Theorem lgs1

Description: The Legendre symbol at 1 . See definition in ApostolNT p. 188. (Contributed by Mario Carneiro, 28-Apr-2016)

	Ref	Expression
Assertion	lgs1	⊢ ( 𝐴 ∈ ℤ → ( 𝐴 /L 1 ) = 1 )

Proof

Step	Hyp	Ref	Expression
1		sq1	⊢ ( 1 ↑ 2 ) = 1
2	1	oveq2i	⊢ ( 𝐴 /L ( 1 ↑ 2 ) ) = ( 𝐴 /L 1 )
3		gcd1	⊢ ( 𝐴 ∈ ℤ → ( 𝐴 gcd 1 ) = 1 )
4		1nn	⊢ 1 ∈ ℕ
5		lgssq2	⊢ ( ( 𝐴 ∈ ℤ ∧ 1 ∈ ℕ ∧ ( 𝐴 gcd 1 ) = 1 ) → ( 𝐴 /L ( 1 ↑ 2 ) ) = 1 )
6	4 5	mp3an2	⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝐴 gcd 1 ) = 1 ) → ( 𝐴 /L ( 1 ↑ 2 ) ) = 1 )
7	3 6	mpdan	⊢ ( 𝐴 ∈ ℤ → ( 𝐴 /L ( 1 ↑ 2 ) ) = 1 )
8	2 7	eqtr3id	⊢ ( 𝐴 ∈ ℤ → ( 𝐴 /L 1 ) = 1 )