Step |
Hyp |
Ref |
Expression |
1 |
|
ceim1l |
⊢ ( 𝐴 ∈ ℝ → ( - ( ⌊ ‘ - 𝐴 ) − 1 ) < 𝐴 ) |
2 |
1
|
adantr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ) → ( - ( ⌊ ‘ - 𝐴 ) − 1 ) < 𝐴 ) |
3 |
|
ceicl |
⊢ ( 𝐴 ∈ ℝ → - ( ⌊ ‘ - 𝐴 ) ∈ ℤ ) |
4 |
|
zre |
⊢ ( - ( ⌊ ‘ - 𝐴 ) ∈ ℤ → - ( ⌊ ‘ - 𝐴 ) ∈ ℝ ) |
5 |
|
peano2rem |
⊢ ( - ( ⌊ ‘ - 𝐴 ) ∈ ℝ → ( - ( ⌊ ‘ - 𝐴 ) − 1 ) ∈ ℝ ) |
6 |
3 4 5
|
3syl |
⊢ ( 𝐴 ∈ ℝ → ( - ( ⌊ ‘ - 𝐴 ) − 1 ) ∈ ℝ ) |
7 |
6
|
adantr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ) → ( - ( ⌊ ‘ - 𝐴 ) − 1 ) ∈ ℝ ) |
8 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ) → 𝐴 ∈ ℝ ) |
9 |
|
zre |
⊢ ( 𝐵 ∈ ℤ → 𝐵 ∈ ℝ ) |
10 |
9
|
adantl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ) → 𝐵 ∈ ℝ ) |
11 |
|
ltletr |
⊢ ( ( ( - ( ⌊ ‘ - 𝐴 ) − 1 ) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( ( - ( ⌊ ‘ - 𝐴 ) − 1 ) < 𝐴 ∧ 𝐴 ≤ 𝐵 ) → ( - ( ⌊ ‘ - 𝐴 ) − 1 ) < 𝐵 ) ) |
12 |
7 8 10 11
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ) → ( ( ( - ( ⌊ ‘ - 𝐴 ) − 1 ) < 𝐴 ∧ 𝐴 ≤ 𝐵 ) → ( - ( ⌊ ‘ - 𝐴 ) − 1 ) < 𝐵 ) ) |
13 |
2 12
|
mpand |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 ≤ 𝐵 → ( - ( ⌊ ‘ - 𝐴 ) − 1 ) < 𝐵 ) ) |
14 |
|
zlem1lt |
⊢ ( ( - ( ⌊ ‘ - 𝐴 ) ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( - ( ⌊ ‘ - 𝐴 ) ≤ 𝐵 ↔ ( - ( ⌊ ‘ - 𝐴 ) − 1 ) < 𝐵 ) ) |
15 |
3 14
|
sylan |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ) → ( - ( ⌊ ‘ - 𝐴 ) ≤ 𝐵 ↔ ( - ( ⌊ ‘ - 𝐴 ) − 1 ) < 𝐵 ) ) |
16 |
13 15
|
sylibrd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 ≤ 𝐵 → - ( ⌊ ‘ - 𝐴 ) ≤ 𝐵 ) ) |
17 |
16
|
3impia |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵 ) → - ( ⌊ ‘ - 𝐴 ) ≤ 𝐵 ) |