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ) 𝑟 ) ⊆ 𝐴 → 𝑤 ∈ ∪ ( [,] “ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) ) |
| 129 |
128
|
rexlimdva |
⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) → ( ∃ 𝑟 ∈ ℝ+ ( 𝑤 ( ball ‘ ( ( abs ∘ − ) ↾ ( ℝ × ℝ ) ) ) 𝑟 ) ⊆ 𝐴 → 𝑤 ∈ ∪ ( [,] “ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) ) |
| 130 |
35 129
|
mpd |
⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑤 ∈ 𝐴 ) → 𝑤 ∈ ∪ ( [,] “ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ) |
| 131 |
29 130
|
eqelssd |
⊢ ( 𝐴 ∈ ( topGen ‘ ran (,) ) → ∪ ( [,] “ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) = 𝐴 ) |
| 132 |
|
fveq2 |
⊢ ( 𝑐 = 𝑎 → ( [,] ‘ 𝑐 ) = ( [,] ‘ 𝑎 ) ) |
| 133 |
132
|
sseq1d |
⊢ ( 𝑐 = 𝑎 → ( ( [,] ‘ 𝑐 ) ⊆ ( [,] ‘ 𝑏 ) ↔ ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑏 ) ) ) |
| 134 |
|
equequ1 |
⊢ ( 𝑐 = 𝑎 → ( 𝑐 = 𝑏 ↔ 𝑎 = 𝑏 ) ) |
| 135 |
133 134
|
imbi12d |
⊢ ( 𝑐 = 𝑎 → ( ( ( [,] ‘ 𝑐 ) ⊆ ( [,] ‘ 𝑏 ) → 𝑐 = 𝑏 ) ↔ ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑏 ) → 𝑎 = 𝑏 ) ) ) |
| 136 |
135
|
ralbidv |
⊢ ( 𝑐 = 𝑎 → ( ∀ 𝑏 ∈ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑐 ) ⊆ ( [,] ‘ 𝑏 ) → 𝑐 = 𝑏 ) ↔ ∀ 𝑏 ∈ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑏 ) → 𝑎 = 𝑏 ) ) ) |
| 137 |
136
|
cbvrabv |
⊢ { 𝑐 ∈ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ∣ ∀ 𝑏 ∈ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑐 ) ⊆ ( [,] ‘ 𝑏 ) → 𝑐 = 𝑏 ) } = { 𝑎 ∈ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ∣ ∀ 𝑏 ∈ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑏 ) → 𝑎 = 𝑏 ) } |
| 138 |
14
|
a1i |
⊢ ( 𝐴 ∈ ( topGen ‘ ran (,) ) → { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ⊆ ran 𝐹 ) |
| 139 |
1 137 138
|
dyadmbl |
⊢ ( 𝐴 ∈ ( topGen ‘ ran (,) ) → ∪ ( [,] “ { 𝑧 ∈ ran 𝐹 ∣ ( [,] ‘ 𝑧 ) ⊆ 𝐴 } ) ∈ dom vol ) |
| 140 |
131 139
|
eqeltrrd |
⊢ ( 𝐴 ∈ ( topGen ‘ ran (,) ) → 𝐴 ∈ dom vol ) |