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 |
|
simp2 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → 𝐵 ∈ ℝ ) |
5 |
|
simp3 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → 𝑀 ∈ ℝ+ ) |
6 |
|
modabs2 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( ( 𝐴 mod 𝑀 ) mod 𝑀 ) = ( 𝐴 mod 𝑀 ) ) |
7 |
6
|
3adant2 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( ( 𝐴 mod 𝑀 ) mod 𝑀 ) = ( 𝐴 mod 𝑀 ) ) |
8 |
|
eqidd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( 𝐵 mod 𝑀 ) = ( 𝐵 mod 𝑀 ) ) |
9 |
2 3 4 4 5 7 8
|
modsub12d |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( ( ( 𝐴 mod 𝑀 ) − 𝐵 ) mod 𝑀 ) = ( ( 𝐴 − 𝐵 ) mod 𝑀 ) ) |