Step |
Hyp |
Ref |
Expression |
1 |
|
neg0 |
⊢ - 0 = 0 |
2 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ) ∧ 𝑁 = 0 ) → 𝑁 = 0 ) |
3 |
2
|
negeqd |
⊢ ( ( ( 𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ) ∧ 𝑁 = 0 ) → - 𝑁 = - 0 ) |
4 |
1 3 2
|
3eqtr4a |
⊢ ( ( ( 𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ) ∧ 𝑁 = 0 ) → - 𝑁 = 𝑁 ) |
5 |
4
|
oveq2d |
⊢ ( ( ( 𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ) ∧ 𝑁 = 0 ) → ( 𝐴 /L - 𝑁 ) = ( 𝐴 /L 𝑁 ) ) |
6 |
|
nn0z |
⊢ ( 𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ ) |
7 |
|
lgsneg |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → ( 𝐴 /L - 𝑁 ) = ( if ( 𝐴 < 0 , - 1 , 1 ) · ( 𝐴 /L 𝑁 ) ) ) |
8 |
6 7
|
syl3an1 |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → ( 𝐴 /L - 𝑁 ) = ( if ( 𝐴 < 0 , - 1 , 1 ) · ( 𝐴 /L 𝑁 ) ) ) |
9 |
|
nn0nlt0 |
⊢ ( 𝐴 ∈ ℕ0 → ¬ 𝐴 < 0 ) |
10 |
9
|
3ad2ant1 |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → ¬ 𝐴 < 0 ) |
11 |
10
|
iffalsed |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → if ( 𝐴 < 0 , - 1 , 1 ) = 1 ) |
12 |
11
|
oveq1d |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → ( if ( 𝐴 < 0 , - 1 , 1 ) · ( 𝐴 /L 𝑁 ) ) = ( 1 · ( 𝐴 /L 𝑁 ) ) ) |
13 |
6
|
3ad2ant1 |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → 𝐴 ∈ ℤ ) |
14 |
|
simp2 |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → 𝑁 ∈ ℤ ) |
15 |
|
lgscl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝐴 /L 𝑁 ) ∈ ℤ ) |
16 |
13 14 15
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → ( 𝐴 /L 𝑁 ) ∈ ℤ ) |
17 |
16
|
zcnd |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → ( 𝐴 /L 𝑁 ) ∈ ℂ ) |
18 |
17
|
mulid2d |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → ( 1 · ( 𝐴 /L 𝑁 ) ) = ( 𝐴 /L 𝑁 ) ) |
19 |
8 12 18
|
3eqtrd |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → ( 𝐴 /L - 𝑁 ) = ( 𝐴 /L 𝑁 ) ) |
20 |
19
|
3expa |
⊢ ( ( ( 𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ) ∧ 𝑁 ≠ 0 ) → ( 𝐴 /L - 𝑁 ) = ( 𝐴 /L 𝑁 ) ) |
21 |
5 20
|
pm2.61dane |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 /L - 𝑁 ) = ( 𝐴 /L 𝑁 ) ) |