Description: The sum of a positive integer and a nonnegative integer less than the positive integer is equal to the nonnegative integer modulo the positive integer. (Contributed by AV, 19-Mar-2021)
Ref | Expression | ||
---|---|---|---|
Assertion | addmodidr | ⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀 ) → ( ( 𝐴 + 𝑀 ) mod 𝑀 ) = 𝐴 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0cn | ⊢ ( 𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ ) | |
2 | nncn | ⊢ ( 𝑀 ∈ ℕ → 𝑀 ∈ ℂ ) | |
3 | addcom | ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑀 ∈ ℂ ) → ( 𝐴 + 𝑀 ) = ( 𝑀 + 𝐴 ) ) | |
4 | 1 2 3 | syl2an | ⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ) → ( 𝐴 + 𝑀 ) = ( 𝑀 + 𝐴 ) ) |
5 | 4 | 3adant3 | ⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀 ) → ( 𝐴 + 𝑀 ) = ( 𝑀 + 𝐴 ) ) |
6 | 5 | oveq1d | ⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀 ) → ( ( 𝐴 + 𝑀 ) mod 𝑀 ) = ( ( 𝑀 + 𝐴 ) mod 𝑀 ) ) |
7 | addmodid | ⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀 ) → ( ( 𝑀 + 𝐴 ) mod 𝑀 ) = 𝐴 ) | |
8 | 6 7 | eqtrd | ⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀 ) → ( ( 𝐴 + 𝑀 ) mod 𝑀 ) = 𝐴 ) |