Definition df-lgs 15677

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 15676	. 2 class /L
2		va	. . 3 setvar a
3		vn	. . 3 setvar n
4		cz 9446	. . 3 class ZZ
5	3	cv 1394	. . . . 5 class n
6		cc0 7999	. . . . 5 class 0
7	5, 6	wceq 1395	. . . 4 wff n = 0
8	2	cv 1394	. . . . . . 7 class a
9		c2 9161	. . . . . . 7 class 2
10		cexp 10760	. . . . . . 7 class ^
11	8, 9, 10	co 6001	. . . . . 6 class ( a ^ 2 )
12		c1 8000	. . . . . 6 class 1
13	11, 12	wceq 1395	. . . . 5 wff ( a ^ 2 ) = 1
14	13, 12, 6	cif 3602	. . . 4 class if ( ( a ^ 2 ) = 1 , 1 , 0 )
15		clt 8181	. . . . . . . 8 class <
16	5, 6, 15	wbr 4083	. . . . . . 7 wff n < 0
17	8, 6, 15	wbr 4083	. . . . . . 7 wff a < 0
18	16, 17	wa 104	. . . . . 6 wff ( n < 0 /\ a < 0 )
19	12	cneg 8318	. . . . . 6 class -u 1
20	18, 19, 12	cif 3602	. . . . 5 class if ( ( n < 0 /\ a < 0 ) , -u 1 , 1 )
21		cabs 11508	. . . . . . 7 class abs
22	5, 21	cfv 5318	. . . . . 6 class ( abs ` n )
23		cmul 8004	. . . . . . 7 class x.
24		vm	. . . . . . . 8 setvar m
25		cn 9110	. . . . . . . 8 class NN
26	24	cv 1394	. . . . . . . . . 10 class m
27		cprime 12629	. . . . . . . . . 10 class Prime
28	26, 27	wcel 2200	. . . . . . . . 9 wff m e. Prime
29	26, 9	wceq 1395	. . . . . . . . . . 11 wff m = 2
30		cdvds 12298	. . . . . . . . . . . . 13 class ||
31	9, 8, 30	wbr 4083	. . . . . . . . . . . 12 wff 2 || a
32		c8 9167	. . . . . . . . . . . . . . 15 class 8
33		cmo 10544	. . . . . . . . . . . . . . 15 class mod
34	8, 32, 33	co 6001	. . . . . . . . . . . . . 14 class ( a mod 8 )
35		c7 9166	. . . . . . . . . . . . . . 15 class 7
36	12, 35	cpr 3667	. . . . . . . . . . . . . 14 class { 1 , 7 }
37	34, 36	wcel 2200	. . . . . . . . . . . . 13 wff ( a mod 8 ) e. { 1 , 7 }
38	37, 12, 19	cif 3602	. . . . . . . . . . . 12 class if ( ( a mod 8 ) e. { 1 , 7 } , 1 , -u 1 )
39	31, 6, 38	cif 3602	. . . . . . . . . . 11 class if ( 2 || a , 0 , if ( ( a mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) )
40		cmin 8317	. . . . . . . . . . . . . . . . 17 class -
41	26, 12, 40	co 6001	. . . . . . . . . . . . . . . 16 class ( m - 1 )
42		cdiv 8819	. . . . . . . . . . . . . . . 16 class /
43	41, 9, 42	co 6001	. . . . . . . . . . . . . . 15 class ( ( m - 1 ) / 2 )
44	8, 43, 10	co 6001	. . . . . . . . . . . . . 14 class ( a ^ ( ( m - 1 ) / 2 ) )
45		caddc 8002	. . . . . . . . . . . . . 14 class +
46	44, 12, 45	co 6001	. . . . . . . . . . . . 13 class ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 )
47	46, 26, 33	co 6001	. . . . . . . . . . . 12 class ( ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m )
48	47, 12, 40	co 6001	. . . . . . . . . . 11 class ( ( ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m ) - 1 )
49	29, 39, 48	cif 3602	. . . . . . . . . 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 12807	. . . . . . . . . . 11 class pCnt
51	26, 5, 50	co 6001	. . . . . . . . . 10 class ( m pCnt n )
52	49, 51, 10	co 6001	. . . . . . . . 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 3602	. . . . . . . 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 4145	. . . . . . 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 10669	. . . . . 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 5318	. . . . 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 6001	. . . 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 3602	. . 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 6003	. 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 1395	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  15683