Definition df-lgs 14066

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 14065	. 2 class /L
2		va	. . 3 setvar a
3		vn	. . 3 setvar n
4		cz 9242	. . 3 class ZZ
5	3	cv 1352	. . . . 5 class n
6		cc0 7802	. . . . 5 class 0
7	5, 6	wceq 1353	. . . 4 wff n = 0
8	2	cv 1352	. . . . . . 7 class a
9		c2 8959	. . . . . . 7 class 2
10		cexp 10505	. . . . . . 7 class ^
11	8, 9, 10	co 5869	. . . . . 6 class ( a ^ 2 )
12		c1 7803	. . . . . 6 class 1
13	11, 12	wceq 1353	. . . . 5 wff ( a ^ 2 ) = 1
14	13, 12, 6	cif 3534	. . . 4 class if ( ( a ^ 2 ) = 1 , 1 , 0 )
15		clt 7982	. . . . . . . 8 class <
16	5, 6, 15	wbr 4000	. . . . . . 7 wff n < 0
17	8, 6, 15	wbr 4000	. . . . . . 7 wff a < 0
18	16, 17	wa 104	. . . . . 6 wff ( n < 0 /\ a < 0 )
19	12	cneg 8119	. . . . . 6 class -u 1
20	18, 19, 12	cif 3534	. . . . 5 class if ( ( n < 0 /\ a < 0 ) , -u 1 , 1 )
21		cabs 10990	. . . . . . 7 class abs
22	5, 21	cfv 5212	. . . . . 6 class ( abs ` n )
23		cmul 7807	. . . . . . 7 class x.
24		vm	. . . . . . . 8 setvar m
25		cn 8908	. . . . . . . 8 class NN
26	24	cv 1352	. . . . . . . . . 10 class m
27		cprime 12090	. . . . . . . . . 10 class Prime
28	26, 27	wcel 2148	. . . . . . . . 9 wff m e. Prime
29	26, 9	wceq 1353	. . . . . . . . . . 11 wff m = 2
30		cdvds 11778	. . . . . . . . . . . . 13 class ||
31	9, 8, 30	wbr 4000	. . . . . . . . . . . 12 wff 2 || a
32		c8 8965	. . . . . . . . . . . . . . 15 class 8
33		cmo 10308	. . . . . . . . . . . . . . 15 class mod
34	8, 32, 33	co 5869	. . . . . . . . . . . . . 14 class ( a mod 8 )
35		c7 8964	. . . . . . . . . . . . . . 15 class 7
36	12, 35	cpr 3592	. . . . . . . . . . . . . 14 class { 1 , 7 }
37	34, 36	wcel 2148	. . . . . . . . . . . . 13 wff ( a mod 8 ) e. { 1 , 7 }
38	37, 12, 19	cif 3534	. . . . . . . . . . . 12 class if ( ( a mod 8 ) e. { 1 , 7 } , 1 , -u 1 )
39	31, 6, 38	cif 3534	. . . . . . . . . . 11 class if ( 2 || a , 0 , if ( ( a mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) )
40		cmin 8118	. . . . . . . . . . . . . . . . 17 class -
41	26, 12, 40	co 5869	. . . . . . . . . . . . . . . 16 class ( m - 1 )
42		cdiv 8618	. . . . . . . . . . . . . . . 16 class /
43	41, 9, 42	co 5869	. . . . . . . . . . . . . . 15 class ( ( m - 1 ) / 2 )
44	8, 43, 10	co 5869	. . . . . . . . . . . . . 14 class ( a ^ ( ( m - 1 ) / 2 ) )
45		caddc 7805	. . . . . . . . . . . . . 14 class +
46	44, 12, 45	co 5869	. . . . . . . . . . . . 13 class ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 )
47	46, 26, 33	co 5869	. . . . . . . . . . . 12 class ( ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m )
48	47, 12, 40	co 5869	. . . . . . . . . . 11 class ( ( ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m ) - 1 )
49	29, 39, 48	cif 3534	. . . . . . . . . 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 12267	. . . . . . . . . . 11 class pCnt
51	26, 5, 50	co 5869	. . . . . . . . . 10 class ( m pCnt n )
52	49, 51, 10	co 5869	. . . . . . . . 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 3534	. . . . . . . 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 4061	. . . . . . 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 10431	. . . . . 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 5212	. . . . 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 5869	. . . 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 3534	. . 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 5871	. 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 1353	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  14072