Step |
Hyp |
Ref |
Expression |
1 |
|
flltp1 |
⊢ ( 𝐴 ∈ ℝ → 𝐴 < ( ( ⌊ ‘ 𝐴 ) + 1 ) ) |
2 |
1
|
ad3antrrr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) ∧ 𝐵 < 𝐴 ) → 𝐴 < ( ( ⌊ ‘ 𝐴 ) + 1 ) ) |
3 |
|
flval |
⊢ ( 𝐵 ∈ ℝ → ( ⌊ ‘ 𝐵 ) = ( ℩ 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐵 ∧ 𝐵 < ( 𝑥 + 1 ) ) ) ) |
4 |
3
|
ad3antlr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) ∧ 𝐵 < 𝐴 ) → ( ⌊ ‘ 𝐵 ) = ( ℩ 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐵 ∧ 𝐵 < ( 𝑥 + 1 ) ) ) ) |
5 |
|
simplr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) ∧ 𝐵 < 𝐴 ) → ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) |
6 |
1
|
adantr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → 𝐴 < ( ( ⌊ ‘ 𝐴 ) + 1 ) ) |
7 |
|
reflcl |
⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℝ ) |
8 |
|
peano2re |
⊢ ( ( ⌊ ‘ 𝐴 ) ∈ ℝ → ( ( ⌊ ‘ 𝐴 ) + 1 ) ∈ ℝ ) |
9 |
7 8
|
syl |
⊢ ( 𝐴 ∈ ℝ → ( ( ⌊ ‘ 𝐴 ) + 1 ) ∈ ℝ ) |
10 |
9
|
adantl |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( ( ⌊ ‘ 𝐴 ) + 1 ) ∈ ℝ ) |
11 |
|
lttr |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ ( ( ⌊ ‘ 𝐴 ) + 1 ) ∈ ℝ ) → ( ( 𝐵 < 𝐴 ∧ 𝐴 < ( ( ⌊ ‘ 𝐴 ) + 1 ) ) → 𝐵 < ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ) |
12 |
10 11
|
mpd3an3 |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( ( 𝐵 < 𝐴 ∧ 𝐴 < ( ( ⌊ ‘ 𝐴 ) + 1 ) ) → 𝐵 < ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ) |
13 |
12
|
ancoms |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐵 < 𝐴 ∧ 𝐴 < ( ( ⌊ ‘ 𝐴 ) + 1 ) ) → 𝐵 < ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ) |
14 |
6 13
|
mpan2d |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐵 < 𝐴 → 𝐵 < ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ) |
15 |
14
|
imp |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐵 < 𝐴 ) → 𝐵 < ( ( ⌊ ‘ 𝐴 ) + 1 ) ) |
16 |
15
|
adantlr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) ∧ 𝐵 < 𝐴 ) → 𝐵 < ( ( ⌊ ‘ 𝐴 ) + 1 ) ) |
17 |
|
flcl |
⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℤ ) |
18 |
|
rebtwnz |
⊢ ( 𝐵 ∈ ℝ → ∃! 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐵 ∧ 𝐵 < ( 𝑥 + 1 ) ) ) |
19 |
|
breq1 |
⊢ ( 𝑥 = ( ⌊ ‘ 𝐴 ) → ( 𝑥 ≤ 𝐵 ↔ ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) ) |
20 |
|
oveq1 |
⊢ ( 𝑥 = ( ⌊ ‘ 𝐴 ) → ( 𝑥 + 1 ) = ( ( ⌊ ‘ 𝐴 ) + 1 ) ) |
21 |
20
|
breq2d |
⊢ ( 𝑥 = ( ⌊ ‘ 𝐴 ) → ( 𝐵 < ( 𝑥 + 1 ) ↔ 𝐵 < ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ) |
22 |
19 21
|
anbi12d |
⊢ ( 𝑥 = ( ⌊ ‘ 𝐴 ) → ( ( 𝑥 ≤ 𝐵 ∧ 𝐵 < ( 𝑥 + 1 ) ) ↔ ( ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ∧ 𝐵 < ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ) ) |
23 |
22
|
riota2 |
⊢ ( ( ( ⌊ ‘ 𝐴 ) ∈ ℤ ∧ ∃! 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐵 ∧ 𝐵 < ( 𝑥 + 1 ) ) ) → ( ( ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ∧ 𝐵 < ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ↔ ( ℩ 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐵 ∧ 𝐵 < ( 𝑥 + 1 ) ) ) = ( ⌊ ‘ 𝐴 ) ) ) |
24 |
17 18 23
|
syl2an |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ∧ 𝐵 < ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ↔ ( ℩ 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐵 ∧ 𝐵 < ( 𝑥 + 1 ) ) ) = ( ⌊ ‘ 𝐴 ) ) ) |
25 |
24
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) ∧ 𝐵 < 𝐴 ) → ( ( ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ∧ 𝐵 < ( ( ⌊ ‘ 𝐴 ) + 1 ) ) ↔ ( ℩ 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐵 ∧ 𝐵 < ( 𝑥 + 1 ) ) ) = ( ⌊ ‘ 𝐴 ) ) ) |
26 |
5 16 25
|
