Step |
Hyp |
Ref |
Expression |
1 |
|
dvdsadd2b |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ ( 𝐶 ∈ ℤ ∧ 𝐴 ∥ 𝐶 ) ) → ( 𝐴 ∥ 𝐵 ↔ 𝐴 ∥ ( 𝐶 + 𝐵 ) ) ) |
2 |
1
|
a1d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ ( 𝐶 ∈ ℤ ∧ 𝐴 ∥ 𝐶 ) ) → ( 𝐵 ∈ ℝ → ( 𝐴 ∥ 𝐵 ↔ 𝐴 ∥ ( 𝐶 + 𝐵 ) ) ) ) |
3 |
2
|
3exp |
⊢ ( 𝐴 ∈ ℤ → ( 𝐵 ∈ ℤ → ( ( 𝐶 ∈ ℤ ∧ 𝐴 ∥ 𝐶 ) → ( 𝐵 ∈ ℝ → ( 𝐴 ∥ 𝐵 ↔ 𝐴 ∥ ( 𝐶 + 𝐵 ) ) ) ) ) ) |
4 |
3
|
com24 |
⊢ ( 𝐴 ∈ ℤ → ( 𝐵 ∈ ℝ → ( ( 𝐶 ∈ ℤ ∧ 𝐴 ∥ 𝐶 ) → ( 𝐵 ∈ ℤ → ( 𝐴 ∥ 𝐵 ↔ 𝐴 ∥ ( 𝐶 + 𝐵 ) ) ) ) ) ) |
5 |
4
|
3imp |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℤ ∧ 𝐴 ∥ 𝐶 ) ) → ( 𝐵 ∈ ℤ → ( 𝐴 ∥ 𝐵 ↔ 𝐴 ∥ ( 𝐶 + 𝐵 ) ) ) ) |
6 |
5
|
com12 |
⊢ ( 𝐵 ∈ ℤ → ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℤ ∧ 𝐴 ∥ 𝐶 ) ) → ( 𝐴 ∥ 𝐵 ↔ 𝐴 ∥ ( 𝐶 + 𝐵 ) ) ) ) |
7 |
|
dvdszrcl |
⊢ ( 𝐴 ∥ 𝐵 → ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ) |
8 |
|
pm2.24 |
⊢ ( 𝐵 ∈ ℤ → ( ¬ 𝐵 ∈ ℤ → 𝐴 ∥ ( 𝐶 + 𝐵 ) ) ) |
9 |
7 8
|
simpl2im |
⊢ ( 𝐴 ∥ 𝐵 → ( ¬ 𝐵 ∈ ℤ → 𝐴 ∥ ( 𝐶 + 𝐵 ) ) ) |
10 |
9
|
com12 |
⊢ ( ¬ 𝐵 ∈ ℤ → ( 𝐴 ∥ 𝐵 → 𝐴 ∥ ( 𝐶 + 𝐵 ) ) ) |
11 |
10
|
adantr |
⊢ ( ( ¬ 𝐵 ∈ ℤ ∧ ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℤ ∧ 𝐴 ∥ 𝐶 ) ) ) → ( 𝐴 ∥ 𝐵 → 𝐴 ∥ ( 𝐶 + 𝐵 ) ) ) |
12 |
|
dvdszrcl |
⊢ ( 𝐴 ∥ ( 𝐶 + 𝐵 ) → ( 𝐴 ∈ ℤ ∧ ( 𝐶 + 𝐵 ) ∈ ℤ ) ) |
13 |
|
zcn |
⊢ ( 𝐶 ∈ ℤ → 𝐶 ∈ ℂ ) |
14 |
13
|
adantr |
⊢ ( ( 𝐶 ∈ ℤ ∧ ( 𝐵 ∈ ℝ ∧ ¬ 𝐵 ∈ ℤ ) ) → 𝐶 ∈ ℂ ) |
15 |
|
recn |
⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ ) |
16 |
15
|
ad2antrl |
⊢ ( ( 𝐶 ∈ ℤ ∧ ( 𝐵 ∈ ℝ ∧ ¬ 𝐵 ∈ ℤ ) ) → 𝐵 ∈ ℂ ) |
17 |
14 16
|
addcomd |
⊢ ( ( 𝐶 ∈ ℤ ∧ ( 𝐵 ∈ ℝ ∧ ¬ 𝐵 ∈ ℤ ) ) → ( 𝐶 + 𝐵 ) = ( 𝐵 + 𝐶 ) ) |
18 |
|
eldif |
⊢ ( 𝐵 ∈ ( ℝ ∖ ℤ ) ↔ ( 𝐵 ∈ ℝ ∧ ¬ 𝐵 ∈ ℤ ) ) |
19 |
|
nzadd |
⊢ ( ( 𝐵 ∈ ( ℝ ∖ ℤ ) ∧ 𝐶 ∈ ℤ ) → ( 𝐵 + 𝐶 ) ∈ ( ℝ ∖ ℤ ) ) |
20 |
19
|
eldifbd |
⊢ ( ( 𝐵 ∈ ( ℝ ∖ ℤ ) ∧ 𝐶 ∈ ℤ ) → ¬ ( 𝐵 + 𝐶 ) ∈ ℤ ) |
21 |
20
|
expcom |
⊢ ( 𝐶 ∈ ℤ → ( 𝐵 ∈ ( ℝ ∖ ℤ ) → ¬ ( 𝐵 + 𝐶 ) ∈ ℤ ) ) |
22 |
