Step |
Hyp |
Ref |
Expression |
1 |
|
modcl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( 𝐴 mod 𝑀 ) ∈ ℝ ) |
2 |
1
|
3adant2 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( 𝐴 mod 𝑀 ) ∈ ℝ ) |
3 |
|
simp1 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → 𝐴 ∈ ℝ ) |
4 |
|
modcl |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( 𝐵 mod 𝑀 ) ∈ ℝ ) |
5 |
4
|
3adant1 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( 𝐵 mod 𝑀 ) ∈ ℝ ) |
6 |
|
simp2 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → 𝐵 ∈ ℝ ) |
7 |
|
simp3 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → 𝑀 ∈ ℝ+ ) |
8 |
|
modabs2 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( ( 𝐴 mod 𝑀 ) mod 𝑀 ) = ( 𝐴 mod 𝑀 ) ) |
9 |
8
|
3adant2 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( ( 𝐴 mod 𝑀 ) mod 𝑀 ) = ( 𝐴 mod 𝑀 ) ) |
10 |
|
modabs2 |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( ( 𝐵 mod 𝑀 ) mod 𝑀 ) = ( 𝐵 mod 𝑀 ) ) |
11 |
10
|
3adant1 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( ( 𝐵 mod 𝑀 ) mod 𝑀 ) = ( 𝐵 mod 𝑀 ) ) |
12 |
2 3 5 6 7 9 11
|
modsub12d |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( ( ( 𝐴 mod 𝑀 ) − ( 𝐵 mod 𝑀 ) ) mod 𝑀 ) = ( ( 𝐴 − 𝐵 ) mod 𝑀 ) ) |