Step |
Hyp |
Ref |
Expression |
1 |
|
resqrtcl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( √ ‘ 𝐴 ) ∈ ℝ ) |
2 |
|
nn0z |
⊢ ( 𝐵 ∈ ℕ0 → 𝐵 ∈ ℤ ) |
3 |
|
flbi |
⊢ ( ( ( √ ‘ 𝐴 ) ∈ ℝ ∧ 𝐵 ∈ ℤ ) → ( ( ⌊ ‘ ( √ ‘ 𝐴 ) ) = 𝐵 ↔ ( 𝐵 ≤ ( √ ‘ 𝐴 ) ∧ ( √ ‘ 𝐴 ) < ( 𝐵 + 1 ) ) ) ) |
4 |
1 2 3
|
syl2an |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐵 ∈ ℕ0 ) → ( ( ⌊ ‘ ( √ ‘ 𝐴 ) ) = 𝐵 ↔ ( 𝐵 ≤ ( √ ‘ 𝐴 ) ∧ ( √ ‘ 𝐴 ) < ( 𝐵 + 1 ) ) ) ) |
5 |
|
nn0re |
⊢ ( 𝐵 ∈ ℕ0 → 𝐵 ∈ ℝ ) |
6 |
|
nn0ge0 |
⊢ ( 𝐵 ∈ ℕ0 → 0 ≤ 𝐵 ) |
7 |
5 6
|
jca |
⊢ ( 𝐵 ∈ ℕ0 → ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) |
8 |
|
sqrtsq |
⊢ ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) → ( √ ‘ ( 𝐵 ↑ 2 ) ) = 𝐵 ) |
9 |
8
|
eqcomd |
⊢ ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) → 𝐵 = ( √ ‘ ( 𝐵 ↑ 2 ) ) ) |
10 |
7 9
|
syl |
⊢ ( 𝐵 ∈ ℕ0 → 𝐵 = ( √ ‘ ( 𝐵 ↑ 2 ) ) ) |
11 |
10
|
breq1d |
⊢ ( 𝐵 ∈ ℕ0 → ( 𝐵 ≤ ( √ ‘ 𝐴 ) ↔ ( √ ‘ ( 𝐵 ↑ 2 ) ) ≤ ( √ ‘ 𝐴 ) ) ) |
12 |
11
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐵 ∈ ℕ0 ) → ( 𝐵 ≤ ( √ ‘ 𝐴 ) ↔ ( √ ‘ ( 𝐵 ↑ 2 ) ) ≤ ( √ ‘ 𝐴 ) ) ) |
13 |
|
nn0sqcl |
⊢ ( 𝐵 ∈ ℕ0 → ( 𝐵 ↑ 2 ) ∈ ℕ0 ) |
14 |
13
|
nn0red |
⊢ ( 𝐵 ∈ ℕ0 → ( 𝐵 ↑ 2 ) ∈ ℝ ) |
15 |
5
|
sqge0d |
⊢ ( 𝐵 ∈ ℕ0 → 0 ≤ ( 𝐵 ↑ 2 ) ) |
16 |
14 15
|
jca |
⊢ ( 𝐵 ∈ ℕ0 → ( ( 𝐵 ↑ 2 ) ∈ ℝ ∧ 0 ≤ ( 𝐵 ↑ 2 ) ) ) |
17 |
16
|
anim2i |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐵 ∈ ℕ0 ) → ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( ( 𝐵 ↑ 2 ) ∈ ℝ ∧ 0 ≤ ( 𝐵 ↑ 2 ) ) ) ) |
18 |
17
|
ancomd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐵 ∈ ℕ0 ) → ( ( ( 𝐵 ↑ 2 ) ∈ ℝ ∧ 0 ≤ ( 𝐵 ↑ 2 ) ) ∧ ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ) ) |
19 |
|
sqrtle |
⊢ ( ( ( ( 𝐵 ↑ 2 ) ∈ ℝ ∧ 0 ≤ ( 𝐵 ↑ 2 ) ) ∧ ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ) → ( ( 𝐵 ↑ 2 ) ≤ 𝐴 ↔ ( √ ‘ ( 𝐵 ↑ 2 ) ) ≤ ( √ ‘ 𝐴 ) ) ) |
20 |
18 19
|
syl |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐵 ∈ ℕ0 ) → ( ( 𝐵 ↑ 2 ) ≤ 𝐴 ↔ ( √ ‘ ( 𝐵 ↑ 2 ) ) ≤ ( √ ‘ 𝐴 ) ) ) |
21 |
12 20
