Step |
Hyp |
Ref |
Expression |
1 |
|
dyadmbl.1 |
⊢ 𝐹 = ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ 〈 ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 ) |
2 |
|
zre |
⊢ ( 𝑥 ∈ ℤ → 𝑥 ∈ ℝ ) |
3 |
2
|
adantr |
⊢ ( ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0 ) → 𝑥 ∈ ℝ ) |
4 |
3
|
lep1d |
⊢ ( ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0 ) → 𝑥 ≤ ( 𝑥 + 1 ) ) |
5 |
|
peano2re |
⊢ ( 𝑥 ∈ ℝ → ( 𝑥 + 1 ) ∈ ℝ ) |
6 |
3 5
|
syl |
⊢ ( ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0 ) → ( 𝑥 + 1 ) ∈ ℝ ) |
7 |
|
2nn |
⊢ 2 ∈ ℕ |
8 |
|
nnexpcl |
⊢ ( ( 2 ∈ ℕ ∧ 𝑦 ∈ ℕ0 ) → ( 2 ↑ 𝑦 ) ∈ ℕ ) |
9 |
7 8
|
mpan |
⊢ ( 𝑦 ∈ ℕ0 → ( 2 ↑ 𝑦 ) ∈ ℕ ) |
10 |
9
|
adantl |
⊢ ( ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0 ) → ( 2 ↑ 𝑦 ) ∈ ℕ ) |
11 |
10
|
nnred |
⊢ ( ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0 ) → ( 2 ↑ 𝑦 ) ∈ ℝ ) |
12 |
10
|
nngt0d |
⊢ ( ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0 ) → 0 < ( 2 ↑ 𝑦 ) ) |
13 |
|
lediv1 |
⊢ ( ( 𝑥 ∈ ℝ ∧ ( 𝑥 + 1 ) ∈ ℝ ∧ ( ( 2 ↑ 𝑦 ) ∈ ℝ ∧ 0 < ( 2 ↑ 𝑦 ) ) ) → ( 𝑥 ≤ ( 𝑥 + 1 ) ↔ ( 𝑥 / ( 2 ↑ 𝑦 ) ) ≤ ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ) ) |
14 |
3 6 11 12 13
|
syl112anc |
⊢ ( ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0 ) → ( 𝑥 ≤ ( 𝑥 + 1 ) ↔ ( 𝑥 / ( 2 ↑ 𝑦 ) ) ≤ ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ) ) |
15 |
4 14
|
mpbid |
⊢ ( ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0 ) → ( 𝑥 / ( 2 ↑ 𝑦 ) ) ≤ ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ) |
16 |
|
df-br |
⊢ ( ( 𝑥 / ( 2 ↑ 𝑦 ) ) ≤ ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ↔ 〈 ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 ∈ ≤ ) |
17 |
15 16
|
sylib |
⊢ ( ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0 ) → 〈 ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 ∈ ≤ ) |
18 |
|
nndivre |
⊢ ( ( 𝑥 ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ∈ ℕ ) → ( 𝑥 / ( 2 ↑ 𝑦 ) ) ∈ ℝ ) |
19 |
2 9 18
|
syl2an |
⊢ ( ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0 ) → ( 𝑥 / ( 2 ↑ 𝑦 ) ) ∈ ℝ ) |
20 |
2 5
|
syl |
⊢ ( 𝑥 ∈ ℤ → ( 𝑥 + 1 ) ∈ ℝ ) |
21 |
|
nndivre |
⊢ ( ( ( 𝑥 + 1 ) ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ∈ ℕ ) → ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ∈ ℝ ) |
22 |
20 9 21
|
syl2an |
⊢ ( ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0 ) → ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ∈ ℝ ) |
23 |
19 22
|
opelxpd |
⊢ ( ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0 ) → 〈 ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 ∈ ( ℝ × ℝ ) ) |
24 |
17 23
|
elind |
⊢ ( ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0 ) → 〈 ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 ∈ ( ≤ ∩ ( ℝ × ℝ ) ) ) |
25 |
24
|
rgen2 |
⊢ ∀ 𝑥 ∈ ℤ ∀ 𝑦 ∈ ℕ0 〈 ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 ∈ ( ≤ ∩ ( ℝ × ℝ ) ) |
26 |
1
|
fmpo |
⊢ ( ∀ 𝑥 ∈ ℤ ∀ 𝑦 ∈ ℕ0 〈 ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 ∈ ( ≤ ∩ ( ℝ × ℝ ) ) ↔ 𝐹 : ( ℤ × ℕ0 ) ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ) |
27 |
25 26
|
mpbi |
⊢ 𝐹 : ( ℤ × ℕ0 ) ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) |