18 21
|
syl5bir |
⊢ ( 𝐶 ∈ ℤ → ( ( 𝐵 ∈ ℝ ∧ ¬ 𝐵 ∈ ℤ ) → ¬ ( 𝐵 + 𝐶 ) ∈ ℤ ) ) |
23 |
22
|
imp |
⊢ ( ( 𝐶 ∈ ℤ ∧ ( 𝐵 ∈ ℝ ∧ ¬ 𝐵 ∈ ℤ ) ) → ¬ ( 𝐵 + 𝐶 ) ∈ ℤ ) |
24 |
17 23
|
eqneltrd |
⊢ ( ( 𝐶 ∈ ℤ ∧ ( 𝐵 ∈ ℝ ∧ ¬ 𝐵 ∈ ℤ ) ) → ¬ ( 𝐶 + 𝐵 ) ∈ ℤ ) |
25 |
24
|
exp32 |
⊢ ( 𝐶 ∈ ℤ → ( 𝐵 ∈ ℝ → ( ¬ 𝐵 ∈ ℤ → ¬ ( 𝐶 + 𝐵 ) ∈ ℤ ) ) ) |
26 |
|
pm2.21 |
⊢ ( ¬ ( 𝐶 + 𝐵 ) ∈ ℤ → ( ( 𝐶 + 𝐵 ) ∈ ℤ → 𝐴 ∥ 𝐵 ) ) |
27 |
25 26
|
syl8 |
⊢ ( 𝐶 ∈ ℤ → ( 𝐵 ∈ ℝ → ( ¬ 𝐵 ∈ ℤ → ( ( 𝐶 + 𝐵 ) ∈ ℤ → 𝐴 ∥ 𝐵 ) ) ) ) |
28 |
27
|
adantr |
⊢ ( ( 𝐶 ∈ ℤ ∧ 𝐴 ∥ 𝐶 ) → ( 𝐵 ∈ ℝ → ( ¬ 𝐵 ∈ ℤ → ( ( 𝐶 + 𝐵 ) ∈ ℤ → 𝐴 ∥ 𝐵 ) ) ) ) |
29 |
28
|
com12 |
⊢ ( 𝐵 ∈ ℝ → ( ( 𝐶 ∈ ℤ ∧ 𝐴 ∥ 𝐶 ) → ( ¬ 𝐵 ∈ ℤ → ( ( 𝐶 + 𝐵 ) ∈ ℤ → 𝐴 ∥ 𝐵 ) ) ) ) |
30 |
29
|
a1i |
⊢ ( 𝐴 ∈ ℤ → ( 𝐵 ∈ ℝ → ( ( 𝐶 ∈ ℤ ∧ 𝐴 ∥ 𝐶 ) → ( ¬ 𝐵 ∈ ℤ → ( ( 𝐶 + 𝐵 ) ∈ ℤ → 𝐴 ∥ 𝐵 ) ) ) ) ) |
31 |
30
|
3imp |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℤ ∧ 𝐴 ∥ 𝐶 ) ) → ( ¬ 𝐵 ∈ ℤ → ( ( 𝐶 + 𝐵 ) ∈ ℤ → 𝐴 ∥ 𝐵 ) ) ) |
32 |
31
|
impcom |
⊢ ( ( ¬ 𝐵 ∈ ℤ ∧ ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℤ ∧ 𝐴 ∥ 𝐶 ) ) ) → ( ( 𝐶 + 𝐵 ) ∈ ℤ → 𝐴 ∥ 𝐵 ) ) |
33 |
32
|
com12 |
⊢ ( ( 𝐶 + 𝐵 ) ∈ ℤ → ( ( ¬ 𝐵 ∈ ℤ ∧ ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℤ ∧ 𝐴 ∥ 𝐶 ) ) ) → 𝐴 ∥ 𝐵 ) ) |
34 |
12 33
|
simpl2im |
⊢ ( 𝐴 ∥ ( 𝐶 + 𝐵 ) → ( ( ¬ 𝐵 ∈ ℤ ∧ ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℤ ∧ 𝐴 ∥ 𝐶 ) ) ) → 𝐴 ∥ 𝐵 ) ) |
35 |
34
|
com12 |
⊢ ( ( ¬ 𝐵 ∈ ℤ ∧ ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℤ ∧ 𝐴 ∥ 𝐶 ) ) ) → ( 𝐴 ∥ ( 𝐶 + 𝐵 ) → 𝐴 ∥ 𝐵 ) ) |
36 |
11 35
|
impbid |
⊢ ( ( ¬ 𝐵 ∈ ℤ ∧ ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℤ ∧ 𝐴 ∥ 𝐶 ) ) ) → ( 𝐴 ∥ 𝐵 ↔ 𝐴 ∥ ( 𝐶 + 𝐵 ) ) ) |
37 |
36
|
ex |
⊢ ( ¬ 𝐵 ∈ ℤ → ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℤ ∧ 𝐴 ∥ 𝐶 ) ) → ( 𝐴 ∥ 𝐵 ↔ 𝐴 ∥ ( 𝐶 + 𝐵 ) ) ) ) |
38 |
6 37
|
pm2.61i |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℤ ∧ 𝐴 ∥ 𝐶 ) ) → ( 𝐴 ∥ 𝐵 ↔ 𝐴 ∥ ( 𝐶 + 𝐵 ) ) ) |