Step |
Hyp |
Ref |
Expression |
1 |
|
dvdszrcl |
⊢ ( 𝑀 ∥ 𝑁 → ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) |
2 |
|
simpr |
⊢ ( ( 𝑀 ∥ 𝑁 ∧ 𝑁 ∥ 𝑀 ) → 𝑁 ∥ 𝑀 ) |
3 |
|
breq1 |
⊢ ( 𝑁 = 0 → ( 𝑁 ∥ 𝑀 ↔ 0 ∥ 𝑀 ) ) |
4 |
|
0dvds |
⊢ ( 𝑀 ∈ ℤ → ( 0 ∥ 𝑀 ↔ 𝑀 = 0 ) ) |
5 |
4
|
adantr |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 0 ∥ 𝑀 ↔ 𝑀 = 0 ) ) |
6 |
|
zcn |
⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ℂ ) |
7 |
6
|
abs00ad |
⊢ ( 𝑀 ∈ ℤ → ( ( abs ‘ 𝑀 ) = 0 ↔ 𝑀 = 0 ) ) |
8 |
7
|
bicomd |
⊢ ( 𝑀 ∈ ℤ → ( 𝑀 = 0 ↔ ( abs ‘ 𝑀 ) = 0 ) ) |
9 |
8
|
adantr |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 = 0 ↔ ( abs ‘ 𝑀 ) = 0 ) ) |
10 |
5 9
|
bitrd |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 0 ∥ 𝑀 ↔ ( abs ‘ 𝑀 ) = 0 ) ) |
11 |
3 10
|
sylan9bb |
⊢ ( ( 𝑁 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → ( 𝑁 ∥ 𝑀 ↔ ( abs ‘ 𝑀 ) = 0 ) ) |
12 |
|
fveq2 |
⊢ ( 𝑁 = 0 → ( abs ‘ 𝑁 ) = ( abs ‘ 0 ) ) |
13 |
|
abs0 |
⊢ ( abs ‘ 0 ) = 0 |
14 |
12 13
|
eqtrdi |
⊢ ( 𝑁 = 0 → ( abs ‘ 𝑁 ) = 0 ) |
15 |
14
|
adantr |
⊢ ( ( 𝑁 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → ( abs ‘ 𝑁 ) = 0 ) |
16 |
15
|
eqeq2d |
⊢ ( ( 𝑁 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → ( ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ↔ ( abs ‘ 𝑀 ) = 0 ) ) |
17 |
11 16
|
bitr4d |
⊢ ( ( 𝑁 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → ( 𝑁 ∥ 𝑀 ↔ ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) ) |
18 |
2 17
|
syl5ib |
⊢ ( ( 𝑁 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → ( ( 𝑀 ∥ 𝑁 ∧ 𝑁 ∥ 𝑀 ) → ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) ) |
19 |
18
|
expd |
⊢ ( ( 𝑁 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → ( 𝑀 ∥ 𝑁 → ( 𝑁 ∥ 𝑀 → ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) ) ) |
20 |
|
simprl |
⊢ ( ( ¬ 𝑁 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → 𝑀 ∈ ℤ ) |
21 |
|
simpr |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → 𝑁 ∈ ℤ ) |
22 |
21
|
adantl |
⊢ ( ( ¬ 𝑁 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → 𝑁 ∈ ℤ ) |
23 |
|
neqne |
⊢ ( ¬ 𝑁 = 0 → 𝑁 ≠ 0 ) |
24 |
23
|
adantr |
⊢ ( ( ¬ 𝑁 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → 𝑁 ≠ 0 ) |
25 |
|
dvdsleabs2 |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → ( 𝑀 ∥ 𝑁 → ( abs ‘ 𝑀 ) ≤ ( abs ‘ 𝑁 ) ) ) |
26 |
20 22 24 25
|
syl3anc |
⊢ ( ( ¬ 𝑁 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → ( 𝑀 ∥ 𝑁 → ( abs ‘ 𝑀 ) ≤ ( abs ‘ 𝑁 ) ) ) |
27 |
|
simpr |
⊢ ( ( 𝑁 ∥ 𝑀 ∧ 𝑀 ∥ 𝑁 ) → 𝑀 ∥ 𝑁 ) |
28 |
|
breq1 |
⊢ ( 𝑀 = 0 → ( 𝑀 ∥ 𝑁 ↔ 0 ∥ 𝑁 ) ) |
