Definition df-lgs 15239

Description: Define the Legendre symbol (actually the Kronecker symbol, which extends the Legendre symbol to all integers, and also the Jacobi symbol, which restricts the Kronecker symbol to positive odd integers). See definition in [ApostolNT] p. 179 resp. definition in [ApostolNT] p. 188. (Contributed by Mario Carneiro, 4-Feb-2015.)

Assertion

Ref	Expression
df-lgs	|- /L = ( a e. ZZ , n e. ZZ |-> if ( n = 0 , if ( ( a ^ 2 ) = 1 , 1 , 0 ) , ( if ( ( n < 0 /\ a < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( m e. NN |-> if ( m e. Prime , ( if ( m = 2 , if ( 2 || a , 0 , if ( ( a mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m ) - 1 ) ) ^ ( m pCnt n ) ) , 1 ) ) ) ` ( abs ` n ) ) ) ) )

Distinct variable group:   m, a, n

Detailed syntax breakdown of Definition df-lgs

Step	Hyp	Ref	Expression
1		clgs 15238	. 2 class /L
2		va	. . 3 setvar a
3		vn	. . 3 setvar n
4		cz 9326	. . 3 class ZZ
5	3	cv 1363	. . . . 5 class n
6		cc0 7879	. . . . 5 class 0
7	5, 6	wceq 1364	. . . 4 wff n = 0
8	2	cv 1363	. . . . . . 7 class a
9		c2 9041	. . . . . . 7 class 2
10		cexp 10630	. . . . . . 7 class ^
11	8, 9, 10	co 5922	. . . . . 6 class ( a ^ 2 )
12		c1 7880	. . . . . 6 class 1
13	11, 12	wceq 1364	. . . . 5 wff ( a ^ 2 ) = 1
14	13, 12, 6	cif 3561	. . . 4 class if ( ( a ^ 2 ) = 1 , 1 , 0 )
15		clt 8061	. . . . . . . 8 class <
16	5, 6, 15	wbr 4033	. . . . . . 7 wff n < 0
17	8, 6, 15	wbr 4033	. . . . . . 7 wff a < 0
18	16, 17	wa 104	. . . . . 6 wff ( n < 0 /\ a < 0 )
19	12	cneg 8198	. . . . . 6 class -u 1
20	18, 19, 12	cif 3561	. . . . 5 class if ( ( n < 0 /\ a < 0 ) , -u 1 , 1 )
21		cabs 11162	. . . . . . 7 class abs
22	5, 21	cfv 5258	. . . . . 6 class ( abs ` n )
23		cmul 7884	. . . . . . 7 class x.
24		vm	. . . . . . . 8 setvar m
25		cn 8990	. . . . . . . 8 class NN
26	24	cv 1363	. . . . . . . . . 10 class m
27		cprime 12275	. . . . . . . . . 10 class Prime
28	26, 27	wcel 2167	. . . . . . . . 9 wff m e. Prime
29	26, 9	wceq 1364	. . . . . . . . . . 11 wff m = 2
30		cdvds 11952	. . . . . . . . . . . . 13 class ||
31	9, 8, 30	wbr 4033	. . . . . . . . . . . 12 wff 2 || a
32		c8 9047	. . . . . . . . . . . . . . 15 class 8
33		cmo 10414	. . . . . . . . . . . . . . 15 class mod
34	8, 32, 33	co 5922	. . . . . . . . . . . . . 14 class ( a mod 8 )
35		c7 9046	. . . . . . . . . . . . . . 15 class 7
36	12, 35	cpr 3623	. . . . . . . . . . . . . 14 class { 1 , 7 }
37	34, 36	wcel 2167	. . . . . . . . . . . . 13 wff ( a mod 8 ) e. { 1 , 7 }
38	37, 12, 19	cif 3561	. . . . . . . . . . . 12 class if ( ( a mod 8 ) e. { 1 , 7 } , 1 , -u 1 )
39	31, 6, 38	cif 3561	. . . . . . . . . . 11 class if ( 2 || a , 0 , if ( ( a mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) )
40		cmin 8197	. . . . . . . . . . . . . . . . 17 class -
41	26, 12, 40	co 5922	. . . . . . . . . . . . . . . 16 class ( m - 1 )
42		cdiv 8699	. . . . . . . . . . . . . . . 16 class /
43	41, 9, 42	co 5922	. . . . . . . . . . . . . . 15 class ( ( m - 1 ) / 2 )
44	8, 43, 10	co 5922	. . . . . . . . . . . . . 14 class ( a ^ ( ( m - 1 ) / 2 ) )
