Theorem mumullem1

Description: Lemma for mumul . A multiple of a non-squarefree number is non-squarefree. (Contributed by Mario Carneiro, 3-Oct-2014)

	Ref	Expression
Assertion	mumullem1	⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( μ ‘ 𝐴 ) = 0 ) → ( μ ‘ ( 𝐴 · 𝐵 ) ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		prmz	⊢ ( 𝑝 ∈ ℙ → 𝑝 ∈ ℤ )
2	1	adantl	⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ 𝑝 ∈ ℙ ) → 𝑝 ∈ ℤ )
3		zsqcl	⊢ ( 𝑝 ∈ ℤ → ( 𝑝 ↑ 2 ) ∈ ℤ )
4	2 3	syl	⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 ↑ 2 ) ∈ ℤ )
5		nnz	⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℤ )
6	5	ad2antrr	⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ 𝑝 ∈ ℙ ) → 𝐴 ∈ ℤ )
7		nnz	⊢ ( 𝐵 ∈ ℕ → 𝐵 ∈ ℤ )
8	7	ad2antlr	⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ 𝑝 ∈ ℙ ) → 𝐵 ∈ ℤ )
9		dvdsmultr1	⊢ ( ( ( 𝑝 ↑ 2 ) ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( ( 𝑝 ↑ 2 ) ∥ 𝐴 → ( 𝑝 ↑ 2 ) ∥ ( 𝐴 · 𝐵 ) ) )
10	4 6 8 9	syl3anc	⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ 𝑝 ∈ ℙ ) → ( ( 𝑝 ↑ 2 ) ∥ 𝐴 → ( 𝑝 ↑ 2 ) ∥ ( 𝐴 · 𝐵 ) ) )
11	10	reximdva	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝐴 → ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ ( 𝐴 · 𝐵 ) ) )
12		isnsqf	⊢ ( 𝐴 ∈ ℕ → ( ( μ ‘ 𝐴 ) = 0 ↔ ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝐴 ) )
13	12	adantr	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( μ ‘ 𝐴 ) = 0 ↔ ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝐴 ) )
14		nnmulcl	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 · 𝐵 ) ∈ ℕ )
15		isnsqf	⊢ ( ( 𝐴 · 𝐵 ) ∈ ℕ → ( ( μ ‘ ( 𝐴 · 𝐵 ) ) = 0 ↔ ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ ( 𝐴 · 𝐵 ) ) )
16	14 15	syl	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( μ ‘ ( 𝐴 · 𝐵 ) ) = 0 ↔ ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ ( 𝐴 · 𝐵 ) ) )
17	11 13 16	3imtr4d	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( μ ‘ 𝐴 ) = 0 → ( μ ‘ ( 𝐴 · 𝐵 ) ) = 0 ) )
18	17	imp	⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( μ ‘ 𝐴 ) = 0 ) → ( μ ‘ ( 𝐴 · 𝐵 ) ) = 0 )