Theorem zdivadd

Description: Property of divisibility: if D divides A and B then it divides A + B . (Contributed by NM, 3-Oct-2008)

	Ref	Expression
Assertion	zdivadd	⊢ ( ( ( 𝐷 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 / 𝐷 ) ∈ ℤ ∧ ( 𝐵 / 𝐷 ) ∈ ℤ ) ) → ( ( 𝐴 + 𝐵 ) / 𝐷 ) ∈ ℤ )

Proof

Step	Hyp	Ref	Expression
1		zcn	⊢ ( 𝐴 ∈ ℤ → 𝐴 ∈ ℂ )
2		zcn	⊢ ( 𝐵 ∈ ℤ → 𝐵 ∈ ℂ )
3		nncn	⊢ ( 𝐷 ∈ ℕ → 𝐷 ∈ ℂ )
4		nnne0	⊢ ( 𝐷 ∈ ℕ → 𝐷 ≠ 0 )
5	3 4	jca	⊢ ( 𝐷 ∈ ℕ → ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) )
6		divdir	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) → ( ( 𝐴 + 𝐵 ) / 𝐷 ) = ( ( 𝐴 / 𝐷 ) + ( 𝐵 / 𝐷 ) ) )
7	1 2 5 6	syl3an	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐷 ∈ ℕ ) → ( ( 𝐴 + 𝐵 ) / 𝐷 ) = ( ( 𝐴 / 𝐷 ) + ( 𝐵 / 𝐷 ) ) )
8	7	3comr	⊢ ( ( 𝐷 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( ( 𝐴 + 𝐵 ) / 𝐷 ) = ( ( 𝐴 / 𝐷 ) + ( 𝐵 / 𝐷 ) ) )
9	8	adantr	⊢ ( ( ( 𝐷 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 / 𝐷 ) ∈ ℤ ∧ ( 𝐵 / 𝐷 ) ∈ ℤ ) ) → ( ( 𝐴 + 𝐵 ) / 𝐷 ) = ( ( 𝐴 / 𝐷 ) + ( 𝐵 / 𝐷 ) ) )
10		zaddcl	⊢ ( ( ( 𝐴 / 𝐷 ) ∈ ℤ ∧ ( 𝐵 / 𝐷 ) ∈ ℤ ) → ( ( 𝐴 / 𝐷 ) + ( 𝐵 / 𝐷 ) ) ∈ ℤ )
11	10	adantl	⊢ ( ( ( 𝐷 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 / 𝐷 ) ∈ ℤ ∧ ( 𝐵 / 𝐷 ) ∈ ℤ ) ) → ( ( 𝐴 / 𝐷 ) + ( 𝐵 / 𝐷 ) ) ∈ ℤ )
12	9 11	eqeltrd	⊢ ( ( ( 𝐷 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 / 𝐷 ) ∈ ℤ ∧ ( 𝐵 / 𝐷 ) ∈ ℤ ) ) → ( ( 𝐴 + 𝐵 ) / 𝐷 ) ∈ ℤ )