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 ) → ( 𝑀 ∥ 𝑁 ↔ 𝑀 ∥ ( 𝑀 − 𝑁 ) ) )

Proof

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