Step |
Hyp |
Ref |
Expression |
1 |
|
etransclem9.k |
⊢ ( 𝜑 → 𝐾 ∈ ℤ ) |
2 |
|
etransclem9.kn0 |
⊢ ( 𝜑 → 𝐾 ≠ 0 ) |
3 |
|
etransclem9.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
4 |
|
etransclem9.n |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
5 |
|
etransclem9.km |
⊢ ( 𝜑 → ¬ 𝐾 ∥ 𝑀 ) |
6 |
|
etransclem9.kn |
⊢ ( 𝜑 → 𝐾 ∥ 𝑁 ) |
7 |
|
dvdsval2 |
⊢ ( ( 𝐾 ∈ ℤ ∧ 𝐾 ≠ 0 ∧ 𝑀 ∈ ℤ ) → ( 𝐾 ∥ 𝑀 ↔ ( 𝑀 / 𝐾 ) ∈ ℤ ) ) |
8 |
1 2 3 7
|
syl3anc |
⊢ ( 𝜑 → ( 𝐾 ∥ 𝑀 ↔ ( 𝑀 / 𝐾 ) ∈ ℤ ) ) |
9 |
5 8
|
mtbid |
⊢ ( 𝜑 → ¬ ( 𝑀 / 𝐾 ) ∈ ℤ ) |
10 |
|
df-neg |
⊢ - 𝑁 = ( 0 − 𝑁 ) |
11 |
10
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑀 + 𝑁 ) = 0 ) → - 𝑁 = ( 0 − 𝑁 ) ) |
12 |
|
oveq1 |
⊢ ( ( 𝑀 + 𝑁 ) = 0 → ( ( 𝑀 + 𝑁 ) − 𝑁 ) = ( 0 − 𝑁 ) ) |
13 |
12
|
eqcomd |
⊢ ( ( 𝑀 + 𝑁 ) = 0 → ( 0 − 𝑁 ) = ( ( 𝑀 + 𝑁 ) − 𝑁 ) ) |
14 |
13
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑀 + 𝑁 ) = 0 ) → ( 0 − 𝑁 ) = ( ( 𝑀 + 𝑁 ) − 𝑁 ) ) |
15 |
3
|
zcnd |
⊢ ( 𝜑 → 𝑀 ∈ ℂ ) |
16 |
4
|
zcnd |
⊢ ( 𝜑 → 𝑁 ∈ ℂ ) |
17 |
15 16
|
pncand |
⊢ ( 𝜑 → ( ( 𝑀 + 𝑁 ) − 𝑁 ) = 𝑀 ) |
18 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 + 𝑁 ) = 0 ) → ( ( 𝑀 + 𝑁 ) − 𝑁 ) = 𝑀 ) |
19 |
11 14 18
|
3eqtrrd |
⊢ ( ( 𝜑 ∧ ( 𝑀 + 𝑁 ) = 0 ) → 𝑀 = - 𝑁 ) |
20 |
19
|
oveq1d |
⊢ ( ( 𝜑 ∧ ( 𝑀 + 𝑁 ) = 0 ) → ( 𝑀 / 𝐾 ) = ( - 𝑁 / 𝐾 ) ) |
21 |
|
dvdsnegb |
⊢ ( ( 𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝐾 ∥ 𝑁 ↔ 𝐾 ∥ - 𝑁 ) ) |
22 |
1 4 21
|
syl2anc |
⊢ ( 𝜑 → ( 𝐾 ∥ 𝑁 ↔ 𝐾 ∥ - 𝑁 ) ) |
23 |
6 22
|
mpbid |
⊢ ( 𝜑 → 𝐾 ∥ - 𝑁 ) |
24 |
4
|
znegcld |
⊢ ( 𝜑 → - 𝑁 ∈ ℤ ) |
25 |
|
dvdsval2 |
⊢ ( ( 𝐾 ∈ ℤ ∧ 𝐾 ≠ 0 ∧ - 𝑁 ∈ ℤ ) → ( 𝐾 ∥ - 𝑁 ↔ ( - 𝑁 / 𝐾 ) ∈ ℤ ) ) |
26 |
1 2 24 25
|
syl3anc |
⊢ ( 𝜑 → ( 𝐾 ∥ - 𝑁 ↔ ( - 𝑁 / 𝐾 ) ∈ ℤ ) ) |
27 |
23 26
|
mpbid |
⊢ ( 𝜑 → ( - 𝑁 / 𝐾 ) ∈ ℤ ) |
28 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 + 𝑁 ) = 0 ) → ( - 𝑁 / 𝐾 ) ∈ ℤ ) |
29 |
20 28
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( 𝑀 + 𝑁 ) = 0 ) → ( 𝑀 / 𝐾 ) ∈ ℤ ) |
30 |
9 29
|
mtand |
⊢ ( 𝜑 → ¬ ( 𝑀 + 𝑁 ) = 0 ) |
31 |
30
|
neqned |
⊢ ( 𝜑 → ( 𝑀 + 𝑁 ) ≠ 0 ) |