Step |
Hyp |
Ref |
Expression |
1 |
|
zre |
⊢ ( 𝐴 ∈ ℤ → 𝐴 ∈ ℝ ) |
2 |
|
flle |
⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ≤ 𝐴 ) |
3 |
1 2
|
syl |
⊢ ( 𝐴 ∈ ℤ → ( ⌊ ‘ 𝐴 ) ≤ 𝐴 ) |
4 |
1
|
leidd |
⊢ ( 𝐴 ∈ ℤ → 𝐴 ≤ 𝐴 ) |
5 |
|
flge |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ∈ ℤ ) → ( 𝐴 ≤ 𝐴 ↔ 𝐴 ≤ ( ⌊ ‘ 𝐴 ) ) ) |
6 |
1 5
|
mpancom |
⊢ ( 𝐴 ∈ ℤ → ( 𝐴 ≤ 𝐴 ↔ 𝐴 ≤ ( ⌊ ‘ 𝐴 ) ) ) |
7 |
4 6
|
mpbid |
⊢ ( 𝐴 ∈ ℤ → 𝐴 ≤ ( ⌊ ‘ 𝐴 ) ) |
8 |
|
reflcl |
⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℝ ) |
9 |
1 8
|
syl |
⊢ ( 𝐴 ∈ ℤ → ( ⌊ ‘ 𝐴 ) ∈ ℝ ) |
10 |
9 1
|
letri3d |
⊢ ( 𝐴 ∈ ℤ → ( ( ⌊ ‘ 𝐴 ) = 𝐴 ↔ ( ( ⌊ ‘ 𝐴 ) ≤ 𝐴 ∧ 𝐴 ≤ ( ⌊ ‘ 𝐴 ) ) ) ) |
11 |
3 7 10
|
mpbir2and |
⊢ ( 𝐴 ∈ ℤ → ( ⌊ ‘ 𝐴 ) = 𝐴 ) |