|
bitr4d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐵 ∈ ℕ0 ) → ( 𝐵 ≤ ( √ ‘ 𝐴 ) ↔ ( 𝐵 ↑ 2 ) ≤ 𝐴 ) ) |
22 |
|
peano2nn0 |
⊢ ( 𝐵 ∈ ℕ0 → ( 𝐵 + 1 ) ∈ ℕ0 ) |
23 |
22
|
nn0red |
⊢ ( 𝐵 ∈ ℕ0 → ( 𝐵 + 1 ) ∈ ℝ ) |
24 |
|
1red |
⊢ ( 𝐵 ∈ ℕ0 → 1 ∈ ℝ ) |
25 |
|
0le1 |
⊢ 0 ≤ 1 |
26 |
25
|
a1i |
⊢ ( 𝐵 ∈ ℕ0 → 0 ≤ 1 ) |
27 |
5 24 6 26
|
addge0d |
⊢ ( 𝐵 ∈ ℕ0 → 0 ≤ ( 𝐵 + 1 ) ) |
28 |
23 27
|
sqrtsqd |
⊢ ( 𝐵 ∈ ℕ0 → ( √ ‘ ( ( 𝐵 + 1 ) ↑ 2 ) ) = ( 𝐵 + 1 ) ) |
29 |
28
|
eqcomd |
⊢ ( 𝐵 ∈ ℕ0 → ( 𝐵 + 1 ) = ( √ ‘ ( ( 𝐵 + 1 ) ↑ 2 ) ) ) |
30 |
29
|
breq2d |
⊢ ( 𝐵 ∈ ℕ0 → ( ( √ ‘ 𝐴 ) < ( 𝐵 + 1 ) ↔ ( √ ‘ 𝐴 ) < ( √ ‘ ( ( 𝐵 + 1 ) ↑ 2 ) ) ) ) |
31 |
30
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐵 ∈ ℕ0 ) → ( ( √ ‘ 𝐴 ) < ( 𝐵 + 1 ) ↔ ( √ ‘ 𝐴 ) < ( √ ‘ ( ( 𝐵 + 1 ) ↑ 2 ) ) ) ) |
32 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
33 |
32
|
a1i |
⊢ ( 𝐵 ∈ ℕ0 → 2 ∈ ℕ0 ) |
34 |
22 33
|
nn0expcld |
⊢ ( 𝐵 ∈ ℕ0 → ( ( 𝐵 + 1 ) ↑ 2 ) ∈ ℕ0 ) |
35 |
34
|
nn0red |
⊢ ( 𝐵 ∈ ℕ0 → ( ( 𝐵 + 1 ) ↑ 2 ) ∈ ℝ ) |
36 |
23
|
sqge0d |
⊢ ( 𝐵 ∈ ℕ0 → 0 ≤ ( ( 𝐵 + 1 ) ↑ 2 ) ) |
37 |
35 36
|
jca |
⊢ ( 𝐵 ∈ ℕ0 → ( ( ( 𝐵 + 1 ) ↑ 2 ) ∈ ℝ ∧ 0 ≤ ( ( 𝐵 + 1 ) ↑ 2 ) ) ) |
38 |
|
sqrtlt |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( ( ( 𝐵 + 1 ) ↑ 2 ) ∈ ℝ ∧ 0 ≤ ( ( 𝐵 + 1 ) ↑ 2 ) ) ) → ( 𝐴 < ( ( 𝐵 + 1 ) ↑ 2 ) ↔ ( √ ‘ 𝐴 ) < ( √ ‘ ( ( 𝐵 + 1 ) ↑ 2 ) ) ) ) |
39 |
37 38
|
sylan2 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐵 ∈ ℕ0 ) → ( 𝐴 < ( ( 𝐵 + 1 ) ↑ 2 ) ↔ ( √ ‘ 𝐴 ) < ( √ ‘ ( ( 𝐵 + 1 ) ↑ 2 ) ) ) ) |
40 |
31 39
|
bitr4d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐵 ∈ ℕ0 ) → ( ( √ ‘ 𝐴 ) < ( 𝐵 + 1 ) ↔ 𝐴 < ( ( 𝐵 + 1 ) ↑ 2 ) ) ) |
41 |
21 40
|
anbi12d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐵 ∈ ℕ0 ) → ( ( 𝐵 ≤ ( √ ‘ 𝐴 ) ∧ ( √ ‘ 𝐴 ) < ( 𝐵 + 1 ) ) ↔ ( ( 𝐵 ↑ 2 ) ≤ 𝐴 ∧ 𝐴 < ( ( 𝐵 + 1 ) ↑ 2 ) ) ) ) |
42 |
4 41
|
bitrd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐵 ∈ ℕ0 ) → ( ( ⌊ ‘ ( √ ‘ 𝐴 ) ) = 𝐵 ↔ ( ( 𝐵 ↑ 2 ) ≤ 𝐴 ∧ 𝐴 < ( ( 𝐵 + 1 ) ↑ 2 ) ) ) ) |