Metamath Proof Explorer

Theorem iddvds

Description: An integer divides itself. Theorem 1.1(a) in ApostolNT p. 14 (reflexive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011)

Ref Expression
Assertion iddvds ⊢ N ∈ ℤ → N ∥ N

Proof

Step Hyp Ref Expression
1 zcn ⊢ N ∈ ℤ → N ∈ ℂ
2 1 mulid2d ⊢ N ∈ ℤ → 1 ⋅ N = N
3 1z ⊢ 1 ∈ ℤ
4 dvds0lem ⊢ 1 ∈ ℤ ∧ N ∈ ℤ ∧ N ∈ ℤ ∧ 1 ⋅ N = N → N ∥ N
5 3 4 mp3anl1 ⊢ N ∈ ℤ ∧ N ∈ ℤ ∧ 1 ⋅ N = N → N ∥ N
6 5 anabsan ⊢ N ∈ ℤ ∧ 1 ⋅ N = N → N ∥ N
7 2 6 mpdan ⊢ N ∈ ℤ → N ∥ N