Theorem lgsfval 25027

Description: Value of the function F which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)

Hypothesis

Ref	Expression
lgsval.1	|- F = ( n e. NN |-> if ( n e. Prime , ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) , 1 ) )

Assertion

Ref	Expression
lgsfval	|- ( M e. NN -> ( F ` M ) = 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 ) )

Distinct variable groups:   A, n   n, M   n, N

Allowed substitution hint:   F( n)

Proof of Theorem lgsfval

Step	Hyp	Ref	Expression
1		eleq1 2689	. . 3 |- ( n = M -> ( n e. Prime <-> M e. Prime ) )
2		eqeq1 2626	. . . . 5 |- ( n = M -> ( n = 2 <-> M = 2 ) )
3		oveq1 6657	. . . . . . . . . 10 |- ( n = M -> ( n - 1 ) = ( M - 1 ) )
4	3	oveq1d 6665	. . . . . . . . 9 |- ( n = M -> ( ( n - 1 ) / 2 ) = ( ( M - 1 ) / 2 ) )
5	4	oveq2d 6666	. . . . . . . 8 |- ( n = M -> ( A ^ ( ( n - 1 ) / 2 ) ) = ( A ^ ( ( M - 1 ) / 2 ) ) )
6	5	oveq1d 6665	. . . . . . 7 |- ( n = M -> ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) = ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) )
7		id 22	. . . . . . 7 |- ( n = M -> n = M )
8	6, 7	oveq12d 6668	. . . . . 6 |- ( n = M -> ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) = ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) )
9	8	oveq1d 6665	. . . . 5 |- ( n = M -> ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) = ( ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) - 1 ) )
10	2, 9	ifbieq2d 4111	. . . 4 |- ( n = M -> if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) = 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 ) ) )
11		oveq1 6657	. . . 4 |- ( n = M -> ( n pCnt N ) = ( M pCnt N ) )
12	10, 11	oveq12d 6668	. . 3 |- ( n = M -> ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) = ( 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 ) ) )
13	1, 12	ifbieq1d 4109	. 2 |- ( n = M -> if ( n e. Prime , ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) , 1 ) = 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 ) )
14		lgsval.1	. 2 |- F = ( n e. NN |-> if ( n e. Prime , ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) , 1 ) )
15		ovex 6678	. . 3 |- ( 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 ) ) e. _V
16		1ex 10035	. . 3 |- 1 e. _V
17	15, 16	ifex 4156	. 2 |- 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 ) e. _V
18	13, 14, 17	fvmpt 6282	1 |- ( M e. NN -> ( F ` M ) = 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 ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ifcif 4086   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   7c7 11075   8c8 11076   mod cmo 12668   ^cexp 12860   || cdvds 14983   Primecprime 15385   pCnt cpc 15541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-1cn 9994

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653

This theorem is referenced by:  lgsval2lem  25032