Step |
Hyp |
Ref |
Expression |
1 |
|
5nn0 |
⊢ 5 ∈ ℕ0 |
2 |
|
flsqrt |
⊢ ( ( ( 𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ) ∧ 5 ∈ ℕ0 ) → ( ( ⌊ ‘ ( √ ‘ 𝑋 ) ) = 5 ↔ ( ( 5 ↑ 2 ) ≤ 𝑋 ∧ 𝑋 < ( ( 5 + 1 ) ↑ 2 ) ) ) ) |
3 |
1 2
|
mpan2 |
⊢ ( ( 𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ) → ( ( ⌊ ‘ ( √ ‘ 𝑋 ) ) = 5 ↔ ( ( 5 ↑ 2 ) ≤ 𝑋 ∧ 𝑋 < ( ( 5 + 1 ) ↑ 2 ) ) ) ) |
4 |
|
5cn |
⊢ 5 ∈ ℂ |
5 |
4
|
sqvali |
⊢ ( 5 ↑ 2 ) = ( 5 · 5 ) |
6 |
|
5t5e25 |
⊢ ( 5 · 5 ) = ; 2 5 |
7 |
5 6
|
eqtri |
⊢ ( 5 ↑ 2 ) = ; 2 5 |
8 |
7
|
breq1i |
⊢ ( ( 5 ↑ 2 ) ≤ 𝑋 ↔ ; 2 5 ≤ 𝑋 ) |
9 |
8
|
a1i |
⊢ ( ( 𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ) → ( ( 5 ↑ 2 ) ≤ 𝑋 ↔ ; 2 5 ≤ 𝑋 ) ) |
10 |
|
5p1e6 |
⊢ ( 5 + 1 ) = 6 |
11 |
10
|
oveq1i |
⊢ ( ( 5 + 1 ) ↑ 2 ) = ( 6 ↑ 2 ) |
12 |
|
6cn |
⊢ 6 ∈ ℂ |
13 |
12
|
sqvali |
⊢ ( 6 ↑ 2 ) = ( 6 · 6 ) |
14 |
|
6t6e36 |
⊢ ( 6 · 6 ) = ; 3 6 |
15 |
13 14
|
eqtri |
⊢ ( 6 ↑ 2 ) = ; 3 6 |
16 |
11 15
|
eqtri |
⊢ ( ( 5 + 1 ) ↑ 2 ) = ; 3 6 |
17 |
16
|
breq2i |
⊢ ( 𝑋 < ( ( 5 + 1 ) ↑ 2 ) ↔ 𝑋 < ; 3 6 ) |
18 |
17
|
a1i |
⊢ ( ( 𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ) → ( 𝑋 < ( ( 5 + 1 ) ↑ 2 ) ↔ 𝑋 < ; 3 6 ) ) |
19 |
9 18
|
anbi12d |
⊢ ( ( 𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ) → ( ( ( 5 ↑ 2 ) ≤ 𝑋 ∧ 𝑋 < ( ( 5 + 1 ) ↑ 2 ) ) ↔ ( ; 2 5 ≤ 𝑋 ∧ 𝑋 < ; 3 6 ) ) ) |
20 |
3 19
|
bitr2d |
⊢ ( ( 𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ) → ( ( ; 2 5 ≤ 𝑋 ∧ 𝑋 < ; 3 6 ) ↔ ( ⌊ ‘ ( √ ‘ 𝑋 ) ) = 5 ) ) |