Definition df-lgs 16322

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 16321	. 2 class /L
2		va	. . 3 setvar a
3		vn	. . 3 setvar n
4		cz 9655	. . 3 class ZZ
5	3	cv 1401	. . . . 5 class n
6		cc0 8185	. . . . 5 class 0
7	5, 6	wceq 1402	. . . 4 wff n = 0
8	2	cv 1401	. . . . . . 7 class a
9		c2 9364	. . . . . . 7 class 2
10		cexp 10996	. . . . . . 7 class ^
11	8, 9, 10	co 6088	. . . . . 6 class ( a ^ 2 )
12		c1 8186	. . . . . 6 class 1
13	11, 12	wceq 1402	. . . . 5 wff ( a ^ 2 ) = 1
14	13, 12, 6	cif 3638	. . . 4 class if ( ( a ^ 2 ) = 1 , 1 , 0 )
15		clt 8366	. . . . . . . 8 class <
16	5, 6, 15	wbr 4131	. . . . . . 7 wff n < 0
17	8, 6, 15	wbr 4131	. . . . . . 7 wff a < 0
18	16, 17	wa 104	. . . . . 6 wff ( n < 0 /\ a < 0 )
19	12	cneg 8505	. . . . . 6 class -u 1
20	18, 19, 12	cif 3638	. . . . 5 class if ( ( n < 0 /\ a < 0 ) , -u 1 , 1 )
21		cabs 11785	. . . . . . 7 class abs
22	5, 21	cfv 5380	. . . . . 6 class ( abs ` n )
23		cmul 8190	. . . . . . 7 class x.
24		vm	. . . . . . . 8 setvar m
25		cn 9313	. . . . . . . 8 class NN
26	24	cv 1401	. . . . . . . . . 10 class m
27		cprime 12910	. . . . . . . . . 10 class Prime
28	26, 27	wcel 2209	. . . . . . . . 9 wff m e. Prime
29	26, 9	wceq 1402	. . . . . . . . . . 11 wff m = 2
30		cdvds 12579	. . . . . . . . . . . . 13 class ||
31	9, 8, 30	wbr 4131	. . . . . . . . . . . 12 wff 2 || a
32		c8 9370	. . . . . . . . . . . . . . 15 class 8
33		cmo 10780	. . . . . . . . . . . . . . 15 class mod
34	8, 32, 33	co 6088	. . . . . . . . . . . . . 14 class ( a mod 8 )
35		c7 9369	. . . . . . . . . . . . . . 15 class 7
36	12, 35	cpr 3710	. . . . . . . . . . . . . 14 class { 1 , 7 }
37	34, 36	wcel 2209	. . . . . . . . . . . . 13 wff ( a mod 8 ) e. { 1 , 7 }
38	37, 12, 19	cif 3638	. . . . . . . . . . . 12 class if ( ( a mod 8 ) e. { 1 , 7 } , 1 , -u 1 )
39	31, 6, 38	cif 3638	. . . . . . . . . . 11 class if ( 2 || a , 0 , if ( ( a mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) )
40		cmin 8504	. . . . . . . . . . . . . . . . 17 class -
41	26, 12, 40	co 6088	. . . . . . . . . . . . . . . 16 class ( m - 1 )
42		cdiv 9010	. . . . . . . . . . . . . . . 16 class /
43	41, 9, 42	co 6088	. . . . . . . . . . . . . . 15 class ( ( m - 1 ) / 2 )
44	8, 43, 10	co 6088	. . . . . . . . . . . . . 14 class ( a ^ ( ( m - 1 ) / 2 ) )
45		caddc 8188	. . . . . . . . . . . . . 14 class +
46	44, 12, 45	co 6088	. . . . . . . . . . . . 13 class ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 )
47	46, 26, 33	co 6088	. . . . . . . . . . . 12 class ( ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m )
48	47, 12, 40	co 6088	. . . . . . . . . . 11 class ( ( ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m ) - 1 )
49	29, 39, 48	cif 3638	. . . . . . . . . 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 13092	. . . . . . . . . . 11 class pCnt
51	26, 5, 50	co 6088	. . . . . . . . . 10 class ( m pCnt n )
52	49, 51, 10	co 6088	. . . . . . . . 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 3638	. . . . . . . 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 4193	. . . . . . 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 10905	. . . . . 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 5380	. . . . 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 6088	. . . 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 3638	. . 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 6090	. 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 1402	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 used by:  lgsval  16328