29 |
|
0dvds |
⊢ ( 𝑁 ∈ ℤ → ( 0 ∥ 𝑁 ↔ 𝑁 = 0 ) ) |
30 |
|
zcn |
⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ℂ ) |
31 |
30
|
abs00ad |
⊢ ( 𝑁 ∈ ℤ → ( ( abs ‘ 𝑁 ) = 0 ↔ 𝑁 = 0 ) ) |
32 |
|
eqcom |
⊢ ( ( abs ‘ 𝑁 ) = 0 ↔ 0 = ( abs ‘ 𝑁 ) ) |
33 |
31 32
|
bitr3di |
⊢ ( 𝑁 ∈ ℤ → ( 𝑁 = 0 ↔ 0 = ( abs ‘ 𝑁 ) ) ) |
34 |
29 33
|
bitrd |
⊢ ( 𝑁 ∈ ℤ → ( 0 ∥ 𝑁 ↔ 0 = ( abs ‘ 𝑁 ) ) ) |
35 |
34
|
adantl |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 0 ∥ 𝑁 ↔ 0 = ( abs ‘ 𝑁 ) ) ) |
36 |
28 35
|
sylan9bb |
⊢ ( ( 𝑀 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → ( 𝑀 ∥ 𝑁 ↔ 0 = ( abs ‘ 𝑁 ) ) ) |
37 |
|
fveq2 |
⊢ ( 𝑀 = 0 → ( abs ‘ 𝑀 ) = ( abs ‘ 0 ) ) |
38 |
37 13
|
eqtrdi |
⊢ ( 𝑀 = 0 → ( abs ‘ 𝑀 ) = 0 ) |
39 |
38
|
adantr |
⊢ ( ( 𝑀 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → ( abs ‘ 𝑀 ) = 0 ) |
40 |
39
|
eqeq1d |
⊢ ( ( 𝑀 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → ( ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ↔ 0 = ( abs ‘ 𝑁 ) ) ) |
41 |
36 40
|
bitr4d |
⊢ ( ( 𝑀 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → ( 𝑀 ∥ 𝑁 ↔ ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) ) |
42 |
27 41
|
syl5ib |
⊢ ( ( 𝑀 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → ( ( 𝑁 ∥ 𝑀 ∧ 𝑀 ∥ 𝑁 ) → ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) ) |
43 |
42
|
a1dd |
⊢ ( ( 𝑀 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → ( ( 𝑁 ∥ 𝑀 ∧ 𝑀 ∥ 𝑁 ) → ( ( abs ‘ 𝑀 ) ≤ ( abs ‘ 𝑁 ) → ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) ) ) |
44 |
43
|
expcomd |
⊢ ( ( 𝑀 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → ( 𝑀 ∥ 𝑁 → ( 𝑁 ∥ 𝑀 → ( ( abs ‘ 𝑀 ) ≤ ( abs ‘ 𝑁 ) → ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) ) ) ) |
45 |
21
|
adantl |
⊢ ( ( ¬ 𝑀 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → 𝑁 ∈ ℤ ) |
46 |
|
simprl |
⊢ ( ( ¬ 𝑀 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → 𝑀 ∈ ℤ ) |
47 |
|
neqne |
⊢ ( ¬ 𝑀 = 0 → 𝑀 ≠ 0 ) |
48 |
47
|
adantr |
⊢ ( ( ¬ 𝑀 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → 𝑀 ≠ 0 ) |
49 |
|
dvdsleabs2 |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ≠ 0 ) → ( 𝑁 ∥ 𝑀 → ( abs ‘ 𝑁 ) ≤ ( abs ‘ 𝑀 ) ) ) |
50 |
45 46 48 49
|
syl3anc |
⊢ ( ( ¬ 𝑀 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → ( 𝑁 ∥ 𝑀 → ( abs ‘ 𝑁 ) ≤ ( abs ‘ 𝑀 ) ) ) |
51 |
|
eqcom |
⊢ ( ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ↔ ( abs ‘ 𝑁 ) = ( abs ‘ 𝑀 ) ) |
52 |
30
|
abscld |
⊢ ( 𝑁 ∈ ℤ → ( abs ‘ 𝑁 ) ∈ ℝ ) |
53 |
6
|
abscld |
⊢ ( 𝑀 ∈ ℤ → ( abs ‘ 𝑀 ) ∈ ℝ ) |
54 |
|
letri3 |