45		caddc 7882	. . . . . . . . . . . . . 14 class +
46	44, 12, 45	co 5922	. . . . . . . . . . . . 13 class ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 )
47	46, 26, 33	co 5922	. . . . . . . . . . . 12 class ( ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m )
48	47, 12, 40	co 5922	. . . . . . . . . . 11 class ( ( ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m ) - 1 )
49	29, 39, 48	cif 3561	. . . . . . . . . 10 class if ( m = 2 , if ( 2 || a , 0 , if ( ( a mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m ) - 1 ) )
50		cpc 12453	. . . . . . . . . . 11 class pCnt
51	26, 5, 50	co 5922	. . . . . . . . . 10 class ( m pCnt n )
52	49, 51, 10	co 5922	. . . . . . . . 9 class ( if ( m = 2 , if ( 2 || a , 0 , if ( ( a mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m ) - 1 ) ) ^ ( m pCnt n ) )
53	28, 52, 12	cif 3561	. . . . . . . 8 class if ( m e. Prime , ( if ( m = 2 , if ( 2 || a , 0 , if ( ( a mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m ) - 1 ) ) ^ ( m pCnt n ) ) , 1 )
54	24, 25, 53	cmpt 4094	. . . . . . 7 class ( m e. NN |-> if ( m e. Prime , ( if ( m = 2 , if ( 2 || a , 0 , if ( ( a mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m ) - 1 ) ) ^ ( m pCnt n ) ) , 1 ) )
55	23, 54, 12	cseq 10539	. . . . . 6 class seq 1 ( x. , ( m e. NN |-> if ( m e. Prime , ( if ( m = 2 , if ( 2 || a , 0 , if ( ( a mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m ) - 1 ) ) ^ ( m pCnt n ) ) , 1 ) ) )
56	22, 55	cfv 5258	. . . . 5 class ( seq 1 ( x. , ( m e. NN |-> if ( m e. Prime , ( if ( m = 2 , if ( 2 || a , 0 , if ( ( a mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m ) - 1 ) ) ^ ( m pCnt n ) ) , 1 ) ) ) ` ( abs ` n ) )
57	20, 56, 23	co 5922	. . . 4 class ( if ( ( n < 0 /\ a < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( m e. NN |-> if ( m e. Prime , ( if ( m = 2 , if ( 2 || a , 0 , if ( ( a mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m ) - 1 ) ) ^ ( m pCnt n ) ) , 1 ) ) ) ` ( abs ` n ) ) )
58	7, 14, 57	cif 3561	. . 3 class if ( n = 0 , if ( ( a ^ 2 ) = 1 , 1 , 0 ) , ( if ( ( n < 0 /\ a < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( m e. NN |-> if ( m e. Prime , ( if ( m = 2 , if ( 2 || a , 0 , if ( ( a mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m ) - 1 ) ) ^ ( m pCnt n ) ) , 1 ) ) ) ` ( abs ` n ) ) ) )
59	2, 3, 4, 4, 58	cmpo 5924	. 2 class ( a e. ZZ , n e. ZZ |-> if ( n = 0 , if ( ( a ^ 2 ) = 1 , 1 , 0 ) , ( if ( ( n < 0 /\ a < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( m e. NN |-> if ( m e. Prime , ( if ( m = 2 , if ( 2 || a , 0 , if ( ( a mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m ) - 1 ) ) ^ ( m pCnt n ) ) , 1 ) ) ) ` ( abs ` n ) ) ) ) )
60	1, 59	wceq 1364	1 wff /L = ( a e. ZZ , n e. ZZ |-> if ( n = 0 , if ( ( a ^ 2 ) = 1 , 1 , 0 ) , ( if ( ( n < 0 /\ a < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( m e. NN |-> if ( m e. Prime , ( if ( m = 2 , if ( 2 || a , 0 , if ( ( a mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m ) - 1 ) ) ^ ( m pCnt n ) ) , 1 ) ) ) ` ( abs ` n ) ) ) ) )

This definition is referenced by:  lgsval  15245