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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 ) → ( 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 ) ) )