Theorem dvdssqf

Description: A divisor of a squarefree number is squarefree. (Contributed by Mario Carneiro, 1-Jul-2015)

	Ref	Expression
Assertion	dvdssqf	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴 ) → ( ( μ ‘ 𝐴 ) ≠ 0 → ( μ ‘ 𝐵 ) ≠ 0 ) )

Proof

Step	Hyp	Ref	Expression
1		simpl3	⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴 ) ∧ 𝑝 ∈ ℙ ) → 𝐵 ∥ 𝐴 )
2		prmz	⊢ ( 𝑝 ∈ ℙ → 𝑝 ∈ ℤ )
3	2	adantl	⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴 ) ∧ 𝑝 ∈ ℙ ) → 𝑝 ∈ ℤ )
4		zsqcl	⊢ ( 𝑝 ∈ ℤ → ( 𝑝 ↑ 2 ) ∈ ℤ )
5	3 4	syl	⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴 ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 ↑ 2 ) ∈ ℤ )
6		simpl2	⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴 ) ∧ 𝑝 ∈ ℙ ) → 𝐵 ∈ ℕ )
7	6	nnzd	⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴 ) ∧ 𝑝 ∈ ℙ ) → 𝐵 ∈ ℤ )
8		simpl1	⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴 ) ∧ 𝑝 ∈ ℙ ) → 𝐴 ∈ ℕ )
9	8	nnzd	⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴 ) ∧ 𝑝 ∈ ℙ ) → 𝐴 ∈ ℤ )
10		dvdstr	⊢ ( ( ( 𝑝 ↑ 2 ) ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ ) → ( ( ( 𝑝 ↑ 2 ) ∥ 𝐵 ∧ 𝐵 ∥ 𝐴 ) → ( 𝑝 ↑ 2 ) ∥ 𝐴 ) )
11	5 7 9 10	syl3anc	⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴 ) ∧ 𝑝 ∈ ℙ ) → ( ( ( 𝑝 ↑ 2 ) ∥ 𝐵 ∧ 𝐵 ∥ 𝐴 ) → ( 𝑝 ↑ 2 ) ∥ 𝐴 ) )
12	1 11	mpan2d	⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴 ) ∧ 𝑝 ∈ ℙ ) → ( ( 𝑝 ↑ 2 ) ∥ 𝐵 → ( 𝑝 ↑ 2 ) ∥ 𝐴 ) )
13	12	reximdva	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴 ) → ( ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝐵 → ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝐴 ) )
14		isnsqf	⊢ ( 𝐵 ∈ ℕ → ( ( μ ‘ 𝐵 ) = 0 ↔ ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝐵 ) )
15	14	3ad2ant2	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴 ) → ( ( μ ‘ 𝐵 ) = 0 ↔ ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝐵 ) )
16		isnsqf	⊢ ( 𝐴 ∈ ℕ → ( ( μ ‘ 𝐴 ) = 0 ↔ ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝐴 ) )
17	16	3ad2ant1	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴 ) → ( ( μ ‘ 𝐴 ) = 0 ↔ ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝐴 ) )
18	13 15 17	3imtr4d	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴 ) → ( ( μ ‘ 𝐵 ) = 0 → ( μ ‘ 𝐴 ) = 0 ) )
19	18	necon3d	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴 ) → ( ( μ ‘ 𝐴 ) ≠ 0 → ( μ ‘ 𝐵 ) ≠ 0 ) )