Theorem zdivmul

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

	Ref	Expression
Assertion	zdivmul	⊢ ( ( ( 𝐷 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( 𝐴 / 𝐷 ) ∈ ℤ ) → ( ( 𝐵 · 𝐴 ) / 𝐷 ) ∈ ℤ )

Proof

Step	Hyp	Ref	Expression
1		zcn	⊢ ( 𝐵 ∈ ℤ → 𝐵 ∈ ℂ )
2	1	3ad2ant2	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐷 ∈ ℕ ) → 𝐵 ∈ ℂ )
3		zcn	⊢ ( 𝐴 ∈ ℤ → 𝐴 ∈ ℂ )
4	3	3ad2ant1	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐷 ∈ ℕ ) → 𝐴 ∈ ℂ )
5		nncn	⊢ ( 𝐷 ∈ ℕ → 𝐷 ∈ ℂ )
6		nnne0	⊢ ( 𝐷 ∈ ℕ → 𝐷 ≠ 0 )
7	5 6	jca	⊢ ( 𝐷 ∈ ℕ → ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) )
8	7	3ad2ant3	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐷 ∈ ℕ ) → ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) )
9		divass	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) → ( ( 𝐵 · 𝐴 ) / 𝐷 ) = ( 𝐵 · ( 𝐴 / 𝐷 ) ) )
10	2 4 8 9	syl3anc	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐷 ∈ ℕ ) → ( ( 𝐵 · 𝐴 ) / 𝐷 ) = ( 𝐵 · ( 𝐴 / 𝐷 ) ) )
11	10	3comr	⊢ ( ( 𝐷 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( ( 𝐵 · 𝐴 ) / 𝐷 ) = ( 𝐵 · ( 𝐴 / 𝐷 ) ) )
12	11	adantr	⊢ ( ( ( 𝐷 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( 𝐴 / 𝐷 ) ∈ ℤ ) → ( ( 𝐵 · 𝐴 ) / 𝐷 ) = ( 𝐵 · ( 𝐴 / 𝐷 ) ) )
13		zmulcl	⊢ ( ( 𝐵 ∈ ℤ ∧ ( 𝐴 / 𝐷 ) ∈ ℤ ) → ( 𝐵 · ( 𝐴 / 𝐷 ) ) ∈ ℤ )
14	13	3ad2antl3	⊢ ( ( ( 𝐷 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( 𝐴 / 𝐷 ) ∈ ℤ ) → ( 𝐵 · ( 𝐴 / 𝐷 ) ) ∈ ℤ )
15	12 14	eqeltrd	⊢ ( ( ( 𝐷 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( 𝐴 / 𝐷 ) ∈ ℤ ) → ( ( 𝐵 · 𝐴 ) / 𝐷 ) ∈ ℤ )