Theorem lgscl

Description: The Legendre symbol is an integer. (Contributed by Mario Carneiro, 4-Feb-2015)

	Ref	Expression
Assertion	lgscl	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝐴 /L 𝑁 ) ∈ ℤ )

Proof

Step	Hyp	Ref	Expression
1		ssrab2	⊢ { 𝑥 ∈ ℤ ∣ ( abs ‘ 𝑥 ) ≤ 1 } ⊆ ℤ
2		eqid	⊢ { 𝑥 ∈ ℤ ∣ ( abs ‘ 𝑥 ) ≤ 1 } = { 𝑥 ∈ ℤ ∣ ( abs ‘ 𝑥 ) ≤ 1 }
3	2	lgscl2	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝐴 /L 𝑁 ) ∈ { 𝑥 ∈ ℤ ∣ ( abs ‘ 𝑥 ) ≤ 1 } )
4	1 3	sselid	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝐴 /L 𝑁 ) ∈ ℤ )