mpbi2and |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) ∧ 𝐵 < 𝐴 ) → ( ℩ 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐵 ∧ 𝐵 < ( 𝑥 + 1 ) ) ) = ( ⌊ ‘ 𝐴 ) ) |
27 |
4 26
|
eqtrd |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) ∧ 𝐵 < 𝐴 ) → ( ⌊ ‘ 𝐵 ) = ( ⌊ ‘ 𝐴 ) ) |
28 |
27
|
oveq1d |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) ∧ 𝐵 < 𝐴 ) → ( ( ⌊ ‘ 𝐵 ) + 1 ) = ( ( ⌊ ‘ 𝐴 ) + 1 ) ) |
29 |
2 28
|
breqtrrd |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) ∧ 𝐵 < 𝐴 ) → 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) |
30 |
29
|
ex |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) → ( 𝐵 < 𝐴 → 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) ) |
31 |
|
lenlt |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴 ) ) |
32 |
|
flltp1 |
⊢ ( 𝐵 ∈ ℝ → 𝐵 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) |
33 |
32
|
adantl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → 𝐵 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) |
34 |
|
reflcl |
⊢ ( 𝐵 ∈ ℝ → ( ⌊ ‘ 𝐵 ) ∈ ℝ ) |
35 |
|
peano2re |
⊢ ( ( ⌊ ‘ 𝐵 ) ∈ ℝ → ( ( ⌊ ‘ 𝐵 ) + 1 ) ∈ ℝ ) |
36 |
34 35
|
syl |
⊢ ( 𝐵 ∈ ℝ → ( ( ⌊ ‘ 𝐵 ) + 1 ) ∈ ℝ ) |
37 |
36
|
adantl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( ⌊ ‘ 𝐵 ) + 1 ) ∈ ℝ ) |
38 |
|
lelttr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( ( ⌊ ‘ 𝐵 ) + 1 ) ∈ ℝ ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) → 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) ) |
39 |
37 38
|
mpd3an3 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) → 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) ) |
40 |
33 39
|
mpan2d |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 → 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) ) |
41 |
31 40
|
sylbird |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ¬ 𝐵 < 𝐴 → 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) ) |
42 |
41
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) → ( ¬ 𝐵 < 𝐴 → 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) ) |
43 |
30 42
|
pm2.61d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) → 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) |
44 |
|
flval |
⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) = ( ℩ 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) ) ) |
45 |
44
|
ad3antrrr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) ∧ 𝐵 < 𝐴 ) → ( ⌊ ‘ 𝐴 ) = ( ℩ 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) ) ) |
46 |
34
|
ad2antlr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐵 < 𝐴 ) → ( ⌊ ‘ 𝐵 ) ∈ ℝ ) |
47 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐵 < 𝐴 ) → 𝐴 ∈ ℝ ) |
48 |
|
simplr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐵 < 𝐴 ) → 𝐵 ∈ ℝ ) |
49 |
|
flle |
⊢ ( 𝐵 ∈ ℝ → ( ⌊ ‘ 𝐵 ) ≤ 𝐵 ) |
50 |
49
|
ad2antlr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐵 < 𝐴 ) → ( ⌊ ‘ 𝐵 ) ≤ 𝐵 ) |
51 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐵 < 𝐴 ) → 𝐵 < 𝐴 ) |
52 |
46 48 47 50 51
|
lelttrd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐵 < 𝐴 ) → ( ⌊ ‘ 𝐵 ) < 𝐴 ) |
53 |
46 47 52
|
ltled |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐵 < 𝐴 ) → ( ⌊ ‘ 𝐵 ) ≤ 𝐴 ) |
54 |
53
|
adantlr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) ∧ 𝐵 < 𝐴 ) → ( ⌊ ‘ 𝐵 ) ≤ 𝐴 ) |
55 |
|
simplr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) ∧ 𝐵 < 𝐴 ) → 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) |
56 |
|
flcl |
⊢ ( 𝐵 ∈ ℝ → ( ⌊ ‘ 𝐵 ) ∈ ℤ ) |
57 |
|
rebtwnz |
