Theorem dvdsleabs

Description: The divisors of a nonzero integer are bounded by its absolute value. Theorem 1.1(i) in ApostolNT p. 14 (comparison property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011) (Proof shortened by Fan Zheng, 3-Jul-2016)

	Ref	Expression
Assertion	dvdsleabs	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → ( 𝑀 ∥ 𝑁 → 𝑀 ≤ ( abs ‘ 𝑁 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dvdsabsb	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ 𝑁 ↔ 𝑀 ∥ ( abs ‘ 𝑁 ) ) )
2	1	3adant3	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → ( 𝑀 ∥ 𝑁 ↔ 𝑀 ∥ ( abs ‘ 𝑁 ) ) )
3		nnabscl	⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → ( abs ‘ 𝑁 ) ∈ ℕ )
4		dvdsle	⊢ ( ( 𝑀 ∈ ℤ ∧ ( abs ‘ 𝑁 ) ∈ ℕ ) → ( 𝑀 ∥ ( abs ‘ 𝑁 ) → 𝑀 ≤ ( abs ‘ 𝑁 ) ) )
5	3 4	sylan2	⊢ ( ( 𝑀 ∈ ℤ ∧ ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) ) → ( 𝑀 ∥ ( abs ‘ 𝑁 ) → 𝑀 ≤ ( abs ‘ 𝑁 ) ) )
6	5	3impb	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → ( 𝑀 ∥ ( abs ‘ 𝑁 ) → 𝑀 ≤ ( abs ‘ 𝑁 ) ) )
7	2 6	sylbid	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → ( 𝑀 ∥ 𝑁 → 𝑀 ≤ ( abs ‘ 𝑁 ) ) )