⊢ ( ( ( abs ‘ 𝑁 ) ∈ ℝ ∧ ( abs ‘ 𝑀 ) ∈ ℝ ) → ( ( abs ‘ 𝑁 ) = ( abs ‘ 𝑀 ) ↔ ( ( abs ‘ 𝑁 ) ≤ ( abs ‘ 𝑀 ) ∧ ( abs ‘ 𝑀 ) ≤ ( abs ‘ 𝑁 ) ) ) ) |
55 |
52 53 54
|
syl2anr |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( abs ‘ 𝑁 ) = ( abs ‘ 𝑀 ) ↔ ( ( abs ‘ 𝑁 ) ≤ ( abs ‘ 𝑀 ) ∧ ( abs ‘ 𝑀 ) ≤ ( abs ‘ 𝑁 ) ) ) ) |
56 |
51 55
|
syl5bb |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ↔ ( ( abs ‘ 𝑁 ) ≤ ( abs ‘ 𝑀 ) ∧ ( abs ‘ 𝑀 ) ≤ ( abs ‘ 𝑁 ) ) ) ) |
57 |
56
|
biimprd |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( ( abs ‘ 𝑁 ) ≤ ( abs ‘ 𝑀 ) ∧ ( abs ‘ 𝑀 ) ≤ ( abs ‘ 𝑁 ) ) → ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) ) |
58 |
57
|
expd |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( abs ‘ 𝑁 ) ≤ ( abs ‘ 𝑀 ) → ( ( abs ‘ 𝑀 ) ≤ ( abs ‘ 𝑁 ) → ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) ) ) |
59 |
58
|
adantl |
⊢ ( ( ¬ 𝑀 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → ( ( abs ‘ 𝑁 ) ≤ ( abs ‘ 𝑀 ) → ( ( abs ‘ 𝑀 ) ≤ ( abs ‘ 𝑁 ) → ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) ) ) |
60 |
50 59
|
syld |
⊢ ( ( ¬ 𝑀 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → ( 𝑁 ∥ 𝑀 → ( ( abs ‘ 𝑀 ) ≤ ( abs ‘ 𝑁 ) → ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) ) ) |
61 |
60
|
a1d |
⊢ ( ( ¬ 𝑀 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → ( 𝑀 ∥ 𝑁 → ( 𝑁 ∥ 𝑀 → ( ( abs ‘ 𝑀 ) ≤ ( abs ‘ 𝑁 ) → ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) ) ) ) |
62 |
44 61
|
pm2.61ian |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ 𝑁 → ( 𝑁 ∥ 𝑀 → ( ( abs ‘ 𝑀 ) ≤ ( abs ‘ 𝑁 ) → ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) ) ) ) |
63 |
62
|
com34 |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ 𝑁 → ( ( abs ‘ 𝑀 ) ≤ ( abs ‘ 𝑁 ) → ( 𝑁 ∥ 𝑀 → ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) ) ) ) |
64 |
63
|
adantl |
⊢ ( ( ¬ 𝑁 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → ( 𝑀 ∥ 𝑁 → ( ( abs ‘ 𝑀 ) ≤ ( abs ‘ 𝑁 ) → ( 𝑁 ∥ 𝑀 → ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) ) ) ) |
65 |
26 64
|
mpdd |
⊢ ( ( ¬ 𝑁 = 0 ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → ( 𝑀 ∥ 𝑁 → ( 𝑁 ∥ 𝑀 → ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) ) ) |
66 |
19 65
|
pm2.61ian |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ 𝑁 → ( 𝑁 ∥ 𝑀 → ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) ) ) |
67 |
1 66
|
mpcom |
⊢ ( 𝑀 ∥ 𝑁 → ( 𝑁 ∥ 𝑀 → ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) ) |
68 |
67
|
imp |
⊢ ( ( 𝑀 ∥ 𝑁 ∧ 𝑁 ∥ 𝑀 ) → ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) |