Metamath Proof Explorer


Theorem dvdssub

Description: An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011)

Ref Expression
Assertion dvdssub ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀𝑁𝑀 ∥ ( 𝑀𝑁 ) ) )

Proof

Step Hyp Ref Expression
1 dvdsnegb ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀𝑁𝑀 ∥ - 𝑁 ) )
2 znegcl ( 𝑁 ∈ ℤ → - 𝑁 ∈ ℤ )
3 dvdsadd ( ( 𝑀 ∈ ℤ ∧ - 𝑁 ∈ ℤ ) → ( 𝑀 ∥ - 𝑁𝑀 ∥ ( 𝑀 + - 𝑁 ) ) )
4 2 3 sylan2 ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ - 𝑁𝑀 ∥ ( 𝑀 + - 𝑁 ) ) )
5 zcn ( 𝑀 ∈ ℤ → 𝑀 ∈ ℂ )
6 zcn ( 𝑁 ∈ ℤ → 𝑁 ∈ ℂ )
7 negsub ( ( 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ ) → ( 𝑀 + - 𝑁 ) = ( 𝑀𝑁 ) )
8 5 6 7 syl2an ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 + - 𝑁 ) = ( 𝑀𝑁 ) )
9 8 breq2d ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ ( 𝑀 + - 𝑁 ) ↔ 𝑀 ∥ ( 𝑀𝑁 ) ) )
10 1 4 9 3bitrd ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀𝑁𝑀 ∥ ( 𝑀𝑁 ) ) )