Step |
Hyp |
Ref |
Expression |
1 |
|
eldif |
⊢ ( ( 𝐴 / 𝐵 ) ∈ ( ℝ ∖ ℚ ) ↔ ( ( 𝐴 / 𝐵 ) ∈ ℝ ∧ ¬ ( 𝐴 / 𝐵 ) ∈ ℚ ) ) |
2 |
|
modval |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 mod 𝐵 ) = ( 𝐴 − ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) ) ) |
3 |
2
|
eqeq1d |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 mod 𝐵 ) = 0 ↔ ( 𝐴 − ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) ) = 0 ) ) |
4 |
|
recn |
⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ ) |
5 |
4
|
adantr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → 𝐴 ∈ ℂ ) |
6 |
|
rpre |
⊢ ( 𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ ) |
7 |
6
|
adantl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → 𝐵 ∈ ℝ ) |
8 |
|
refldivcl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ∈ ℝ ) |
9 |
7 8
|
remulcld |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) ∈ ℝ ) |
10 |
9
|
recnd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) ∈ ℂ ) |
11 |
5 10
|
subeq0ad |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 − ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) ) = 0 ↔ 𝐴 = ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) ) ) |
12 |
|
rerpdivcl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 / 𝐵 ) ∈ ℝ ) |
13 |
|
reflcl |
⊢ ( ( 𝐴 / 𝐵 ) ∈ ℝ → ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ∈ ℝ ) |
14 |
13
|
recnd |
⊢ ( ( 𝐴 / 𝐵 ) ∈ ℝ → ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ∈ ℂ ) |
15 |
12 14
|
syl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ∈ ℂ ) |
16 |
|
rpcnne0 |
⊢ ( 𝐵 ∈ ℝ+ → ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) |
17 |
16
|
adantl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) |
18 |
|
divmul2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐴 / 𝐵 ) = ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ↔ 𝐴 = ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) ) ) |
19 |
5 15 17 18
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 / 𝐵 ) = ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ↔ 𝐴 = ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) ) ) |
20 |
|
eqcom |
⊢ ( ( 𝐴 / 𝐵 ) = ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ↔ ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) = ( 𝐴 / 𝐵 ) ) |
21 |
19 20
|
bitr3di |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 = ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) ↔ ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) = ( 𝐴 / 𝐵 ) ) ) |
22 |
3 11 21
|
3bitrd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 mod 𝐵 ) = 0 ↔ ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) = ( 𝐴 / 𝐵 ) ) ) |
23 |
|
flidz |
⊢ ( ( 𝐴 / 𝐵 ) ∈ ℝ → ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) = ( 𝐴 / 𝐵 ) ↔ ( 𝐴 / 𝐵 ) ∈ ℤ ) ) |
24 |
|
zq |
⊢ ( ( 𝐴 / 𝐵 ) ∈ ℤ → ( 𝐴 / 𝐵 ) ∈ ℚ ) |
25 |
23 24
|
syl6bi |
⊢ ( ( 𝐴 / 𝐵 ) ∈ ℝ → ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) = ( 𝐴 / 𝐵 ) → ( 𝐴 / 𝐵 ) ∈ ℚ ) ) |
26 |
12 25
|
syl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) = ( 𝐴 / 𝐵 ) → ( 𝐴 / 𝐵 ) ∈ ℚ ) ) |
27 |
22 26
|
sylbid |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 mod 𝐵 ) = 0 → ( 𝐴 / 𝐵 ) ∈ ℚ ) ) |
28 |
27
|
necon3bd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ¬ ( 𝐴 / 𝐵 ) ∈ ℚ → ( 𝐴 mod 𝐵 ) ≠ 0 ) ) |
29 |
28
|
adantld |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( ( 𝐴 / 𝐵 ) ∈ ℝ ∧ ¬ ( 𝐴 / 𝐵 ) ∈ ℚ ) → ( 𝐴 mod 𝐵 ) ≠ 0 ) ) |
30 |
1 29
|
syl5bi |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 / 𝐵 ) ∈ ( ℝ ∖ ℚ ) → ( 𝐴 mod 𝐵 ) ≠ 0 ) ) |
31 |
30
|
3impia |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ∧ ( 𝐴 / 𝐵 ) ∈ ( ℝ ∖ ℚ ) ) → ( 𝐴 mod 𝐵 ) ≠ 0 ) |