Theorem 2lgs2

Description: The Legendre symbol for 2 at 2 is 0 . (Contributed by AV, 20-Jun-2021)

	Ref	Expression
Assertion	2lgs2	⊢ ( 2 /L 2 ) = 0

Proof

Step	Hyp	Ref	Expression
1		2z	⊢ 2 ∈ ℤ
2		lgs2	⊢ ( 2 ∈ ℤ → ( 2 /L 2 ) = if ( 2 ∥ 2 , 0 , if ( ( 2 mod 8 ) ∈ { 1 , 7 } , 1 , - 1 ) ) )
3	1 2	ax-mp	⊢ ( 2 /L 2 ) = if ( 2 ∥ 2 , 0 , if ( ( 2 mod 8 ) ∈ { 1 , 7 } , 1 , - 1 ) )
4		z2even	⊢ 2 ∥ 2
5	4	iftruei	⊢ if ( 2 ∥ 2 , 0 , if ( ( 2 mod 8 ) ∈ { 1 , 7 } , 1 , - 1 ) ) = 0
6	3 5	eqtri	⊢ ( 2 /L 2 ) = 0