Step |
Hyp |
Ref |
Expression |
1 |
|
modnegd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
modnegd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
modnegd.3 |
⊢ ( 𝜑 → 𝐶 ∈ ℝ+ ) |
4 |
|
modnegd.4 |
⊢ ( 𝜑 → ( 𝐴 mod 𝐶 ) = ( 𝐵 mod 𝐶 ) ) |
5 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
6 |
5
|
znegcld |
⊢ ( 𝜑 → - 1 ∈ ℤ ) |
7 |
|
modmul1 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( - 1 ∈ ℤ ∧ 𝐶 ∈ ℝ+ ) ∧ ( 𝐴 mod 𝐶 ) = ( 𝐵 mod 𝐶 ) ) → ( ( 𝐴 · - 1 ) mod 𝐶 ) = ( ( 𝐵 · - 1 ) mod 𝐶 ) ) |
8 |
1 2 6 3 4 7
|
syl221anc |
⊢ ( 𝜑 → ( ( 𝐴 · - 1 ) mod 𝐶 ) = ( ( 𝐵 · - 1 ) mod 𝐶 ) ) |
9 |
1
|
recnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
10 |
|
1cnd |
⊢ ( 𝜑 → 1 ∈ ℂ ) |
11 |
10
|
negcld |
⊢ ( 𝜑 → - 1 ∈ ℂ ) |
12 |
9 11
|
mulcomd |
⊢ ( 𝜑 → ( 𝐴 · - 1 ) = ( - 1 · 𝐴 ) ) |
13 |
9
|
mulm1d |
⊢ ( 𝜑 → ( - 1 · 𝐴 ) = - 𝐴 ) |
14 |
12 13
|
eqtrd |
⊢ ( 𝜑 → ( 𝐴 · - 1 ) = - 𝐴 ) |
15 |
14
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝐴 · - 1 ) mod 𝐶 ) = ( - 𝐴 mod 𝐶 ) ) |
16 |
2
|
recnd |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
17 |
16 11
|
mulcomd |
⊢ ( 𝜑 → ( 𝐵 · - 1 ) = ( - 1 · 𝐵 ) ) |
18 |
16
|
mulm1d |
⊢ ( 𝜑 → ( - 1 · 𝐵 ) = - 𝐵 ) |
19 |
17 18
|
eqtrd |
⊢ ( 𝜑 → ( 𝐵 · - 1 ) = - 𝐵 ) |
20 |
19
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝐵 · - 1 ) mod 𝐶 ) = ( - 𝐵 mod 𝐶 ) ) |
21 |
8 15 20
|
3eqtr3d |
⊢ ( 𝜑 → ( - 𝐴 mod 𝐶 ) = ( - 𝐵 mod 𝐶 ) ) |