Theorem lgs2

Description: The Legendre symbol at 2 . (Contributed by Mario Carneiro, 4-Feb-2015)

	Ref	Expression
Assertion	lgs2	$$⊢A\in \mathbb{Z}\to A{/}_{L}2=if\left(2\parallel A,0,if\left(A\mathrm{mod}8\in \left\{1,7\right\},1,-1\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		2prm	$$⊢2\in \mathbb{P}$$
2		lgsval2	$$⊢\left(A\in \mathbb{Z}\wedge 2\in \mathbb{P}\right)\to A{/}_{L}2=if\left(2=2,if\left(2\parallel A,0,if\left(A\mathrm{mod}8\in \left\{1,7\right\},1,-1\right)\right),\left(\left(A^{\frac{2-1}{2}}+1\right)\mathrm{mod}2\right)-1\right)$$
3	1 2	mpan2	$$⊢A\in \mathbb{Z}\to A{/}_{L}2=if\left(2=2,if\left(2\parallel A,0,if\left(A\mathrm{mod}8\in \left\{1,7\right\},1,-1\right)\right),\left(\left(A^{\frac{2-1}{2}}+1\right)\mathrm{mod}2\right)-1\right)$$
4		eqid	$$⊢2=2$$
5	4	iftruei	$$⊢if\left(2=2,if\left(2\parallel A,0,if\left(A\mathrm{mod}8\in \left\{1,7\right\},1,-1\right)\right),\left(\left(A^{\frac{2-1}{2}}+1\right)\mathrm{mod}2\right)-1\right)=if\left(2\parallel A,0,if\left(A\mathrm{mod}8\in \left\{1,7\right\},1,-1\right)\right)$$
6	3 5	eqtrdi	$$⊢A\in \mathbb{Z}\to A{/}_{L}2=if\left(2\parallel A,0,if\left(A\mathrm{mod}8\in \left\{1,7\right\},1,-1\right)\right)$$