⊢ ( 𝐴 ∈ ℝ → ∃! 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) ) |
58 |
|
breq1 |
⊢ ( 𝑥 = ( ⌊ ‘ 𝐵 ) → ( 𝑥 ≤ 𝐴 ↔ ( ⌊ ‘ 𝐵 ) ≤ 𝐴 ) ) |
59 |
|
oveq1 |
⊢ ( 𝑥 = ( ⌊ ‘ 𝐵 ) → ( 𝑥 + 1 ) = ( ( ⌊ ‘ 𝐵 ) + 1 ) ) |
60 |
59
|
breq2d |
⊢ ( 𝑥 = ( ⌊ ‘ 𝐵 ) → ( 𝐴 < ( 𝑥 + 1 ) ↔ 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) ) |
61 |
58 60
|
anbi12d |
⊢ ( 𝑥 = ( ⌊ ‘ 𝐵 ) → ( ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) ↔ ( ( ⌊ ‘ 𝐵 ) ≤ 𝐴 ∧ 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) ) ) |
62 |
61
|
riota2 |
⊢ ( ( ( ⌊ ‘ 𝐵 ) ∈ ℤ ∧ ∃! 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) ) → ( ( ( ⌊ ‘ 𝐵 ) ≤ 𝐴 ∧ 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) ↔ ( ℩ 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) ) = ( ⌊ ‘ 𝐵 ) ) ) |
63 |
56 57 62
|
syl2anr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( ( ⌊ ‘ 𝐵 ) ≤ 𝐴 ∧ 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) ↔ ( ℩ 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) ) = ( ⌊ ‘ 𝐵 ) ) ) |
64 |
63
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) ∧ 𝐵 < 𝐴 ) → ( ( ( ⌊ ‘ 𝐵 ) ≤ 𝐴 ∧ 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) ↔ ( ℩ 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) ) = ( ⌊ ‘ 𝐵 ) ) ) |
65 |
54 55 64
|
mpbi2and |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) ∧ 𝐵 < 𝐴 ) → ( ℩ 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) ) = ( ⌊ ‘ 𝐵 ) ) |
66 |
45 65
|
eqtrd |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) ∧ 𝐵 < 𝐴 ) → ( ⌊ ‘ 𝐴 ) = ( ⌊ ‘ 𝐵 ) ) |
67 |
49
|
ad3antlr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) ∧ 𝐵 < 𝐴 ) → ( ⌊ ‘ 𝐵 ) ≤ 𝐵 ) |
68 |
66 67
|
eqbrtrd |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) ∧ 𝐵 < 𝐴 ) → ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) |
69 |
68
|
ex |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) → ( 𝐵 < 𝐴 → ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) ) |
70 |
|
flle |
⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ≤ 𝐴 ) |
71 |
70
|
adantr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ⌊ ‘ 𝐴 ) ≤ 𝐴 ) |
72 |
7
|
adantr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ⌊ ‘ 𝐴 ) ∈ ℝ ) |
73 |
|
letr |
⊢ ( ( ( ⌊ ‘ 𝐴 ) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( ( ⌊ ‘ 𝐴 ) ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) → ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) ) |
74 |
73
|
3coml |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( ⌊ ‘ 𝐴 ) ∈ ℝ ) → ( ( ( ⌊ ‘ 𝐴 ) ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) → ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) ) |
75 |
72 74
|
mpd3an3 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( ( ⌊ ‘ 𝐴 ) ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) → ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) ) |
76 |
71 75
|
mpand |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 → ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) ) |
77 |
31 76
|
sylbird |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ¬ 𝐵 < 𝐴 → ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) ) |
78 |
77
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) → ( ¬ 𝐵 < 𝐴 → ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) ) |
79 |
69 78
|
pm2.61d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) → ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ) |
80 |
43 79
|
impbida |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ↔ 𝐴 < ( ( ⌊ ‘ 𝐵 ) + 1 ) ) ) |