Description: An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011)
Ref | Expression | ||
---|---|---|---|
Assertion | dvdsaddr | ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ 𝑁 ↔ 𝑀 ∥ ( 𝑁 + 𝑀 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdsadd | ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ 𝑁 ↔ 𝑀 ∥ ( 𝑀 + 𝑁 ) ) ) | |
2 | zcn | ⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ℂ ) | |
3 | zcn | ⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ℂ ) | |
4 | addcom | ⊢ ( ( 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ ) → ( 𝑀 + 𝑁 ) = ( 𝑁 + 𝑀 ) ) | |
5 | 2 3 4 | syl2an | ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 + 𝑁 ) = ( 𝑁 + 𝑀 ) ) |
6 | 5 | breq2d | ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ ( 𝑀 + 𝑁 ) ↔ 𝑀 ∥ ( 𝑁 + 𝑀 ) ) ) |
7 | 1 6 | bitrd | ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ 𝑁 ↔ 𝑀 ∥ ( 𝑁 + 𝑀 ) ) ) |