Theorem 1lgs

Description: The Legendre symbol at 1 . See example 1 in ApostolNT p. 180. (Contributed by Mario Carneiro, 28-Apr-2016)

	Ref	Expression
Assertion	1lgs	⊢ ( 𝑁 ∈ ℤ → ( 1 /L 𝑁 ) = 1 )

Proof

Step	Hyp	Ref	Expression
1		sq1	⊢ ( 1 ↑ 2 ) = 1
2	1	oveq1i	⊢ ( ( 1 ↑ 2 ) /L 𝑁 ) = ( 1 /L 𝑁 )
3		1gcd	⊢ ( 𝑁 ∈ ℤ → ( 1 gcd 𝑁 ) = 1 )
4		1z	⊢ 1 ∈ ℤ
5		ax-1ne0	⊢ 1 ≠ 0
6	4 5	pm3.2i	⊢ ( 1 ∈ ℤ ∧ 1 ≠ 0 )
7		lgssq	⊢ ( ( ( 1 ∈ ℤ ∧ 1 ≠ 0 ) ∧ 𝑁 ∈ ℤ ∧ ( 1 gcd 𝑁 ) = 1 ) → ( ( 1 ↑ 2 ) /L 𝑁 ) = 1 )
8	6 7	mp3an1	⊢ ( ( 𝑁 ∈ ℤ ∧ ( 1 gcd 𝑁 ) = 1 ) → ( ( 1 ↑ 2 ) /L 𝑁 ) = 1 )
9	3 8	mpdan	⊢ ( 𝑁 ∈ ℤ → ( ( 1 ↑ 2 ) /L 𝑁 ) = 1 )
10	2 9	eqtr3id	⊢ ( 𝑁 ∈ ℤ → ( 1 /L 𝑁 ) = 1 )