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Metamath Proof Explorer

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 )